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Class 12 MATHS CBSE PYQ Chapter Wise Topic Wise.pdf

The document contains solved papers for Class 12 Mathematics covering topics from 2014 to 2023, specifically focusing on relations, functions, and their types. It includes a variety of questions categorized into objective and long-answer formats, along with explanations for selected answers. Each section is structured to facilitate understanding of mathematical concepts, such as equivalence relations and functions, guiding students in their exam preparations.

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CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
Relations And Functions Previous Year Questions (Topic-wise) 1. RELATIONS AND ITS TYPES Objective Qs (1 mark) 1. Let R be a relation in the set N given by 𝑅 = {(𝑎, 𝑏): 𝑎 = 𝑏 − 2, 𝑏 > 6}, then: (a) (2,4) ∈ 𝑅 (b) (3,8) ∈ 𝑅 (c) (6,8) ∈ 𝑅 (d) (8,7) ∈ 𝑅 [CBSE Term-1 SQP 2021] 2. 𝐴 = {1,2,3,4}. A relation 𝑅 in the set 𝐴 is given by 𝑅 = {(1,1), (2,3), (3,2), (4,3), (3,4)}, then relation 𝑅 is: (a) reflexive (b) symmetric (c) transitive (d) equivalence [Delhi Gov. Term-1 SQP 2021] 3. Let set 𝑋 = {1,2,3} and a relation 𝑅 is defined in 𝑋 as 𝑅 = {(1,3), (2,2), (3,2)}, then minimum ordered pairs which should be added in relation R to make it reflexive and symmetric are: (a) {(1,1), (2,3), (1,4)} (b) {(5,3), (3,1), (1,2)} (c) {(1,1), (3,3), (3,1), (2,3)} (d) {(1,1), (3,3), (3,1), (1,2)} [CBSE Term-1 2021] 4. If 𝑅 = {(𝑥, 𝑦): 𝑥, 𝑦 ∈ 𝑍, 𝑥2 + 𝑦2 ≤ 4} is a relation in set 𝑍. Then domain of 𝑅 is: (a) {0,1,2} (b) {−2, −1,0,1,2} (c) [0, −1, −2] (d) {−1,0,1} [CBSE Term-1 2021] 5. A relation 𝑅 is defined on 𝑁. Which of the following is the reflexive relation? CH-1
(a) R = {(𝑥, 𝑦): 𝑥 > 𝑦; 𝑥, 𝑦 ∈ N} (b) R = {(𝑥, 𝑦): 𝑥 + 𝑦 = 10; 𝑥, 𝑦 ∈ N} (c) R = {(𝑥, 𝑦): 𝑥𝑦 is the square number; 𝑥, 𝒚 ∈ N} (d) 𝑅 = {(𝑥, 𝑦): 𝑥 + 4𝑦 = 10; 𝑥, 𝑦 ∈ 𝑁} [CBSE Term-1 2021] 6. The number of equivalence relations in the set {1,2,3} containing the elements (1,2) and (2,1) is: (a) 0 (b) 1 (c) 2 (d) 3 [CBSE Term-1 2021] 7. A relation 𝑅 is defined on 𝑍 as: 𝑎R𝑏 if and only if 𝑎2 − 7𝑎𝑏 + 6𝑏2 = 0 Then, 𝑅 is: (a) reflexive and symmetric (b) symmetric but not reflexive (c) transitive but not reflexive (d) reflexive but not symmetric [CBSE Term-1 2021] Very Short & Short Qs (𝟏 - 3 marks) 8. Let 𝑅 be the equivalence relation on the set 𝑍 of integers given by 𝑅 = {(𝑎, 𝑏): 2 divides 𝑎 − 𝑏}. Write the equivalence class {0}. [CBSE 2021] 9. Check if the relation 𝑅 in the set 𝑅 of real numbers defined as 𝑅 = {(𝑎, 𝑏): 𝑎 < 𝑏} is: (A) symmetric; (B) transitive [CBSE 2020] 10. Let 𝐴 = {𝑥 ∈ 𝑍: 0 ≤ 𝑥 ≤ 12}. Show that R = {(𝑎, 𝑏): 𝑎, 𝑏 ∈ A, |𝑎 − 𝑏| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also, write the equivalence class [2]. [CBSE 2018] 11. If 𝑅 = {(𝑥, 𝑦): 2𝑥 + 𝑦 = 8} is a relation on 𝑁, write the range of 𝑅.
[CBSE 2014] 12. Let 𝑅 = {(𝑎, 𝑎3): 𝑎 is a prime number less than 5} be a relation. Find the range of 𝑅. [CBSE 2014] 13. Let 𝑅 be a relation defined on the set of natural numbers 𝑁 as 𝑅 = {(𝑥, 𝑦): 𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 and 2𝑥 + 𝑦 = 24}. Then, find the domain and range of the relation 𝑅. Also, find whether 𝑅 is an equivalence relation or not. [CBSE 2014] Long Qs (𝟒 - 5 marks) 14. Let N be the set of all natural numbers and R be a relation on N × N defined by (𝑎, 𝑏)R(𝑐, 𝑑) ⇔ 𝑎𝑑 = 𝑏𝑐 for all (𝑎, 𝑏), (𝑐, 𝑑) ∈ N × N. Show that R is an equivalence relation on N × N. Also, find the equivalence class of (2,6), i.e., [(2,6)]. [CBSE SQP 2023] 15. Given a non-empty set 𝑋, define the relation 𝑅 in 𝑃(𝑋) as follows: For 𝐴, 𝐵 ∈ 𝑃(𝑋), (𝐴, 𝐵) ∈ 𝑅 if 𝐴 ⊂ 𝐵. Prove that 𝑅 is reflexive, transitive and not symmetric. [CBSE SQP 2022] 16. Define the relation 𝑅 in the set 𝑁 × 𝑁 as follows: For (𝑎, 𝑏), (𝑐, 𝑑) ∈ N × N, (𝑎, 𝑏)R(𝑐, 𝑑) if 𝑎𝑑 = 𝑏𝑐. Prove that 𝑅 is an equivalence relation in N × N. [CBSE SQP 2022] 17. Show that the relation 𝑆 in the set 𝐴 = [𝑥 ∈ 𝑍 : 0 ≤ 𝐗 ≤ 12] given by 𝑆 = [(𝑎, 𝑏): 𝑎, 𝑏 ∈ 𝑍, |𝑎 − 𝑏| is divisible by 3 ] is an equivalence relation. [CBSE 2019] 18. Let 𝐍 denote the set of all natural numbers and R be a relation on N × N defined by (𝑎, 𝑏)R(𝑐, 𝑑) if 𝑎𝑑(𝑏 + 𝑐) = 𝑏𝑐(𝑎 + 𝑑). Prove that 𝑅 is an equivalence relation. [CBSE 2015] 19. Show that the relation 𝑅 in the set 𝐴 = {1,2,3,4,5} given by 𝑅 = {(𝑎, 𝑏): |𝑎 − 𝑏| is divisible by 2} is an equivalence relation. Write all the equivalence classes of 𝑅. [CBSE 2015]
2. FUNCTIONS AND ITS TYPES Objective Qs (1 mark) 20. Let A = {1,2,3, … , 𝑛} and B = {𝑎, 𝑏}. Then the number of surjections from 𝐴 to 𝐵 is: (a) 𝑛 P2 (b) 2𝑛 − 2 (c) 2𝑛 − 1 (d) None of these [Delhi Gov. SQP 2022] 21. Let 𝑋 = {𝑥2 : 𝑥 ∈ 𝑁} and the function 𝑓: 𝑁 → 𝑋 is defined by 𝑓(𝑥) = 𝑥2 , 𝑥 ∈ 𝑁. Then this function is: (a) injective only (b) not bijective (c) surjective (d) bijective [CBSE Term-1 2021] 22. A function 𝑓: 𝑅 → 𝑅 defined by 𝑓(𝑥) = 2 + 𝑥2 is: (a) not one-one (b) one-one (c) not onto (d) neither one-one nor onto [CBSE Term-1 2021] 23. The function 𝑓: 𝑅 → 𝑅 defined by 𝑓(𝑥) = 4 + 3cos 𝑥 is: (a) bijective (b) one-one but not onto (c) onto but not one-one (d) neither one-one nor onto [CBSE Term-1 2021] 24. The number of functions defined from {1,2, 3,4,5} → {𝑎, 𝑏} which are one-one is: (a) 5 (b) 3 (c) 2 (d) 0 [CBSE Term-1 2021]
25. Let 𝑓: 𝑅 → 𝑅 be defined by 𝑓(𝑥) = 1 𝑥 , 𝑥 ∈ 𝑅. Then 𝑓 is: (a) one-one (b) onto (c) bijective (d) f is not defined [CBSE Term-1 2021] 26. Assertion (A): The relation 𝑓: {1,2,3,4} → {𝑥, 𝑦, 𝑧, 𝑝} defined by 𝑓 = {(1, 𝑥), (2, 𝑦), (3, 𝑧)} is a bijective function. Reason (R): The function 𝑓: {1,2,3} → {𝑥, 𝑦, 𝑧, 𝑝} such that 𝑓 = {(1, 𝑥), (2, 𝑦), (3, 𝑧)} is one-one. [CBSE SQP 2023] Very Short & Short Qs (1 - 3 marks) 27. Let 𝑓: N → N be defined as: 𝑓(𝑛) = � 𝑛+1 2 , if 𝑛 is odd 𝑛 2 , if 𝑛 is even for all 𝑛 ∈ 𝑁. Find whether the function 𝑓 is bijective or not. [Delhi Gov. SQP 2022] 28. Let 𝑓: 𝑋 → 𝑌 be a function. Define a relation 𝑅 on 𝑋 given by 𝑅 = {(𝑎, 𝑏): 𝑓(𝑎) = 𝑓(𝑏)}. Show that 𝑅 is an equivalence relation on 𝑋. [Delhi Gov. SQP 2022] 29. Prove that the function 𝑓 is surjective, where 𝑓: N → N such that 𝑓(𝑛) = � 𝑛 + 1 2 , if 𝑛 is 𝑛2 , if 𝑛 is even Is the function injective? Justify your answer. [CBSE SQP 2022] 30. Let 𝐴 = {1,2,3}, 𝐵 = {4,5,6,7} and 𝑓 = {(1,4), (2,5), (3,6)} be a function from A to 𝐵. State whether ' 𝑓 ' is one-one or not. [CBSE 2014]
Long Qs (4 - 5 marks) 31. Show that the function 𝑓: 𝑅 → {𝑥 ∈ 𝑅 : −1 < 𝑥 < 1} defined by 𝑓(𝑥) = 𝑥 1+|𝑥| , 𝑥 ∈ 𝑅 is one-one and onto function. [CBSE SQP 2023] 32. A function 𝑓: [−4,4] → [0,4] is given by 𝑓(𝑥) = √16 − 𝑥2. Show that 𝑓 is an onto function but not a one-one function. Further, find all possible values of ' 𝑎 ' for which 𝑓(𝑎) = √7. [CBSE 2023] 33. Show that the function 𝑓: (−∞, 0) → (−1,0) and 𝑓(𝑥) = 𝑥 1+|𝑥| , 𝑥 ∈ (−∞, 0) is one-one and onto. [CBSE 2020]
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLUTIONS (FROM 2014 TO 2023) PYQ
SOLUTIONS 1. RELATIONS AND ITS TYPES 1. (c) (6,8) ∈ 𝑅 Explanation: Given, 𝑎 = 𝑏 − 2 and 𝑏 > 6 ⇒ (6,8) ∈ R 2. (b) symmetric Explanation: We have, 𝑅 = {(1,1), (2,3), (3,2), (4,3), (3,4)} ∵ (2,2), (3,3), (4,4) ∉ 𝑅 ∴ 𝑅 is not reflexive. For (2,3) ∈ 𝑅, we have (3,2) ∈ 𝑅 Similarly for (4,3) ∈ 𝑅, we have (3,4) ∈ 𝑅. ∴ 𝑅 is symmetric. For (2,3) ∈ 𝑅 and (3,2) ∈ 𝑅, we have (2,2) ∉ 𝑅 ∴ 𝑅 is not transitive ∵ 𝑅 is not reflexive and transitive so it is not an equivalence relation. NOTE: A relation 𝑅 in a set 𝐴 is called (i) reflexive, if (𝑎, 𝑎) ∈ 𝑅, for every 𝑎 ∈ 𝐴, (ii) symmetric, if (𝑎1, 𝑎2) ∈ 𝑅 implies that (𝑎2, 𝑎1) ∈ 𝑅, For all 𝑎1, 𝑎2 ∈ 𝐴. (iii) transitive, if (𝑎1, 𝑎2) ∈ 𝑅 and (𝑎2, 𝑎3) ∈ 𝑅 implies that (𝑎1, 𝑎3) ∈ 𝑅 for all 𝑎1, 𝑎2, 𝑎3 ∈ 𝑅. 3. (c) {(1,1), (3,3), (3,1), (2,3)} Explanation: The ordered pairs to be added in R are: (1,1), (3,3) {needed to make R reflexive} CH-1
(3,1), (2,3) {needed to make 𝑅 symmetric} So, {(1,1)(3,3), (3,1), (2,3)} are required. 4. (b) {−2, −1,0,1,2} Explanation: Given, 𝑥2 + 𝑦2 ≤ 4 ⇒ 𝑦2 ≤ 4 − 𝑥2 ⇒ 𝑦 ≤ �4 − 𝑥2 For domain: 4 − 𝑥2 ≥ 0 ⇒ 𝑥2 ≤ 4 ∴ −2 ≤ 𝑥 ≤ 2 So, Domain = {−2, −1,0,1,2} 5. (c) 𝑅 = {(𝑥, 𝑦) : 𝑥𝑦 is the square number; 𝑥, 𝑦 ∈ 𝑁} Explanation: When 𝑥 ∈ 𝑁, 𝑥2 is a square number So, (𝑥, 𝑥) ∈ R for all 𝑥 ∈ N. Therefore, R is reflexive. 6. (c) 2 Explanation: Total possible pairs = (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) Reflexive means (𝑎, 𝑎) should be in relation. So, (1,1), (2,2), (3,3) should be in relation. Symmetric means if (𝑎, 𝑏) is in relation, then (𝑏, 𝑎) should be in relation. So, since (1,2) is in relation, (2,1) should also be in relation. Transitive means if (𝑎, 𝑏) is in relation and (𝑏, 𝑐) is in relation, then (𝑎, 𝑐) is in relation. So, if (1,2) is in relation and (2,1) is in relation, then (1,1) should be in relation. Relation R1 = {(1,2), (2,1), (1,1), (2,2), (3,3)} Total possible pairs = (1,1), (1,2), (1,3), (2,1), (2, 2), (2,3), (3,1), (3,2), (3,3) So, smallest relation is 𝑅1 = {(1,2), (2,1), (1,1), (2,2), (3,3)}. If we add (2,3) then we have to add (3,2) also, as it is symmetric but, as (1,2) and (2,3) are there, we need to add (1,3) also, as it is transitive.
As we are adding (1,3) we should add (3,1) also, as it is symmetric. Relation 𝑅2 = {(1,2), (2,1), (1,1), (2,2), (3,3), (2,3), (3,2), (1,3), (3,1)} Hence, only two possible relations are there which are equivalence. 7. (d) reflexive but not symmetric Explanation: We have, 𝑎2 − 7𝑎𝑎 + 6𝑎2 = 7𝑎2 − 7𝑎2 = 0 Therefore, 𝑎 Ra for all 𝑎 in Z 𝑆𝑜, 𝑅 is reflexive Let 𝑎𝑅𝑏, then 𝑎2 − 7𝑎𝑏 + 6𝑏2 = 0 Consider 𝑏2 − 7𝑏𝑎 + 6𝑎2 = 𝑏2 − (𝑎2 + 6𝑏2) + 6𝑎2 = 5𝑎2 − 5𝑏2 So, 𝑅 is not symmetric. 8. 𝑅 = {(𝑎, 𝑏): 2 divides (𝑎 − 𝑏)} ⇒ (𝑎 − 𝑏) is a multiple of 2. To find equivalence class 0, put 𝑏 = 0 So, (𝑎 − 0) is a multiple of 2 ⇒ 𝑎 is a multiple of 2 So, in set 𝑍 of integers, all the multiple of 2 will come in equivalence class {0} Hence, equivalence class {0} = {2𝑥} where 𝑥 = integer (Z). NOTE → An equivalence class of 𝑎 is denoted as [𝑎] = {𝑥 ∈ 𝐴 : (𝑎, 𝑥) ∈ 𝑅}. This comprises all of 𝐴 's elements related to letter ' 𝑎 '. 9. 𝑅 = {(𝑎, 𝑏): 𝑎 < 𝑏} (A) Checking for symmetric, if (𝑎, 𝑏) ∈ R such that 𝑎 < 𝑏 then, (𝑏, 𝑎) ∈ 𝑅 not possible (2,3) ∈ R but, (3,2) ∈ R ∴ Relation 𝑅 is not symmetric. (B) Checking for transitive,
if (𝑎, 𝑏) ∈ 𝑅 and (𝑏, 𝑐) ∈ 𝑅 such that 𝑎 < 𝑏 and 𝑏 < 𝑐 then clearly, 𝑎 < 𝑐 i.e., (𝑎, 𝑐) ∈ R ∴ Relation R is transitive. 10. For reflexive: Let 𝑎 ∈ 𝑍, then |𝑎 − 𝑎| = 0, which is divisible by 4. So, (𝑎, 𝑎) ∈ 𝑅. Thus, 𝑅 is a reflexive relation. For symmetric: Let 𝑎, 𝑏 ∈ 𝑍 such that |𝑎 − 𝑏| is divisible by 4. Then |𝑏 − 𝑎| is also divisible by 4 [∵ |𝑏 − 𝑎| = |𝑎 − 𝑏|] So, (𝑎, 𝑏) ∈ 𝑅 ⇒ (𝑏, 𝑎) ∈ 𝑅. Thus, 𝑅 is a symmetric relation. For transitive: Let 𝑎, 𝑏, 𝑐 ∈ 𝑍 and (𝑎, 𝑏) ∈ 𝑅 and (𝑏, 𝑐) ∈ 𝑅. Since (𝑎, 𝑏) ∈ 𝐑 and (𝑏, 𝑐) ∈ 𝑅 Therefore, |𝑎 − 𝑏| = 4𝑘 for some 𝑘 ∈ 𝑍 ⇒ (𝑎 − 𝑏) = ±4𝑘 and |𝑏 − 𝑐| = 4𝑙 for some 𝑙 ∈ 𝑍 ⇒ (𝑏 − 𝑐) = ±4𝑙 Now, (𝑎 − 𝑐) = (𝑎 − 𝑏) + (𝑏 − 𝑐) = ±4𝑘 ± 4𝑙 = 4(±𝑘 ± 𝑙), which is divisible by 4. So, (𝑎, 𝑏) ∈ R and (𝑏, 𝑐) ∈ R ⇒ (𝑎, 𝑐) ∈ 𝑅. Thus, 𝑅 is transitive. Since, 𝑅 is reflexive, symmetric and transitive, hence, it is an equivalence relation. Set of elements related to 1 is {1,5,9} Set of elements related to 2 is {2,6,10} So, equivalence class of [2] is {2,6,10}. 11. 𝑅 = {(𝑥, 𝑦): 2𝑥 + 𝑦 = 8} is a relation on 𝑁.
Therefore, 𝑅 = {(3,2), (2,4), (1,6)} So, Range = {2,4,6}. 12. Relation 𝑅 = {(𝑎, 𝑎3): 𝑎 is a prime number less than 5}. Therefore, 𝑅 = {(2,8), (3,27)}, ⇒ Range = {8,27} 13. We have, 2𝑥 + 𝑦 = 24 ⇒ 𝑦 = 24 − 2𝑥 For 𝑥 = 1, 𝑦 = 24 − 2 × 1 = 24 − 2 = 22 For 𝑥 = 2, 𝑦 = 24 − 2 × 2 = 24 − 4 = 20 For 𝑥 = 3, 𝑦 = 24 − 2 × 3 = 24 − 6 = 18 For 𝑥 = 4, 𝑦 = 24 − 2 × 4 = 24 − 8 = 16 For 𝑥 = 5, 𝑦 = 24 − 2 × 5 = 24 − 10 = 14 For 𝑥 = 6, 𝑦 = 24 − 2 × 6 = 24 − 12 = 12 For 𝑥 = 7, 𝑦 = 24 − 2 × 7 = 24 − 14 = 10 For 𝑥 = 8, 𝑦 = 24 − 2 × 8 = 24 − 16 = 8 For 𝑥 = 9, 𝑦 = 24 − 2 × 9 = 24 − 18 = 6 For 𝑥 = 10, 𝑦 = 24 − 2 × 10 = 24 − 20 = 4 For 𝑥 = 11, 𝑦 = 24 − 2 × 11 = 24 − 22 = 2 In ordered pair, 𝑅 = {(1,22), (2,20), (3,18), (4,16), (5,14), (6,12), (7,10), (8,8), (9,6), (10,4), (11,2)} ∴ Domain of 𝑅 = {1,2,3,4,5,6,7,8,9,10,11} Range of 𝑅 = {22,20,18,16,14,12,10,8,6,4,2} Here, 1 ∈ 𝑁 but (1,1) ∉ 𝑅, hence 𝑅 is not reflexive. Hence, R is not an equivalence relation.
NOTE: A relation is said to be equivalence if it is reflexive, symmetric and transitive. 14. Let (𝑎, 𝑏) be an arbitrary element of 𝑁 × 𝑁. Then, (𝑎, 𝑏) ∈ 𝑁 × 𝑁 and 𝑎, 𝑏 ∈ 𝑁 We have, 𝑎𝑏 = 𝑏𝑎; ( As 𝑎, 𝑏 ∈ 𝑁 and multiplication is commutative on N ) ⇒ (𝑎, 𝑏)R(𝑎, 𝑏), according to the definition of the relation R on N × N. Thus, (𝑎, 𝑏)𝑅(𝑎, 𝑏), ∀(𝑎, 𝑏) ∈ N × N. So, 𝑅 is reflexive relation on N × N. Let (𝑎, 𝑏), (𝑐, 𝑑) be arbitrary elements of N × N such that (𝑎, 𝑏)𝑅(𝑐, 𝑑). Then, (𝑎, 𝑏)R(𝑐, 𝑑) ⇒ 𝑎𝑑 = 𝑏𝑐 ⇒ 𝑏𝑐 = 𝑎𝑑 (changing LHS and RHS) ⇒ 𝑐𝑏 = 𝑑𝑎; (As 𝑎, 𝑏, 𝑐, 𝑑 ∈ N and multiplication is commutative on N ) ⇒ (𝑐, 𝑑)𝑅(𝑎, 𝑏); according to the definition of the relation R on N × N Thus, (𝑎, 𝑏)R(𝑐, 𝑑) ⇒ (𝑐, 𝑑)𝑅(𝑎, 𝑏) So, R is symmetric relation on N × N. Let (𝑎, 𝑏), (𝑐, 𝑑), (𝑒, 𝑓) be arbitrary elements of N × N such that (𝑎, 𝑏)R(𝑐, 𝑑) and (𝑐, 𝑑)R(𝑒, 𝑓). Then (𝑎, 𝑏)R(𝑐, 𝑑) ⇒ 𝑎𝑑 = 𝑏𝑐 and (𝑐, 𝑑)𝑅(𝑒, 𝑓) ⇒ 𝑐𝑓 = 𝑑𝑒 ⇒ (𝑎𝑑)(𝑐𝑓) = (𝑏𝑐)(𝑑𝑒) ⇒ 𝑎𝑓 = 𝑏𝑒 ⇒ (𝑎, 𝑏)R(𝑒, 𝑓); (according to the definition of the relation R on N × N ) Thus, (𝑎, 𝑏)R(𝑐, 𝑑) and (𝑐, 𝑑)R(𝑒, 𝑓)
⇒ (𝑎, 𝑏)R(𝑒, 𝑓) So, 𝑅 is transitive relation on N × N. As the relation 𝑅 is reflexive, symmetric and transitive so, it is equivalence relation on N × N. [(2,6)] = {(𝑥, 𝑦) ∈ N × N: (𝑥, 𝑦)𝑅(2,6)} = {(𝑥, 𝑦) ∈ N × N: 3𝑥 = 𝑦} = {(𝑥, 3𝑥): 𝑥 ∈ N} = {(1,3), (2,6), (3,9), … … . . } 15. Let 𝐴 ∈ 𝑃(𝑋). Then 𝐴 ⊂ 𝐴 ⇒ (𝐴, 𝐴) ∈ 𝑅 Hence, 𝑅 is reflexive. Let 𝐴, 𝐵, 𝐶, ∈ 𝑃(𝑋) such that (𝐴, 𝐵), (𝐵, 𝐶) ∈ 𝑅 ⇒ 𝐴 ⊂ 𝐵, 𝐵, ⊂ 𝐶 ⇒ 𝐴 ⊂ 𝐶 ⇒ (𝐴, 𝐶) ∈ 𝑅 Hence, 𝑅 is transitive. 𝜙, 𝑋 ∈ 𝑃(𝑋) such that 𝜙 ⊂ 𝑋. Hence, (𝜙, 𝑋) ∈ 𝑅. But X ⊄ 𝜙 which implies that (𝑋, 𝜙) ∉ 𝑅. Thus, 𝑅 is not symmetric. 16. Let (𝑎, 𝑏) ∈ 𝑁 × 𝑁. Then we have, 𝑎𝑏 = 𝑏𝑎 (by commutative property of multiplication of natural numbers) ⇒ (𝑎, 𝑏)R(𝑎, 𝑏) Hence, 𝑅 is reflexive. Let (𝑎, 𝑏), (𝑐, 𝑑) ∈ 𝑁 × 𝑁 such that (𝑎, 𝑏)𝑅(𝑐, 𝑑). Then 𝑎𝑑 = 𝑏𝑐 ⇒ 𝑐𝑏 = 𝑑𝑎 (by commutative property of multiplication of natural numbers
⇒ (𝑐, 𝑑)R(𝑎, 𝑏) Hence, 𝑅 is symmetric. Let (𝑎, 𝑏), (𝑐, 𝑑), (𝑒, 𝑓) ∈ 𝑁 × 𝑁 such that (𝑎, 𝑏)R(𝑐, 𝑑) and (𝑐, 𝑑)R(𝑒, 𝑓). Then, 𝑎𝑑 = 𝑏𝑐, 𝑐𝑓 = 𝑑𝑒 ⇒ 𝑎𝑑𝑐𝑓 = 𝑏𝑐𝑑𝑒 ⇒ 𝑎𝑓 = 𝑏𝑒 ⇒ (𝑎, 𝑏)R(𝑒, 𝑓) Hence, R is transitive. Since, 𝑅 is reflexive, symmetric and transitive, 𝑅 is an equivalence relation on 𝑁 × 𝑁. 17. A = {𝑥 ∈ 𝑍: 0 ≤ 𝑥 ≤ 12} = {1,2,3,4,5,6,7,8,9, 10,11,12} 𝑅 = {(𝑎, 𝑏): |𝑎 − 𝑏| is divisible by 3} For any element 𝑎 ∈ A, we have (𝑎, 𝑎) ∈ R as |𝑎 − 𝑎| = 0 is divisible by 3. ∴ R is reflexive. Now, let (𝑎, 𝑏) ∈ 𝑅 ⇒ |𝑎 − 𝑏| is divisible 3. ⇒∣ 𝑏 − 𝑎)| = |𝑎 − 𝑏 ∣ is divisible by 3 ⇒ (𝑏, 𝑎) ∈ R ∴ R is symmetric. Now, let (𝑎, 𝑏), (𝑏, 𝑐) ∈ 𝑅. ⇒ |𝑎 − 𝑏| is divisible by 3 and |𝑏 − 𝑐| is divisible by 3. ⇒ ±(𝑎 − 𝑏) is divisible by 3 and ±(𝑏 − 𝑐) is divisible by 3. ⇒ (𝑎 − 𝑐) = (𝑎 − 𝑏) + (𝑏 − 𝑐) is divisible by 3. and −(𝑎 − 𝑐) = −{(𝑎 − 𝑏) + (𝑏 − 𝑐)} is divisible by 3. ⇒ |𝑎 − 𝑐| is divisible by 3. ⇒ (𝑎, 𝑐) ∈ 𝑅
∴ R is transitive. Hence, R is an equivalence relation. 18. Reflexive: Let (𝑎, 𝑏) ∈ 𝑁 × 𝑁. ∵ 𝑎𝑏(𝑏 + 𝑎) = 𝑏𝑎(𝑎 + 𝑏) ⇒ (𝑎, 𝑏)R(𝑎, 𝑏) ⇒ 𝑅 is reflexive. Symmetric: For (𝑎, 𝑏), (𝑐, 𝑑) ∈ 𝑁 × 𝑁 such that (𝑎, 𝑏)𝑅(𝑐, 𝑑). ∴ 𝑎𝑑(𝑏 + 𝑐) = 𝑏𝑐(𝑎 + 𝑑) or, 𝑏𝑐(𝑎 + 𝑑) = 𝑎𝑑(𝑏 + 𝑐) or, 𝑐𝑏(𝑑 + 𝑎) = 𝑑𝑎(𝑐 + 𝑏) ⇒ (𝑐, 𝑑)𝑅(𝑎, 𝑏) ⇒ 𝑅 is symmetric. Transitive: Let (𝑎, 𝑏), (𝑐, 𝑑), (𝑒, 𝑓) ∈ 𝑁 × 𝑁 such that (𝑎, 𝑏)R(𝑐, 𝑑) and (𝑐, 𝑑)R(𝑒, 𝑓). ∴ 𝑎𝑑(𝑏 + 𝑐) = 𝑏𝑐(𝑎 + 𝑑) and 𝑐𝑓(𝑑 + 𝑒) = 𝑑𝑒(𝑐 + 𝑓) ⇒ 𝑎𝑑𝑏 + 𝑎𝑑𝑐 = 𝑏𝑐𝑎 + 𝑏𝑐𝑑 ….(i) and 𝑐𝑓𝑑 + 𝑐𝑓𝑒 = 𝑑𝑒𝑐 + 𝑑𝑒𝑓 ….(ii) Multiplying (i) by 𝑒𝑓 and (ii) by 𝑎𝑏 and then adding them, we get 𝑎𝑑𝑏𝑒𝑓 + 𝑎𝑑𝑐𝑒𝑓 + 𝑐𝑓𝑑𝑎𝑏 + 𝑐𝑓𝑒𝑎𝑏 = 𝑏𝑐𝑎𝑒𝑓 + 𝑏𝑐𝑑𝑒𝑓 + 𝑑𝑒𝑐𝑎𝑏 + 𝑑𝑒𝑓𝑎𝑏 ⇒ 𝑎𝑑𝑐𝑒𝑓 + 𝑎𝑑𝑐𝑓𝑏 = 𝑏𝑐𝑑𝑒𝑎 + 𝑏𝑐𝑑𝑒𝑓 ⇒ adcf (𝑒 + 𝑏) = 𝑏𝑐𝑑𝑒(𝑎 + 𝑓) ⇒ 𝑎𝑓(𝑏 + 𝑒) = 𝑏𝑒(𝑎 + 𝑓) ⇒ (𝑎, 𝑏)R(𝑒, 𝑓) ⇒ 𝑅 is transitive. Hence, 𝑅 is an equivalence relation. 19. Given 𝑅 = {(𝑎, 𝑏): |𝑎 − 𝑏| is divisible by 2}
and 𝐴 = {1,2,3,4,5} 𝑅 = {(1,1), (2,2), (3,3), (4,4), (5,5)(1,3), (1,5), (2,4), (3,5), (3,1), (5,1), (4,2), (5,3)} Reflexive: ∀𝑎 ∈ A, (𝑎, 𝑎) ∈ R, ∴ R is reflexive ∴ [As {(1,1), (2,2), (3,3), (5,5)} ∈ 𝑅] Symmetric ∀(𝑎, 𝑏) ∈ R, (𝑏, 𝑎) ∈ R, ∴ R is symmetric. [ As {(1,3), (1,5), (2,4), (3,5), (3,1), (5,1), (4,2), (5,3)} ∈ 𝑅] Transitive ∀(𝑎, 𝑏), (𝑏, 𝑐) ∈ 𝑅, (𝑎, 𝑐) ∈ R ∴ R is transitive. [As (1,3), (3,1) ∈ 𝑅 ⇒ (1,1) ∈ 𝑅 and similarly others] ∴ R is an equivalence relation. Equivalence classes are and [1] = {1,3,5} [2] = {2,4} [3] = {1,3,5} [4] = {2,4} [5] = {1,3,5} 2. FUNCTIONS AND ITS TYPES 20. (b) 2𝑛 − 2 Explanation: Given that, 𝐴 = {1,2,3, … 𝑛} and 𝐵 = {𝑎, 𝑏} If function is surjective then its range must be set 𝐵 = {𝑎, 𝑏}
Now number of onto functions = Number of ways ' 𝑛 ' distinct objects can be distributed in two boxes ' 𝑎 ' and ' 𝑏 ' in such a way that no box remains empty. Now for each object there are two options, either it is put in box 'a' or in box 'b' So total number of ways of ' 𝑛 ' different objects put in two boxes = 2 × 2 × 2 … 𝑛 times = 2𝑛 But in one case all the objects are put in box ' 𝑎 ' and in one case all the objects are put in box 'b'. So, number of surjective functions = 2𝑛 − 2 21. (d) bijective Explanation: Given, 𝑓(𝑥) = 𝑥2 For injective: Let 𝑓(𝑥1) = 𝑓(𝑥2) ⇒ 𝑥1 2 = 𝑥2 2 ⇒ 𝑥1 = 𝑥2 ( ∵ - ve rejected) So, it is injective. For subjective: Range = 𝑋 Co-domain = 𝑋 So, range = co-domain So, it is subjective. Hence, it is bijective. 22. (d) neither one-one nor onto Explanation: For one-one: 𝑓(𝑥1) = 𝑓(𝑥2) ⇒ 2 + 𝑥1 2 = 2 + 𝑥2 2 ⇒ 𝑥1 2 = 𝑥2 2 ⇒ 𝑥1 = ±𝑥2 [∵ 𝑥1, 𝑥2 ∈ 𝑅] So, it is not one-one. For onto:
Range = Positive real numbers Co-domain = R ∴ Range ≠ co-domain So, it is not onto. Hence, 𝑓(𝑥) is neither one-one nor onto. 23. (d) neither one-one nor onto Explanation: Given: 𝑓(𝑥) = 4 + 3cos 𝑥 Since, cos 𝜋 2 = cos �− 𝜋 2 � ⇒ 4 + 3cos 𝜋 2 = 4 + cos �− 𝜋 2 � ⇒ 𝑓 � 𝜋 2 � = 𝑓 �− 𝜋 2 � But 𝜋 2 ≠ − 𝜋 2 So, 𝑓 is not one-one. Range of cos 𝑥 is [−1,1] ⇒ −1 ≤ cos 𝑥 ≤ 1 ⇒ −3 ≤ 3cos 𝑥 ≤ 3 ⇒ 1 ≤ 4 + 3cos 𝑥 ≤ 7 ⇒ 1 ≤ 𝑓(𝑥) ≤ 7 So, the range of 𝑓 is [1,7] Thus, 𝑓 is not onto. Hence, 𝑓 is neither one-one nor onto. 24. (d) 0 Explanation: If we consider one-one function, only two elements of the set {1,2,3,4,5} can have images. Therefore, there can't be a one-one function from {1,2,3,4,5} → {𝑎, 𝑏} Hence, the number of one-one functions is 0. 25. (d) fis not defined
Explanation: We have, 𝑓(𝑥) = 1 𝑥 , ∀𝑥 ∈ 𝑅 For 𝑥 = 0, 𝑓(𝑥) is not defined. Hence, 𝑓(𝑥) is not a defined function. 26. (d) (A) is false but (R) is true. Explanation: Assertion is false. As element 4 has no image under 𝑓, so relation 𝑓 is not a function. Reason is true. The given function 𝑓: {1, 2,3} → {𝑥, 𝑦, 𝑧, 𝑝} is one - one, as for each 𝑎 ∈ {1,2,3}, there is a different image in {𝑥, 𝑦, 𝑧, 𝑝} under 𝑓. 27. The given function is 𝑓: 𝑁 → 𝑁 such that 𝑓(𝑛) = � 𝑛 + 1 2 , if 𝑛 is odd 𝑛 2 , if 𝑛 is even for all 𝑛 ∈ N We shall verify whether 𝑓(𝑥) is one-one and onto. One-One: From the definition of 𝑓(𝑛) 𝑓(1) = 1+1 2 = 1 and 𝑓(2) = 2 2 = 1 𝑓(𝑛) is not an one-one function because at two distinct values from domain (N), 𝑓(𝑛) has same image. Onto: For onto function, we check whether Range of 𝑓(𝑛) = Co-domain of 𝑓(𝑛) Now, if 𝑛 is an odd natural number, then (2𝑛 − 1) is also an odd natural number. Now, 𝑓(2𝑛 − 1) = 2𝑛−1+1 2 = 𝑛 … . (𝑖) Again, if 𝑛 is an even natural number, then 2𝑛 is also an even natural number. Then, 𝑓(2𝑛) = 2𝑛 2 = 𝑛 … . (𝑖𝑖)#(𝑖𝑖) From equations, (i) and (ii), we observe that for each 𝑛 (whether even or odd), there exists its pre- image in N. i.e., Range of 𝑓(𝑛) = Co-domain of 𝑓(𝑛).
Hence, 𝑓 is onto. Since, 𝑓(𝑛) is onto but not one-one, it is not a bijective function. 28. The given function is 𝑓: 𝑋 → 𝑌 and relation on 𝑋 is 𝑅 = {(𝑎, 𝑏): 𝑓(𝑎) = 𝑓(𝑏)} Reflexive: Since, for every 𝑥 ∈ 𝑋, we have 𝑓(𝑥) = 𝑓(𝑥) ⇒ (𝑥, 𝑥) ∈ 𝑅, ∀𝑥 ∈ 𝑋 Therefore, 𝑅 is reflexive. Symmetric: Let (𝑥, 𝑦) ∈ 𝑅 Then, 𝑓(𝑥) = 𝑓(𝑦) ⇒ 𝑓(𝑦) = 𝑓(𝑥) ⇒ (𝑦, 𝑥) ∈ 𝑅 Thus, (𝑥, 𝑦) ∈ 𝑅 ⇒ (𝑦, 𝑥) ∈ 𝑅, ∀𝑥, 𝑦 ∈ 𝑋 Therefore, 𝑅 is symmetric. Transitive: Let 𝑥, 𝑦, 𝑧 ∈ 𝑋 such that (𝑥, 𝑦) ∈ 𝑅 and (𝑦, 𝑧) ∈ 𝑅 Then, 𝑓(𝑥) = 𝑓(𝑦) … (𝑖) And 𝑓(𝑦) = 𝑓(𝑧) … . (𝑖𝑖) From eqs. (i) and (ii), we get 𝑓(𝑥) = 𝑓(𝑧) ⇒ (𝑥, 𝑧) ∈ 𝑅 Thus, (𝑥, 𝑦) ∈ 𝑅 and (𝑦, 𝑧) ∈ 𝑅 ⇒ (𝑥, 𝑧) ∈ 𝑅, ∀𝑥, 𝑦, 𝑧 ∈ 𝑅 Therefore, 𝑅 is transitive.
Since, 𝑅 is reflexive symmetric and transitive, so it is an equivalence relation. 29. Let 𝑦 ∈ 𝑁 (co-domain). Then ∃2𝑦 ∈ 𝑁 (domain) such that 𝑓(2𝑦) = 2𝑦 2 = 𝑦. Hence, 𝑓 is surjective. 1,2 ∈ 𝑁 (domain) such that 𝑓(1) = 1 = 𝑓(2) Hence, 𝑓 is not injective. 30. Given that 𝐴 = {1,2,3}, 𝐵 = {4,5,6,7} Now, 𝑓: 𝐴 → 𝐵 is defined as 𝑓 = {(1,4), (2,5), (3,6)}. 𝑓(1) = 4, 𝑓(2) = 5, 𝑓(3) = 6, so 𝑓 is one-one. 31. We have, 𝑓(𝑥) = � 𝑥 1+𝑥 , if 𝑥 ≥ 0 𝑥 1−𝑥 , if 𝑥 < 0 Now, we consider the following cases Case 1: when 𝑥 ≥ 0, we have 𝑓(𝑥) = 𝑥 1+𝑥 Injectivity: let 𝑥, 𝑦 ∈ 𝑅+ ∪ {0} such that 𝑓(𝑥) = 𝑓(𝑦), then ⇒ 𝑥 1 + 𝑥 = 𝑦 1 + 𝑦 ⇒ 𝑥 + 𝑥𝑦 = 𝑦 + 𝑥𝑦 ⇒ 𝑥 = 𝑦 So, 𝑓 is injective function. Surjectivity: when 𝑥 ≥ 0, we have 𝑓(𝑥) = 𝑥 1+𝑥 ≥ 0 and 𝑓(𝑥) = 1 − 1 1+𝑥 < 1, as 𝑥 ≥ 0 Let 𝑦 ∈ [0,1), thus for each 𝑦 ∈ [0,1) there exists 𝑥 = 𝑦 1−𝑦 ≥ 0 such that 𝑓(𝑥) = 𝑦 1−𝑦 1+ 𝑦 1−𝑦 = 𝑦. So, 𝑓 is onto function on [0, ∞) to [0,1). Case 2: when 𝑥 < 0, we have 𝑓(𝑥) = 𝑥 1−𝑥 Injectivity: Let 𝑥, 𝑦 ∈ 𝑅− i.e., 𝑥, 𝑦 < 0, such that 𝑓(𝑥) = 𝑓(𝑦), then ⇒ 𝑥 1 − 𝑥 = 𝑦 1 − 𝑦 ⇒ 𝑥 − 𝑥𝑦 = 𝑦 − 𝑥𝑦 ⇒ 𝑥 = 𝑦 So, 𝑓 is injective function. Surjectivity: 𝑥 < 0,
we have 𝑓(𝑥) = 𝑥 1−𝑥 < 0 also, 𝑓(𝑥) = 𝑥 1 − 𝑥 = −1 + 1 1 − 𝑥 > −1 −1 < 𝑓(𝑥) < 0. Let 𝑦 ∈ (−1,0) be an arbitrary real number and there exists 𝑥 = 𝑦 1+𝑦 < 0 such that, 𝑓(𝑥) = 𝑓 � 𝑦 1 + 𝑦 � = 𝑦 1 + 𝑦 1 − 𝑦 1 + 𝑦 = 𝑦 So, for 𝑦 ∈ (−1,0), there exists 𝑥 = 𝑦 1+𝑦 < 0 such that 𝑓(𝑥) = 𝑦. Hence, 𝑓 is onto function on (−∞, 0) to (−1,0). Case 3: (Injectivity): Let 𝑥 > 0 and 𝑦 < 0 such that 𝑓(𝑥) = 𝑓(𝑦) ⇒ 𝑥 1 + 𝑥 = 𝑦 1 − 𝑦 ⇒ 𝑥 − 𝑥𝑦 = 𝑦 + 𝑥𝑦 ⇒ 𝑥 − 𝑦 = 2𝑥𝑦, Here LHS > 0 but RHS < 0, which is inadmissible. Hence, 𝑓(𝑥) ≠ 𝑓(𝑦) when 𝑥 ≠ 𝑦. Hence, 𝑓 is one-one and onto function. 32. 𝑓(𝑥) = �16 − 𝑥2 for 𝑥 = 2, 𝑓(𝑥) = √12 for 𝑥 = −2, 𝑓(𝑥) = √12 Since, for 𝑥 = 2 and -2, the function has same image
∴ The given function is not one-one. Let 𝑦 ∈ [0,4] ∴ 𝑦 ≥ 0 𝑦 = �16 − 𝑥2 𝑦2 = 16 − 𝑥2 𝑥 = �16 − 𝑦2 For 𝑥 ∈ 𝑅, 16 − 𝑦2 ≥ 0 (4 − 𝑦)(4 + 𝑦) ≥ 0 ⇒ Either (4 − 𝑦) ≥ 0 and (4 + 𝑦) ≥ 0 or (4 − 𝑦) < 0 and (4 + 𝑦) < 0. ∴ For every 𝑦 ∈ [0,4]∃𝑥 ∈ [−4,4] such that 𝑦 = 𝑓(𝑥) ∴ The given function is onto 𝑓(𝑎) = √7 �16 − 𝑎2 = √7 Squaring on both sides: 16 − 𝑎2 = 7 𝑎2 = 9 𝑎 = ±3 33. Given, 𝑓(𝑥) = 𝑥 1 + |𝑥| , 𝑥 ∈ (−∞, 0) = 𝑥 1 − 𝑥 [∵ 𝑥 ∈ (−∞, 0), |𝑥| = −𝑥] For one-one: Let 𝑓(𝑥1) = 𝑓(𝑥2), 𝑥1, 𝑥2 ∈ (−∞, 0)
⇒ 𝑥1 1 − 𝑥1 = 𝑥2 1 − 𝑥2 ⇒ 𝑥1(1 − 𝑥2) = 𝑥2(1 − 𝑥1) ⇒ 𝑥1 − 𝑥1𝑥2 = 𝑥2 − 𝑥1𝑥2 ⇒ 𝑥1 = 𝑥2 Hence, if 𝑓(𝑥1) = 𝑓(𝑥2), then 𝑥1 = 𝑥2 ∴ f is one-one For onto: Let 𝑓(𝑥) = 𝑦 ⇒ 𝑦 = 𝑥 1 − 𝑥 ⇒ 𝑦(1 − 𝑥) = 𝑥 ⇒ 𝑦 − 𝑥𝑦 = 𝑥 ⇒ 𝑥 + 𝑥𝑦 = 𝑦 ⇒ 𝑥(1 + 𝑦) = 𝑦 ⇒ 𝑥 = 𝑦 1 + 𝑦 So, x is defined for all values of y. ∴ 𝑓 is onto.
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
Inverse Trigonometric Functions 1. INVERSE TRIGONOMETRIC FUNCTIONS Objective Qs (1 mark) 1. The value of sin �sin−1 � 1 2 � + cos−1 � 1 2 �� is: (a) 1 (b) 2 (c) 1 2 (d) 1 4 [Delhi Gov. SQP 2022] 2. The value of sin−1 �cos 13𝜋 5 � is: (a) − 3𝜋 5 (b) − 𝜋 10 (c) 3𝜋 10 (d) 𝜋 10 [CBSE Term-1 2021] 3. The value of the expression sec−1 (2) + sin−1 � 1 2 � + tan−1 (−√3) is: (a) 5𝜋 3 (b) 𝜋 3 (c) −𝜋 3 (d) 𝜋 6 [Delhi Gov. Term-1 2021] 4. If 𝑎 ≤ 2sin−1 𝑥 + cos−1 𝑥 ≤ 𝑏, then: (a) 𝑎 = 0, 𝑏 = 𝜋 (b) 𝑎 = 𝜋, 𝑏 = 2𝜋 (c) 𝑎 = −𝜋 2 , 𝑏 = 𝜋 2 (d) 𝑎 = 0, 𝑏 = 𝜋 2 [Delhi Gov. Term-1 SQP 2021] 5. If tan−1 𝑥 = 𝑦, then: (a) −1 < 𝑦 < 1 (b) −𝜋 2 ≤ 𝑦 ≤ 𝜋 2 CH-2 CLICK HERE FOR SOLUTIONS
(c) −𝜋 2 < 𝑦 < 𝜋 2 (d) 𝑦 ∈ � −𝜋 2 , 𝜋 2 � [CBSE Term-1 SQP 2021] 6. If sin−1 𝑥 + sin−1 𝑦 = 2𝜋 3 , then cos−1 𝑥 + cos−1 𝑦 is: (a) −𝜋 3 (b) 𝜋 3 (c) 𝜋 (d) 𝜋 2 [Delhi Gov. Term - 1 SQP 2021] 7. The principal value of �tan−1 √3 − cot−1 (−√3)� is: (a) 𝜋 (b) − 𝜋 2 (c) 0 (d) 2√3 [CBSE Term-1 2021] In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices. (a) Both (A) and (R) are true and (R) is the correct explanation of (A). (b) Both (A) and (R) are true but (R) is not the correct explanation of (𝐴). (c) (A) is true but (R) is false. (d) (A) is false but (R) is true. 8. Assertion (A): Maximum value of (cos−1 𝑥)2 is 𝜋2 . Reason (R): Range of the principal value branch of cos−1 𝑥 is � −𝜋 2 , 𝜋 2 �. [CBSE 2023] 9. Assertion (A): The domain of the function sec−1 2𝑥 is �−∞, − 1 2 � ∪ � 1 2 , ∞�. Reason (R): sec−1 (−2) = − 𝜋 4 [CBSE SQP 2022] 10. Assertion (A): sin−1 �sin � 2𝜋 3 �� = 2𝜋 3 Reason (R): sin−1 (sin 𝜃) = 𝜃 if 𝜃 ∈ �� −𝜋 2 � , 𝜋 2 � CLICK HERE FOR SOLUTIONS
[Delhi Gov. SQP 2022] Very Short & Short Qs (1 - 3 marks) 11. Find the value of sin−1 �cos � 33𝜋 5 ��. [CBSE SQP 2023] 12. Find the domain of sin−1 (𝑥2 − 4). [CBSE SQP 2023] 13. Evaluate sin−1 �sin 3𝜋 4 � + cos−1 (cos 𝜋) + tan−1 (1). [CBSE 2023] 14. Draw the graph of cos−1 𝑥, where 𝑥 ∈ [−1,0]. Also, write its range. [CBSE 2023] 15. Find the value of sin−1 �sin � 13𝜋 7 ��. 16. Prove that: sin−1 �2𝑥�1 − 𝑥2� = 2cos−1 𝑥, 1 √2 ≤ 𝑥 ≤ 1 [CBSE 2020] 17. Prove that 3sin−1 𝑥 = sin−1 (3𝑥 − 4𝑥3), 𝑥 ∈ �− 1 2 , 1 2 � [CBSE 2018] 18. Find the value of tan−1 √3 − sec−1 (−2). [CBSE 2018] 19. Prove that: 3cos−1 𝑥 = cos−1 (4𝑥3 − 3𝑥), 𝑥 ∈ � 1 2 , 1� [CBSE 2018] 20. Solve for 𝑥: tan−1 (𝑥 − 1) + tan−1 𝑥 + tan−1 (𝑥 + 1) = tan−1 3𝑥. [CBSE 2014] 21. If (tan−1 𝑥)2 + (cot−1 𝑥)2 = 5𝜋2 8 , find 𝑥. [CBSE 2015] 22. Prove that: 2tan−1 � 1 5 � + sec−1 � 5√2 7 � + 2 CLICK HERE FOR SOLUTIONS
tan−1 � 1 8 � = 𝜋 4 [CBSE 2014] 23. If tan−1 𝑥tan−1 𝑦 = 𝜋 4 , 𝑥𝑦 < 1, then write the value of 𝑥 + 𝑦 + 𝑥𝑦. [CBSE 2014] 24. Write the principal value of tan−1 �sin � −𝜋 2 �� [CBSE 2014] 25. Find the value of cot � 𝜋 2 − 2cot−1 √3�. [CBSE 2014] 26. Write the value of cos−1 �− 1 2 � + 2sin−1 � 1 2 � [CBSE 2014] 27. Solve for 𝑥: tan−1 𝑥 + 2cot−1 𝑥 = 2𝜋 3 [CBSE 2014] 28. Prove that: cot−1 � √1+sin 𝑥+√1−sin 𝑥 √1+sin 𝑥−√1−sin 𝑥 � = 𝑥 2 , 𝑥 ∈ �0, 𝜋 4 � [CBSE 2014] 29. If sin �sin−1 1 5 + cos−1 𝑥� = 1, then find the value of 𝑥. [CBSE 2014] 30. Prove that: sin−1 � 8 17 � + sin−1 � 3 5 � = cos−1 � 36 85 �. Long Qs (4 - 5 marks) 31. Solve tan−1 2𝑥 + tan−1 3𝑥 = 𝜋 4 . [CBSE 2019, 15] 32. Find the value of sin �cos−1 4 5 + tan−1 2 3 �. [CBSE 2014] 33. Prove that: cos−1 12 13 + sin−1 3 5 = sin−1 56 65 CLICK HERE FOR SOLUTIONS
[CBSE 2019] 34. Solve: tan−1 (𝑥 + 1) + tan−1 (𝑥 − 1) = tan−1 8 31 [CBSE 2019, 15] 35. Prove that: 2tan−1 � 1 2 � + tan−1 � 1 7 � = sin−1 � 31 25√2 � [CBSE 2019] CLICK HERE FOR SOLUTIONS
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
Matrices 1. MATRIX AND ITS TYPES Objective Qs (1 mark) 1. If A = �𝑎𝑖𝑗� is a square matrix of order 2 such that 𝑎𝑖𝑗 = � 1, when 𝑖 ≠ 𝑗 0, when 𝑖 = 𝑗 , then 𝐴2 is: (a) � 1 0 1 0 � 2×2 (b) � 1 1 0 0 � 2×2 (c) � 1 1 1 0 � 2×2 (d) � 1 0 0 1 � 2×2 [CBSE SQP 2023] 2. If 𝐴 is a 2 × 3 matrix such that 𝐴𝐵 and 𝐴𝐵′ both are defined, then order of the matrix 𝐵 is: (a) 2 × 2 (b) 2 × 1 (c) 3 × 2 (d) 3 × 3 [CBSE 2023] 3. If � 2𝑎 + 𝑏 𝑎 − 2𝑏 5𝑐 − 𝑑 4𝑐 + 3𝑑 � = � 4 −3 11 24 �, then the value of 𝑎 + 𝑏 − 𝑐 + 2𝑑 is: (a) 8 (b) 10 (c) 4 (d) -8 [CBSE Term-1 SQP 2021] Very Short & Short Qs (1 - 3 marks) 4. Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3. [CBSE 2016] 5. Write the element 𝑎23 of a 3 × 3 matrix A = �𝑎𝑖𝑗� whose elements 𝑎𝑖𝑗 are given by 𝑎𝑖𝑗 = |𝑖−𝑗| 2 [CBSE 2015] 6. Find the element 𝑎32 of a 3 × 3 matrix: if 𝑎𝑖𝑗 is given by 𝑎𝑖𝑗 = 1 2 | − 3𝑖 + 𝑗|. [CBSE 2014] CH-3 CLICK HERE FOR SOLUTIONS
7. If � 𝑥𝑦 4 𝑧 + 6 𝑥 + 𝑦 � = � 8 𝑤 0 6 �, then write the value of (𝑥 + 𝑦 + 𝑧). [CBSE 2014] 8. If � 𝑥 − 𝑦 𝑧 2𝑥 − 𝑦 𝑤� = � −1 4 0 5 �, find the value of 𝑥 + 𝑦. [CBSE 2014] 9. If � 𝑎 + 4 3𝑏 8 −6 � = � 2𝑎 + 2 𝑏 + 2 8 𝑎 − 8𝑏 �, then find the value of (𝑎 − 2𝑏). [CBSE 2014] 2. OPERATIONS ON MATRICES Objective Qs (1 mark) 10. If � 1 −1 2 3 � + 𝑋 = � 3 4 5 6 �, where X = � 𝑎 𝑏 𝑐 𝑑 � , then 𝑎 + 𝑐 − 𝑏 − 𝑑 = (a) 13 (b) 5 (c) -8 (d) -3 [Delhi Gov. Term-1 SQP 2021] 11. Given that matrices 𝐴 and 𝐵 are of order 3 × 𝑛 and 𝑚 × 5 respectively, then the order of matrix 𝐶 = 5𝐴 + 3𝐵 is: (a) 3 × 5 and 𝑚 = 𝑛 (b) 3 × 5 (c) 3 × 3 (d) 5 × 5 [CBSE Term-1 SQP 2021] 12. If 𝐴 = �𝑎𝑖𝑗� is a square matrix of order 2 such that 𝑎𝑖𝑗 = � 1, when 𝑖 ≠ 𝑗 0, when 𝑖 = 𝑗 , then 𝐴2 is: (a) � 1 0 1 0 � (b) � 1 1 0 0 � (c) � 1 1 1 0 � (d) � 1 0 0 1 � [CBSE Term-1 SQP 2021] 13. If 𝐴 is square matrix such that 𝐴2 = 𝐴, then (𝐼 + 𝐴)3 − 7𝐴 is equal to: (a) 𝐴 (b) I + A CLICK HERE FOR SOLUTIONS
(c) I- A (d) I [CBSE Term-1 SQP 2021] 14. If 𝐴 = � 0 2 3 −4 � and 𝑘𝐴 = � 0 3𝑎 2𝑏 24 �, then the values of 𝑘, 𝑎 and 𝑏 respectively are: (a) −6, −12, −18 (b) −6, −4, −9 (c) −6,4,9 (d) −6,12,18 [CBSE Term-1 SQP 2021] 15. If [𝑥 − 25 + 𝑦] � 0 1 1 0 � = 0, then 𝑥 + 𝑦 is: (a) 0 (b) -2 (c) -1 (d) -3 [Delhi Gov. Term-1 SQP 2021] 16. For the matrix 𝑋 = � 0 1 1 1 0 1 1 1 0 � , (𝑋2 − 𝑋) is: (a) 21 (b) 31 (c) 1 (d) 51 [CBSE Term-1 2021] 17. If 𝐴 is a diagonal matrix of order 3 × 3 such that 𝐴2 = 𝐴, then number of possible matrices 𝐴 are: (a) 4 (b) 8 (c) 16 (d) 32 [Delhi Gov. Term-1 2021] Very Short & Short Qs (1 - 3 marks) 18. If 3𝐴 − 𝐵 = � 5 0 1 1 � and 𝐵 = � 4 3 2 5 � and find the value of matrix 𝐴. [CBSE 2019] 19. If 𝐴 = � 2 0 1 2 1 3 1 −1 0 �, then find the value of (𝐴2 − 5𝐴). [CBSE 2019] CLICK HERE FOR SOLUTIONS
20. If 𝐴 = � 4 2 −1 1 �, show that (𝐴 − 2𝐼)(𝐴 − 3𝐼) = 0. [CBSE 2019] 21. Let 𝐴 = � 2 −1 3 4 � , 𝐵 = � 5 2 7 4 � , 𝐶 = � 2 5 3 8 �. Find a matrix 𝐷 such that 𝐶𝐷 − 𝐴𝐵 = 0. [CBSE 2017] Long Qs (4 - 5 marks) 22. If 𝐴 = � 1 0 2 0 2 1 2 0 3 � and 𝐴3 − 6𝐴2 + 7𝐴 + 𝑘|3 = 0, find the value of 𝑘. [CBSE 2016] 23. If A = � 1 −1 2 −1 � , B = � 𝑎 1 𝑏 −1 � and (𝐴 + 𝐵)2 = 𝐴2 + 𝐵2 , then find the values of 𝑎 and 𝑏. [CBSE 2015] 24. If [2𝑥 3] � 1 2 −3 0 � � 𝑥 3 � = 0, find 𝑥. [CBSE 2015] 25. A trust fund has ₹35,000 is to be invested in two different types of bonds. The first bond pays 8% interest per annum which will be given to orphanage and second bond pays 10% interest per annum which will be given to an N.G.O. (Cancer Aid Society). Using matrix multiplication, determine how to divide ₹35,000 among two types of bonds if the trust fund obtains an annual total interest of ₹3,200. What are the values reflected in this question? [CBSE 2015] 26. In a parliament election, a political party hired a public relations firm to promote its candidates in three ways - telephone, house calls and letters. The cost per contact (in paisa) is given in matrix 𝐴 as 𝐴 = � 140 200 150 � Telephone House calls Letters The number of contacts of each type made in two cities 𝑋 and 𝑌 is given in the matrix 𝐵 as Telephone Housecalls Letters 𝐵 = � 1000 500 5000 3000 1000 10000 � City 𝑋 City 𝑌 Find the total amount spent by the party in the two cities. What should one consider before casting his/her vote - party's promotional activity or their social activities? [CBSE 2015] CLICK HERE FOR SOLUTIONS
27. To promote making of toilets for women, an organisation tried to generate awareness through (i) house calls, (ii) letters, and (iii) announcements. The cost for each mode per attempt is given below: (i) ₹ 50 (ii) ₹ 20 (iii) ₹ 40 The number of attempts made in three villages 𝑋, 𝑌 and 𝑍 are given below: (i) (ii) (iii) 𝑋 400 300 100 𝑌 300 250 75 𝑍 500 400 150 Find the total cost incurred by the organisation for three villages separately, using matrices. [CBSE 2015] 28. Solve the matrix equation for 𝑥. [𝑥 1] � 1 0 −2 0 � = 0 [CBSE 2014] 3. TRANSPOSE OF A MATRIX, SYMMETRIC AND SKEW-SYMMETRIC MATRICES Objective Qs (1 mark) 29. If � 2 0 5 4 � = 𝑃 + 𝑄, where 𝑃 is a symmetric and 𝑄 is a skew symmetric matrix, then 𝑄 is equal to: (a) � 2 5 2 5 2 4 � (b) � 0 − 5 2 5 2 0 � (c) � 0 5 2 − 5 2 0 � (d) � 2 − 5 2 5 2 4 � [CBSE 2023] CLICK HERE FOR SOLUTIONS
30. If 𝐴 = �𝑎𝑖𝑗� is a skew-symmetric matrix of order 𝑛, then (a) 𝑎𝑖𝑗 = 1 𝑎𝑗𝑖 ∀𝑖, 𝑗 (b) 𝑎𝑖𝑗 ≠ 0∀𝑖, 𝑗 (c) 𝑎𝑖𝑗 = 0, where 𝑖 = 𝑗 (d) 𝑎𝑖𝑗 ≠ 0 where 𝑖 = 𝑗 [CBSE SQP 2022] 31. For two matrices 𝑃 = � 3 4 −1 2 0 1 � and 𝑄⊤ = � −1 2 1 1 2 3 � 𝑃 − 𝑄 is: (a) � 2 3 −3 0 0 −3 � (b) � 4 3 −3 0 −1 −2 � (c) � 4 3 0 −3 −1 −2 � (d) � 4 3 0 −3 0 −3 � [CBSE Term-1 2021] 32. If 𝐴 = � cos 𝛼 −sin 𝛼 sin 𝛼 cos 𝛼 � and 𝐴 + 𝐴′ = 1, then the value of 𝛼 is: (a) 𝜋 6 (b) 𝜋 3 (c) 𝜋 (d) 3𝜋 2 [CBSE Term-1 2021] 33. If a matrix 𝐴 is both symmetric and skew symmetric, then 𝐴 is necessarily a/an: (a) diagonal matrix (b) zero square matrix (c) square matrix (d) identity matrix [CBSE Term-1 2021] Very Short & Short Qs (1 - 3 marks) 34. If 𝑃 is a 3 × 3 matrix such that 𝑃′ = 2𝑃 + 𝐼, where 𝑃′ is the transpose of 𝑃, then: (a) P = I (b) 𝑃 = −1 CLICK HERE FOR SOLUTIONS
(c) 𝑃 = 21 (d) 𝑃 = −21 [CBSE Term-1 2021] 35. If is a matrix of order 3 × 2, then find the order of the matrix 𝐴′ . [CBSE 2020] 36. If 𝐴 and 𝐵 are symmetric matrices, such that 𝐴𝐵 and 𝐵𝐴 are both defined, then prove that 𝐴𝐵 − 𝐵𝐴 is a skew symmetric matrix. [CBSE 2019] 37. If 𝐴 = � 1 2 2 2 1 𝑥 −2 2 −1 � is a matrix satisfying 𝐴𝐴′ = 91, find 𝑥. [CBSE 2018] 38. If the matrix 𝐴 = � 0 𝑎 −3 2 0 −1 𝑏 1 0 � is skew symmetric, find the values of 𝑎 and 𝑏. [CBSE 2018] 39. Show that all the diagonal elements of a skew-symmetric matrix are zero. [CBSE 2017] 40. If 𝐴 = � 3 5 7 9 � is written as 𝐴 = 𝑃 + 𝑄, where 𝑃 is a symmetric matrix and 𝑄 is a skewsymmetric matrix, then write matrix 𝑃. [CBSE 2016] 41. Matrix 𝐴 = � 0 2𝑏 −2 3 1 3 3𝑎 3 −1 � is given to be symmetric, then find the values of 𝑎 and 𝑏. [CBSE 2016] 42. If 𝐴 = � cos 𝛼 sin 𝛼 −sin 𝛼 cos 𝛼 �, find 𝛼 satisfying 0 < 𝛼 < 𝜋 2 when 𝐴 + 𝐴⊤ = √2𝐼2; where 𝐴⊤ is transpose of 𝐴. [CBSE 2016] 43. Express the matrix 𝐴 = � 2 4 −6 7 3 5 1 −2 4 � as the sum of a symmetric and skew-symmetric matrix. [CBSE 2015] 4. INVERTIBLE MATRICES Objective Qs (1 mark) CLICK HERE FOR SOLUTIONS
44. If 𝐴 = � 1 −1 0 2 3 4 0 1 2 � and 𝐵 = � 2 2 −4 −4 2 −4 2 −1 5 �, then: (a) A−1 = B (b) A−1 = 6 B (c) B−1 = B (d) 𝐵−1 = 1 6 𝐴 45 If 𝐴 = � 3 1 −1 2 �, show that 𝐴2 − 5𝐴 + 7𝐼 = 0. Hence find 𝐴−1 . [CBSE 2020] CLICK HERE FOR SOLUTIONS
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
Determinants 1. DETERMINANTS OF A SQUARE MATRIX AND AREA OF A TRIANGLE Objective Qs (1 mark) 1. If |𝐴| = |𝑘𝐴|, where 𝐴 is a square matrix of order 2, then sum of all possible values of 𝑘 is: (a) 1 (b) -1 (c) 2 (d) 0 [CBSE 2023] 2. If � 1 2 1 2 3 1 3 𝑎 1 � is non-singular matrix and 𝑎 ∈ 𝐴, then the set 𝐴 is: (a) R (b) {0} (c) {4} (d) R − {4} [CBSE 2023] 3. If (𝑎, 𝑏), (𝑐, 𝑑) and (𝑒, 𝑓) are the vertices of △ 𝐴𝐵𝐶 and △ denotes the area of △ 𝐴𝐵𝐶, then � 𝑎 𝑐 𝑒 𝑏 𝑑 𝑓 1 1 1 � 2 is equal to: (a) 2Δ2 (b) 4Δ2 (c) 2Δ (d) 4Δ [CBSE 2023] 4. If 𝐴 and 𝐵 are invertible square matrices of the same order, then which of the following is not correct? (a) |𝐴𝐵−1| = |𝐴| |𝐵| (b) |(𝐴𝐵)−1| = 1 |𝐴||𝐵| (c) (𝐴𝐵)−1 = 𝐵−1 𝐴−1 (d) (𝐴 + 𝐵)−1 = 𝐵−1 + 𝐴−1 [CBSE SQP 2023] CH-4 CLICK HERE FOR SOLUTIONS
5. If the area of the triangle with vertices (−3,0), (3,0) and (0, 𝑘) is 9 sq. units, then the value/s of 𝑘 will be: (a) 9 (b) pm 3 (c) -9 (d) 6 [CBSE SQP 2023] 6. The value of |𝐴|, if 𝐴 = � 0 2𝑥 − 1 √𝑥 1 − 2𝑥 0 2√𝑥 −√𝑥 −2√𝑥 0 �, where 𝑥 ∈ R+ , is: (a) (2𝑥 + 1)2 (b) 0 (c) (2𝑥 + 1)3 (d) (2𝑥 − 1)2 [CBSE SQP 2023] 7. Given that 𝐴 is a square matrix of order 3 and |𝐴| = −2, then |adj (2𝐴)| is equal to: (a) −26 (b) +4 (c) −28 (d) 28 [CBSE SQP 2023] 8. If 𝐴 is a square matrix of order 3, |𝐴′| = −3, then |𝐴𝐴′| = (a) 9 (b) -9 (c) 3 (d) -3 [CBSE SQP 2022] 9. The area of a triangle with vertices 𝐴, 𝐵, 𝐶 is given by: (a) |𝐴𝐵 �����⃗ × 𝐴𝐶 �����⃗| (b) 1 2 |𝐴𝐵 �����⃗ × 𝐴𝐶 �����⃗| (c) 1 4 |𝐴𝐶 �����⃗ × 𝐴𝐵 �����⃗| (d) 1 8 |𝐴𝐶 �����⃗ × 𝐴𝐵 �����⃗| [CBSE SQP 2022] 10. Value of 𝑘, for which 𝐴 = � 𝑘 8 4 2𝑘 � is a singular matrix, is: (a) 4 (b) -4 (c) ± 4 (d) 0 [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
11. Given that 𝐴 is a non-singular matrix of order 3, such that 𝐴2 = 2𝐴, then value of |2𝐴| is: (a) 4 (b) 8 (c) 64 (d) 16 [CBSE Term-1 SQP 2021] 12. Let, 𝐴 = � 1 sin 𝛼 1 −sin 𝛼 1 sin 𝛼 −1 −sin 𝛼 1 � , where 0 ≤ 𝛼 ≤ 2𝜋, then: (a) |A| = 0 (b) |A| ∈ (2, ∞) (c) |A| ∈ (2,4) (d) |𝐴| ∈ [2,4] [CBSE Term-1 SQP 2021] 13. Three points 𝑃(2𝑥, 𝑥 + 3), 𝑄(0, 𝑥) and R(𝑥 + 3, 𝑥 + 6) are collinear, then 𝑥 is: (a) 0 (b) 2 (c) 3 (d) 1 [CBSE Term-1 2021] 14. If, 𝐴 is a skew-symmetric matrix of order 3 , then the value of |A| is: (a) 3 (b) 0 (c) 9 (d) 27 [CBSE 2020] 15. If 𝐴 is a 3 × 3 matrix such that |𝐴| = 8, then |3 A| equals: (a) 8 (b) 24 (c) 72 (d) 216 [CBSE 2020] Very Short & Short Qs (1 - 3 marks) 16. If 𝐴 is a square matrix satisfying 𝐴′ 𝐴 = 𝐼, write the value of |𝐴|. [CBSE 2019] 17. If 𝐴 = � 𝑝 2 2 𝑝 � and |𝐴3| = 125, then find the value of 𝑝. [CBSE 2019] CLICK HERE FOR SOLUTIONS
18. If � 𝑥 sin 𝜃 cos 𝜃 −sin 𝜃 −𝑥 1 cos 𝜃 1 𝑥 � = 8, write the value of 𝑥. [CBSE 2016] 19. Find the maximum value of � 1 1 1 1 1 + sin 𝜃 1 1 1 1 + cos 𝜃 � [CBSE 2016] 20. Write the value of the determinant � 𝑝 𝑝 + 1 𝑝 − 1 𝑝 � [CBSE 2014] 21. Write the value of � 2 7 65 3 8 75 5 9 86 �. [CBSE 2014] 2. MINORS AND COFACTORS Objective Qs (1 mark) 22. If 𝐴 is any square matrix of order 3 × 3 such that |adj 𝐴| = 256, then the sum of all possible values of |A| is: (a) 256 (b) 16 (c) -16 (d) 0 [Delhi Gov. Term-1 SQP 2021] 23. Given that 𝐴 = �𝑎𝑖𝑗� is a square matrix of order 3 × 3 and |𝐴| = −7, then the value of ∑𝑖=1 3 𝑎𝑖2𝐴𝑖2, where 𝐴𝑖𝑗 denotes the cofactor of element 𝑎𝑖𝑗 is: (a) 7 (b) -7 (c) 0 (d) 49 [CBSE Term-1 SQP 2021] 24. Given that 𝐴 is a square matrix of order 3 and |𝐴| = −4, then |adj 𝐴| is: (a) -4 (b) 4 (c) -16 (d) 16 [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
25. For matrix 𝐴 = � 2 5 −11 7 � , (adj 𝐴)⊤ is: (a) � −2 −5 11 −7 � (b) � 7 5 11 2 � (c) � 7 11 −5 2 � (d) � 7 −5 11 2 � [CBSE Term-1 SQP 2021] 26. For 𝐴 = � 3 1 −1 2 �, then 14𝐴−1 is given by: (a) 14 � 2 −1 1 3 � (b) � 4 −2 2 6 � (c) 2 � 2 −1 1 −3 � (d) 2 � −3 −1 1 −2 � [CBSE Term-1 SQP 2021] 27. If 𝐴 = � 1 −2 4 2 −1 3 4 2 0 � is the adjoint of a square matrix 𝐵, then 𝐵−1 is equal to: (a) ±A (b) ±√2 A (c) ± 1 √2 𝐵 (d) ± 1 √2 A [CBSE Term-1 2021] 28. If � 1 −tan 𝜃 tan 𝜃 1 � � 1 tan 𝜃 −tan 𝜃 1 � −1 = � 𝑎 −𝑏 𝑏 𝑎 �, then (a) 𝑎 = 1 = 𝑏 (b) 𝑎 = cos 2𝜃, 𝑏 = sin 2𝜃 (c) 𝑎 = sin 2𝜃, 𝑏 = cos 𝜃 (d) 𝑎 = cos 𝜃, 𝑏 = sin 𝜃 [CBSE Term-1 2021] Very Short & Short Qs (1 - 3 marks) 29. If 𝐴 = � 2 3 5 −2 � be such that 𝐴−1 = 𝑘𝐴, then find the value of 𝑘. [CBSE 2018] CLICK HERE FOR SOLUTIONS
30. Given 𝐴 = � 2 −3 −4 7 �, compute 𝐴−1 and show that 2𝐴−1 = 91 − 𝐴. [CBSE 2018] 31. Find the inverse of the matrix 𝐴 = � 1 3 3 1 4 3 1 3 4 �. [CBSE 2018] 32. If for any 2 × 2 square matrix 𝐴, 𝐴(adj 𝐴) = � 8 0 0 8 �, then write the value of |𝐴|. 33. If 𝐴 = � 5 6 −3 −4 3 2 −4 −7 3 �, then write the cofactor of the element 𝑎21 of its 2nd row. [CBSE 2015] 34. In the interval 𝜋 2 < 𝑥 < 𝜋, find the value of 𝑥 for which the matrix, � 2sin 𝑥 3 1 2sin 𝑥 � is singular. [CBSE 2015] 35. If 𝐴 = � 1 −2 3 0 −1 4 −2 2 1 �, then find (𝐴⊤)−1 . [CBSE 2015] 36. Find the adjoint of the matrix 𝐴 = � −1 −2 −2 2 1 −2 2 −2 1 � and hence show that 𝐴(adj 𝐴) = |𝐴|𝐼3. [CBSE 2015] 37. If 𝐴 = � 2 3 1 −4 � and 𝐵 = � 1 −2 −1 3 �, then verify that (𝐴𝐵)−1 = 𝐵−1 𝐴−1 [CBSE 2015] Long Qs (4 - 5 marks) 38. Show that for the matrix 𝐴 = � 1 1 1 1 2 −3 2 −1 3 �, 𝐴3 − 6𝐴2 + 5𝐴 + 11𝐼 = 0. Hence, find 𝐴−1 . [CBSE 2019] 39. If 𝐴 = � cos 𝛼 −sin 𝛼 0 sin 𝛼 cos 𝛼 0 0 0 1 �, find adj 𝐴 and verify that 𝐴(adj 𝐴) = (adj 𝐴)𝐴 = |𝐴|𝐼3. [CBSE 2016] 3. APPLICATIONS OF DETERMINANTS AND MATRICES CLICK HERE FOR SOLUTIONS
Objective Qs (1 mark) 40. The system of linear equations 5𝑥 + 𝑘𝑦 = 5, 3𝑥 + 3𝑦 = 5; will be consistent if: (a) 𝑘 ≠ −3 (b) 𝑘 = −5 (c) 𝑘 = 5 (d) 𝑘 ≠ 5 [CBSE Term-1 2021] 41. If 𝐴 = � 1 −1 0 2 3 4 0 1 2 � and 𝐵 = � 2 2 −4 −4 2 −4 2 −1 5 �, then: (a) A−1 = B (b) A−1 = 6 B (c) B−1 = B (d) B−1 = 1 6 A [CBSE Term-1 SQP 2021] Very Short & Short Qs (1-3 marks) 42. For what values of 𝑘, the system of linear equations 𝑥 + 𝑦 + 𝑧 = 2 2𝑥 + 𝑦 − 𝑧 = 3 3𝑥 + 2𝑦 + 𝑘𝑧 = 4 has a unique solution? [CBSE 2016] 43. The monthly incomes of Aryan and Babban are in the ratio 3: 4 and their monthly expenditures are in the ratio 5: 7. If each saves ₹ 15,000 per month, then find their monthly incomes using matrices. [CBSE 2016] Longs Qs (4 - 5 marks) 44. Using the matrix method, solve the following system of linear equations: 2 𝑥 + 3 𝑦 + 10 𝑧 = 4, 4 𝑥 − 6 𝑦 + 5 𝑧 = 1, 6 𝑥 + 9 𝑦 + 20 𝑧 = 2. [CBSE SQP 2023] 45. If 𝐴 = � −3 −2 −4 2 1 2 2 1 3 � and 𝐵 = � 1 2 0 −2 −1 −2 0 −1 1 �, then find 𝐴𝐵 and use it to solve the following system of equations: CLICK HERE FOR SOLUTIONS
𝑥 − 2𝑦 = 3 2𝑥 − 𝑦 − 𝑧 = 2 −2𝑦 + 𝑧 = 3 [CBSE 2023] 46. If 𝑓(𝛼) = � cos 𝛼 −sin 𝛼 0 sin 𝛼 cos 𝛼 0 0 0 1 �, then prove that 𝑓(𝛼) ⋅ 𝑓(−𝛽) = 𝑓(𝛼 − 𝛽). [CBSE 2023] 47. If 𝐴 = � 1 2 0 −2 −1 −2 0 −1 1 �, find 𝐴−1 . Using 𝐴−1 , solve the system of linear equations 𝑥 − 2𝑦 = 10,2𝑥 − 𝑦 − 𝑧 = 8, −2𝑦 + 𝑧 = 7. [CBSE 2021] 48. Evaluate the product 𝐴𝐵, where 𝐴 = � 1 −1 0 2 3 4 0 1 2 � and 𝐵 = � 2 2 −4 −4 2 −4 2 −1 5 � Hence the solve system of linear equations 𝑥 − 𝑦 = 3 2𝑥 + 3𝑦 + 4𝑧 = 17 𝑦 + 2𝑧 = 7 [CBSE 2021] 49. If 𝐴 = � 1 2 −3 3 2 −2 2 −1 1 �, then find 𝐴−1 and use it to solve the following systems of the equations: 𝑥 + 2𝑦 − 3𝑧 = 6 3𝑥 + 2𝑦 − 2𝑧 = 3 3𝑥 + 2𝑦 − 𝑧 = 5 [CBSE 2020] 50. If 𝐴 = � 2 3 4 1 −1 0 0 1 2 �, find 𝐴−1 . Hence, solve the system of equations 2𝑥 + 3𝑦 + 4𝑧 = 17, 𝑥 − 𝑦 = 3, 𝑦 + 2𝑧 = 7. [CBSE 2020] CLICK HERE FOR SOLUTIONS
51. Determine the product of � −4 4 4 −7 1 3 5 −3 −1 �. and � 1 −1 1 1 −2 −2 2 1 3 �, and then use this to solve the system of equations 𝑥 − 𝑦 + 𝑧 = 4 𝑥 − 2𝑦 − 2𝑧 = 9 2𝑥 − 𝑦 + 𝑧 = 2 [CBSE 2017] 52. A shopkeeper has 3 varieties of pens ' 𝐴 ', ' 𝐵 ' and ' 𝐶 '. Meena purchased 1 pen of each variety for a total of ₹ 21. Jeevan purchased 4 pens of 'A' variety, 3 pens of 'B' variety and 2 pens of ' 𝐶 ' variety for ₹60, while Shikha purchased 6 pens of 'A' variety 2 pens of ' 𝐵 ' variety and 3 pens of ' 𝐶 ' variety for ₹70. Using matrix method, find the cost of each variety of pen. [CBSE 2016] 53. A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received ₹2,800 as interest, However, if trust had interchanged money in bonds, they would have got ₹ 100 less as interest, Using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Helpage India as donation. Which value is reflected in this question? [CBSE 2016] 54. A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is ₹ 9,000 whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is ₹ 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society? [CBSE 2016] 55. Two schools 𝐴 and 𝐵 decided to award prizes to their students for three values, team spirit truthfulness and tolerance at the rate of ₹𝑥, ₹ 𝑦 and ₹𝑧 per student respectively. School A, decided to award a total of ₹ 1,100 for the three values to 3,1 and 2 students respectively while school 𝐵 decided to award ₹1,400 for the three values to 1,2 and 3 students respectively. If one prize for all the three values together amount to ₹ 600 then (A) Represent the above situation by a matrix equation after forming linear equations. (B) Is it possible to sove the system of equations so obtained using matrices? (C) Which value you prefer to be rewarded most and why? [CBSE 2015] CLICK HERE FOR SOLUTIONS
56. A total amount of ₹ 7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and 8 1 2 % respectively. The total annual interest from these three accounts is ₹ 550. Equal amounts have been deposited in the 5% and 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices. [CBSE 2014] CLICK HERE FOR SOLUTIONS
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
Continuity And Differentiability 1. LIMITS AND CONTINUITY OF A FUNCTION Objective Qs (1 mark) 1. If 𝑓(𝑥) = � 𝑘𝑥 |𝑥| , if 𝑥 < 0 3, if 𝑥 ≥ 0 is continuous at 𝑥 = 0, then the value of 𝑘 is: (a) -3 (b) 0 (c) 3 (d) any real number [CBSE SQP 2023] 2. The function 𝑓(𝑥) = [𝑥], where [𝑥] denotes the greatest integer function, is continuous at: (a) 4 (b) 1.5 (c) 1 (d) -2 [CBSE Term-1 2021] 3. The value of 𝑘(𝑘 < 0) for which the function 𝑓 defined as: 𝑓(𝑥) = � 1 − cos 𝑘𝑥 𝑥sin 𝑥 , 𝑥 ≠ 0 1 2 , 𝑥 = 0 is continuous at 𝑥 = 0 is: (a) ± 1 (b) -1 (c) ± 1 2 (d) 1 2 [CBSE Term-1 SQP 2021] 4. The point(s), at which the function 𝑓 given by 𝑓(𝑥) = � 𝑥 |𝑥| , 𝑥 < 0 −1, 𝑥 ≥ 0 is continuous, is/are: (a) 𝑥 ∈ 𝑅 (b) 𝑥 = 0 (c) 𝑥 ∈ R − {0} (d) 𝑥 = −1 and 1 [CBSE Term-1 SQP 2021] CH-5 CLICK HERE FOR SOLUTIONS
5. 𝑓(𝑥) = � 3𝑥 − 8 if 𝑥 ≤ 5 2𝑘 if 𝑥 > 5 is continuous, find 𝑘. (a) 2 7 (b) 3 7 (c) 4 7 (d) 7 2 [CBSE Term-1 2021] Very Short & Short Qs (1 - 3 marks) 6. Find the value of 𝜆 so that the function 𝑓 defined by 𝑓(𝑥) = � 𝜆𝑥, if 𝑥 ≤ 𝜋 cos 𝑥, if 𝑥 > 𝜋 is continuous at 𝑥 = 𝜋. [CBSE 2020] 7. Determine the value of ' 𝑘 ' for which the following function is continuous at 𝑥 = 3. 𝑓(𝑥) = � (𝑥 + 3)2 − 36 𝑥 − 3 , 𝑥 ≠ 3 𝑘 , 𝑥 = 3 [CBSE 2017] 8. Find the values of 𝑝 and 𝑞, for which 𝑓(𝑥) = ⎩ ⎪ ⎨ ⎪ ⎧ 1 − sin3 𝑥 3cos2 𝑥 , if 𝑥 < 𝜋 2 𝑝 , if 𝑥 = 𝜋 2 𝑞(1 − sin 𝑥) (𝜋 − 2𝑥)2 , if 𝑥 > 𝜋 2 is continuous at 𝑥 = 𝜋 2 . [CBSE 2016] 9. Find the value of the constant 𝑘 so that the function 𝑓, defined below, is continuous at 𝑥 = 0, where 𝑓(𝑥) = � 1 − cos 4𝑥 8𝑥2 , if 𝑥 ≠ 0 𝑘 , if 𝑥 = 0 [CBSE SQP 2014] CLICK HERE FOR SOLUTIONS
Long Qs (4 - 5 marks) 10. If 𝑓(𝑥) = ⎩ ⎪ ⎨ ⎪ ⎧ sin (𝑎 + 1)𝑥 + 2sin 𝑥 𝑥 𝑥 < 0 2 √1 + 𝑏𝑥 − 1 𝑥 𝑥 > 0 is continuous at 𝑥 = 0 then find the values of 𝑎 and 𝑏. [CBSE 2016] 2. DIFFERENTIABILITY Objective Qs (1 mark) 11. The set of all points where the function 𝑓(𝑥) = 𝑥 + |𝑥| is differentiable, is: (a) (0, ∞) (b) (−∞, 0) (c) (−∞, 0) ∪ (0, ∞) (d) (−∞, ∞) [CBSE SQP 2023] 12. The function 𝑓(𝑥) = 𝑥|𝑥| is: (a) continuous-and differentiable at 𝑥 = 0. (b) continuous but not differentiable at 𝑥 = 0. (c) differentiable but not continuous at 𝑥 = 0. (d) neither differentiable nor continuous at 𝑥 = 0. [CBSE 2023] 13. The function given below at 𝑥 = 4 is: 𝑓(𝑥) = � 2𝑥 + 3, 𝑥 ≤ 4 𝑥2 − 5, 𝑥 > 4 (a) continuous but not differentiable (b) differentiable but not continuous (c) continuous as well as differentiable (d) neither continuous nor differentiable CLICK HERE FOR SOLUTIONS
[Delhi. Gov. Term-1 SQP 2021] 14. The function 𝑓(𝑥) = � 𝑥2 for 𝑥 ≤ 1 2 − 𝑥 for 𝑥 ≥ 1 is: (a) Not differentiable at 𝑥 = 1 (b) Differentiable at 𝑥 = 1 (c) Not continuous at 𝑥 = 1 (d) Neither continuous nor differentiable at 𝑥 = 1 [CBSE Term-1 2021] Very Short & Short Qs (1-3 marks) 15. Prove that the greatest integer function defined by 𝑓(𝑥) = [𝑥],0 < 𝑥 < 3 is not differentiable at 𝑥 = 1. [CBSE 2020] 16. Let, 𝑓(𝑥) = 𝑥|𝑥|, for all 𝑥 ∈ 𝑅. Check its differentiability at 𝑥 = 0. [CBSE 2020] 17. Prove that the greatest integer function defined by 𝑓(𝑥) = [𝑥],0 < 𝑥 < 2 is not differentiable at 𝑥 = 1 [CBSE 2020] 18. Find the values of 𝑎 and 𝑏, if the function 𝑓 defined by 𝑓(𝑥) = � 𝑥2 + 3𝑥 + 𝑎 , 𝑥 ≤ 1 𝑏𝑥 + 2 , 𝑥 > 1 is differentiable at 𝑥 = 1. [CBSE 2016] 19. For what value of 𝜆, the function defined by 𝑓(𝑥) = �𝜆(𝑥2 + 2) , if 𝑥 ≤ 0 4𝑥 + 6,𝑥 > 0 is continuous at 𝑥 = 0 ? Hence, check the differentiability of 𝑓(𝑥) at 𝑥 = 0. [CBSE 2015] 20. Find whether the following function is differentiable at 𝑥 = 1 and 𝑥 = 2 or not. 𝑓(𝑥) = � 𝑥, 𝑥 < 1 2 − 𝑥, 1 < 𝑥 ≤ 2 −2 + 3𝑥 − 𝑥2 , 𝑥 > 2 [CBSE 2015] CLICK HERE FOR SOLUTIONS
3. DERIVATIVES Objective Qs (1 mark) 21. If tan � 𝑥+𝑦 𝑥−𝑦 � = 𝑘, then 𝑑𝑦 𝑑𝑥 is equal to: (a) −𝑦 𝑥 (b) 𝑦 𝑥 (c) sec2 � 𝑦 𝑥 � (d) −sec2 � 𝑦 𝑥 � [CBSE 2023] 22. If 𝑒𝑥 + 𝑒𝑦 = 𝑒𝑥+𝑦 , then 𝑑𝑦 𝑑𝑥 is: (a) 𝑒𝑦−𝑥 (b) 𝑒𝑥+𝑦 (c) −𝑒𝑦−𝑥 (d) 2𝑒𝑥−𝑦 [Delhi Gov. SQP 2022, CBSE Term-1 SQP 2021] 23. The derivative of sin−1 �2𝑥√1 − 𝑥2� w.r.t. sin−1 𝑥, 1 √2 < 𝑥 < 1, is: (a) 2 (b) 𝜋 2 − 2 (c) 𝜋 2 (d) -2 [CBSE Term-1 SQP 2021] 24. If (𝑥2 + 𝑦2)2 = 𝑥𝑦, then 𝑑𝑦 𝑑𝑥 is: (a) 𝑥+4𝑥�𝑥2+𝑦2� 4𝑦(𝑥2+𝑦2)−𝑥 (b) 𝑦−4𝑥�𝑥2+𝑦2� 𝑥+4(𝑥2+𝑦2) (c) 𝑦−4𝑥�𝑥2+𝑦2� 4𝑦(𝑥2+𝑦2)−𝑥 (d) 4𝑦�𝑥2+𝑦2�−𝑥 𝑦−4𝑥(𝑥2+𝑦2) [CBSE Term- 1 2021] 25. If 𝑦 = log (cos 𝑒𝑥), then 𝑑𝑦 𝑑𝑥 is: (a) cos 𝑒𝑥−1 (c) 𝑒𝑥 sin 𝑒𝑥 (b) e−𝑥 cos 𝑒𝑥 (d) −𝑒𝑥 tan 𝑒𝑥 [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
26. If 𝑦2 (2 − 𝑥) = 𝑥3 , then � 𝑑𝑦 𝑑𝑥 � (1,1) is equal to: (a) 2 (b) -2 (c) 3 (d) − 3 2 [CBSE Term-1 2021] Very Short & Short Qs (1 - 3 marks) 27. If 𝑓(𝑥) = � 𝑎𝑥 + 𝑏; 0 < 𝑥 ≤ 1 2𝑥2 − 𝑥; 1 < 𝑥 < 2 is a differentiable function in (0,2), then find the values of 𝑎 and b. [CBSE 2023] 28. If 𝑦√1 − 𝑥2 + 𝑥�1 − 𝑦2 = 1, then prove that 𝑑𝑦 𝑑𝑥 = −� 1−𝑦2 1−𝑥2. [CBSE SQP 2022] 29. If 𝑦 = tan−1 𝑥 + cot−1 𝑥, 𝑥 ∈ 𝑅, then find 𝑑𝑦 𝑑𝑥 . [CBSE 2020] 30. If cos (𝑥𝑦) = 𝑘, where 𝑘 is a constant and 𝑥𝑦 ≠ 𝑛𝜋, 𝑛 ∈ 𝑍, then 𝑑𝑦 𝑑𝑥 is equal to ....... [CBSE 2020] 31. If 𝑥 = 𝑎sec 𝜃, 𝑦 = 𝑏 tan 𝜃, then find 𝑑𝑦 𝑑𝑥 at 𝜃 = 𝜋 3 . [CBSE 2020] 32. If 𝑦 = 𝑒𝑥2cos 𝑥 + (cos 𝑥)𝑥 , then find 𝑑𝑦 𝑑𝑥 . [CBSE 2020] 33. Differentiate tan−1 � cos 𝑥−sin 𝑥 cos 𝑥+sin 𝑥 � with respect to 𝑥. [CBSE 2018] 34. If 𝑥 = 𝑎(2𝜃 − sin 2𝜃) and 𝑦 = 𝑎(1 − cos 2𝜃), find 𝑑𝑦 𝑑𝑥 when 𝜃 = 𝜋 3 . [CBSE 2018] 35. Find 𝑑𝑦 𝑑𝑥 at 𝑥 = 1, 𝑦 = 𝜋 4 , if sin2 𝑦 + cos 𝑥𝑦 = 𝑘. [CBSE 2017] 36. If 𝑦 = sin−1 �6𝑥√1 − 9𝑥2�, − 1 3√2 < 𝑥 < 1 3√2 , then find 𝑑𝑦 𝑑𝑥 . [CBSE 2017] CLICK HERE FOR SOLUTIONS
37. If 𝑦 = tan−1 � 𝑎 𝑥 � + log � (𝑥−𝑎) (𝑥+𝑎) , prove that 𝑑𝑦 𝑑𝑥 = 2𝑎3 (𝑥4−𝑎4) [CBSE 2014] Long Qs 38. If log (𝑥2 + 𝑦2) = 2tan−1 � 𝑦 𝑥 � Show that 𝑑𝑦 𝑑𝑥 = 𝑥 + 𝑦 𝑥 − 𝑦 . [CBSE 2019] 39. If 𝑥 = 𝑎cos 𝜃 + 𝑏sin 𝜃 and 𝑦 = 𝑎sin 𝜃 −𝑏cos 𝜃, then show that 𝑦2 𝑑2 𝑦 𝑑𝑥2 − 𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 = 0. [CBSE 2015] 40. If 𝑦 = 𝑥cos−1 𝑥 �(1−𝑥2) − log �(1 − 𝑥2), then prove that 𝑑𝑦 𝑑𝑥 = cos−1 𝑥 (1−𝑥2)3/2. [CBSE 2015] 41. If 𝑥 = 𝑎sin 2𝑡(1 + cos 2𝑡) and 𝑦 = 𝑏cos 2𝑡(1 −cos 2𝑡), show that at 𝑡 = 𝜋 4 , � 𝑑𝑦 𝑑𝑥 � = 𝑏 𝑎 . [CBSE 2015] 42. If 𝑥 = 𝑎𝑒𝑡 (sin 𝑡 + cos 𝑡) and 𝑦 = 𝑎𝑒𝑡 (sin 𝑡 − cos 𝑡), then prove that : 𝑑𝑦 𝑑𝑥 = (𝑥+𝑦) (𝑥−𝑦) . [CBSE 2015] 43. If 𝑒𝑥 + 𝑒𝑦 = 𝑒𝑥+𝑦 , then show that 𝑑𝑦 𝑑𝑥 = −𝑒𝑦−𝑥 . [CBSE 2014] 44. Differentiate tan−1 � 𝑥 √1−𝑥2 � with respect to sin−1 �2𝑥√1 − 𝑥2�. [CBSE 2014] 45. Find the value of 𝑑𝑦 𝑑𝑥 at 𝜃 = 𝑥 4 , if 𝑥 = 𝑎𝑒𝜃 (sin 𝜃 − cos 𝜃) and 𝑦 = 𝑎𝑒𝜃 (sin 𝜃 + cos 𝜃). [CBSE 2014] 4. SECOND ORDER DERIVATIVE 46. If 𝑦 = 5cos 𝑥 − 3sin 𝑥, then 𝑑2𝑦 𝑑𝑥2 is equal to: (a) −𝑦 (b) 𝑦 (c) 25𝑦 (d) 9𝑦 [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
47. If 𝑥 = 𝑎sec 𝜃, 𝑦 = 𝑏tan 𝜃, then 𝑑2𝑦 𝑑𝑥2 at 𝜃 = 𝜋 6 is: (a) −3√3𝑏 𝑎2 (b) −2√3𝑏 𝑎 (c) −3√3𝑏 𝑎 (d) −𝑏 3√3𝑎2 [CBSE Term-1 SQP 2021] 48. If 𝑦 = log𝑒 � 𝑥2 𝑒2�, then 𝑑2𝑦 𝑑𝑥2 is equal to: (a) −1 𝑥 (b) − 1 𝑥2 (c) 2 𝑥2 (d) − 2 𝑥2 [CBSE 2020] Very Short & Short Qs (1 - 3 marks) 49. If 𝑦 = √𝑎𝑥 + 𝑏, prove that 𝑦 � 𝑑2𝑦 𝑑𝑥2� + � 𝑑𝑦 𝑑𝑥 � 2 = 0. [CBSE 2023] 50. If 𝑥 = 𝑎cos 𝜃, 𝑦 = 𝑏sin 𝜃, then find 𝑑2𝑦 𝑑𝑥2. [CBSE 2020] 51. If 𝑥 = 𝑎𝑡2 , 𝑦 = 2𝑎𝑡 then find 𝑑2𝑦 𝑑𝑥2. [CBSE 2020] 52. If 𝑒𝑦 (𝑥 + 1) = 1, then show that 𝑑2𝑦 𝑑𝑥2 = � 𝑑𝑦 𝑑𝑥 � 2 . [CBSE 2017] 53. If 𝑥𝑚 ⋅ 𝑦𝑛 = (𝑥 + 𝑦)𝑚+𝑛 , prove that 𝑑2𝑦 𝑑𝑥2 = 0. [CBSE 2017] 54. If 𝑦 = 2cos (log 𝑥) + 3sin (log 𝑥), prove that 𝑥2 𝑑2 𝑦 𝑑𝑥2 + 𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 = 0 [CBSE 2016] 55. If 𝑦 = 𝐴𝑒𝑚𝑥 + 𝐵𝑒𝑛𝑥 , show that 𝑑2𝑦 𝑑𝑥2 − (𝑚 + 𝑛) 𝑑𝑦 𝑑𝑥 + 𝑚𝑛𝑦 = 0. [CBSE 2015, 2014] CLICK HERE FOR SOLUTIONS
Long Qs (4 - 5 marks) 56. If 𝑥 = sin 𝑡 and 𝑦 = sin 𝑝𝑡, prove that (1 − 𝑥2) 𝑑2 𝑦 𝑑𝑥2 − 𝑥 𝑑𝑦 𝑑𝑥 + 𝑝2 𝑦 = 0 [CBSE 2019] 57. If 𝑦 = (sin−1 𝑥)2 , then prove that: (1 − 𝑥2) 𝑑2 𝑦 𝑑𝑥2 − 𝑥 � 𝑑𝑦 𝑑𝑥 � − 2 = 0 [CBSE 2019] 58. If 𝑦 = 𝑥𝑥 , prove that 𝑑2𝑦 𝑑𝑥2 − 1 𝑦 � 𝑑𝑦 𝑑𝑥 � 2 − 𝑦 𝑥 = 0. [CBSE 2016, 2014] 59. If 𝑦 = 2cos (log 𝑥) + 3sin (log 𝑥), prove that 𝑥2 𝑑2𝑦 𝑑𝑥2 + 𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 = 0 [CBSE 2016] 60. If 𝑥 = 𝑎 �cos 𝑡 + log tan 𝑡 2 � and 𝑦 = 𝑎sin 𝑡, then find 𝑑2𝑦 𝑑𝑥2 at 𝑡 = 𝜋 3 . [CBSE 2014] CLICK HERE FOR SOLUTIONS
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
Application of Derivatives 1. RATE OF CHANGE OF QUANTITIES Case Based Qs (4- 5 marks) 1. The relation between the height of the plant (' 𝑦 ' in cm ) with respect to its exposure to the sunlight is governed by the following equation 𝑦 = 4𝑥 − 1 2 𝑥2 , where ' 𝑥 ' is the number of days exposed to the sunlight, for 𝑥 ≤ 3. Based on the above information, answer the following questions: (A) Find the rate of growth of the plant with respect to the number of days exposed to the sunlight. (B) Does the rate of growth of the plant increase or decrease in the first three days? What will be the height of the plant after 2 days? [CBSE SQP 2023] Very Short & Short Qs (1 − 3 marks) 2. If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius. [CBSE 2023] 3. A man 1.6 m tall walks at the rate of 0.3 m/s away from a street light that is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening? [CBSE SQP 2022] 4. The radius of a circle is increasing at the uniform rate of 3 cm/sec. At the instant when the radius of the circle is 2 cm, find its area increases at the rate of cm2 /sec. [CBSE 2020] CH-6 CLICK HERE FOR SOLUTIONS
5. A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is dragged away from the wall along the ground at the rate of 2 cm/sec. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall? [CBSE 2019] 6. The volume of a cube is increasing at the rate of 8 cm3 /s. How fast is the surface area increasing when the length of its edge is 12 cm ? [CBSE 2019] 7. The total cost 𝐶(𝑥) in Rupees, associated with the production of 𝑥 units of an item is given by 𝐶(𝑥) = 0.005𝑥3 − 0.02𝑥2 + 30𝑥 + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. [CBSE 2018] 8. The length 𝑥 of a rectangle is decreasing at the rate of 5 cm/ minute and the width 𝑦 is increasing at the rate of 4 cm/ minute. When 𝑥 = 8 cm and 𝑦 = 6 cm, find the rate of change of: (A) the perimeter. (B) area of rectangle. [CBSE 2017] 9. The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm. [CBSE 2017] Long Qs (4 - 5 marks) 10. The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing, when the side of the triangle is 20 cm. [CBSE 2015] 2. INCREASING AND DECREASING FUNCTIONS Objective Qs (1 mark) 11. The real function 𝑓(𝑥) = 2𝑥3 − 3𝑥2 − 36𝑥 + 7 is: (a) strictly increasing in (−∞, −2) and strictly decreasing in (−2, ∞). (b) strictly decreasing in (−2,3). (c) strictly decreasing in (−∞, 3) and strictly increasing in (3, ∞). (d) strictly decreasing in (−∞, −2) ∪ (3, ∞). CLICK HERE FOR SOLUTIONS
[CBSE Term-1 SQP 2021] 12. The value of 𝑏 for which the function 𝑓(𝑥) = 𝑥 + cos 𝑥 + 𝑏 is strictly decreasing over 𝑅 is : (a) 𝑏 < 1 (b) no value of 𝑏 exists (c) 𝑏 ≤ 1 (d) 𝑏 ≥ 1 [CBSE Term-1 SQP 2021] 13. Find the intervals in which the function 𝑓 given by 𝑓(𝑥) = 𝑥2 − 4𝑥 + 6 is strictly increasing. (a) (−∞, 2) ∪ (2, ∞) (b) (2, ∞) (c) (−∞, 2) (d) (−∞, 2] ∪ (2, ∞) [CBSE Term-1 SQP 2021] 14. The interval in which 𝑦 = 𝑥2 𝑒−𝑥 is increasing, is: (a) (−∞, ∞) (b) (−2,0) (c) (2, ∞) (d) (0,2) [CBSE Term-1 2021] Case Based Qs ( 4 - 5 marks) 15. The use of electric vehicles will curb air pollution in the long run. The use of electric vehicles is increasing every year and estimated electric vehicles in use at any time 𝑡 is given by the function 𝑉. 𝑉(𝑡) = 1 5 𝑡3 − 5 2 𝑡2 + 25𝑡 − 2 where 𝑡 represents the time and 𝑡 = 1,2,3 … corresponds to year 2001, 2002, 2003, ....... respectively. Based on the above information, answer the following questions: (A) Can the above function be used to estimate number of vehicles in the year 2000? Justify. (B) Prove that the function 𝑉(𝑡) is an increasing function. CLICK HERE FOR SOLUTIONS
[CBSE 2023] 16. The temperature of a person during an intestinal illness is given by 𝑓(𝑥) = −0.1𝑥2 + 𝑚𝑥 + 98.6, 0 ≤ 𝑥 ≤ 12, 𝑚 being a constant, where 𝑓(𝑥) is the temperature in ∘ F at 𝑥 days. Based on the above information, answer the following questions: (A) Is the function differentiable in the interval (0,12) ? Justify your answer. (B) If 6 is the critical point of the function, then find the value of the constant 𝑚. (C) Find the intervals in which the function is strictly increasing/strictly decreasing. [CBSE SQP 2022] Very Short & Short Qs (1 − 3 marks) 17. Find the interval/s in which the function 𝑓: 𝑅 → 𝑅 defined by 𝑓(𝑥) = 𝑥𝑒𝑥 , is increasing. [CBSE SQP 2023] 18. Check whether the function 𝑓: 𝑅 → 𝑅 defined by 𝑓(𝑥) = 𝑥3 + 𝑥, has any critical point/s or not? If yes, then find the point/s. [CBSE SQP 2023] 19. Find the interval in which the function 𝑓 given by 𝑓(𝑥) = 7 − 4𝑥 − 𝑥2 is strictly increasing. [CBSE 2020] 20. Find the intervals in which the function 𝑓 given by 𝑓(𝑥) = tan 𝑥 − 4𝑥, 𝑥 ∈ �0, 𝜋 2 � is : (A) Strictly increasing (B) Strictly decreasing [CBSE 2020] 21. Find the interval in which the function 𝑓(𝑥) = 𝑥4 4 − 𝑥3 − 5𝑥2 + 24𝑥 + 12 is: (A) strictly increasing. (B) strictly decreasing. CLICK HERE FOR SOLUTIONS
[CBSE 2018] 22. Show that the function 𝑓(𝑥) = 4𝑥3 − 18𝑥2 + 27𝑥 − 7 is always increasing on 𝑅. [CBSE 2017] 23. Find the interval in which 𝑓(𝑥) = sin 3𝑥 − cos 3𝑥, 0 < 𝑥 < 𝜋, is strictly increasing or strictly decreasing. [CBSE 2016] 24. Find the value (𝑠) of 𝑥 for which 𝑦 = [𝑥(𝑥 − 2)]2 is an increasing function. [CBSE 2014] 25. Prove that the function 𝑓 defined by 𝑓(𝑥) = 𝑥2 − 𝑥 + 1 is neither increasing nor decreasing in (−1,1). Hence, find the intervals in which 𝑓(𝑥) is: (A) strictly increasing. (B) strictly decreasing. [CBSE 2014] 26. Find intervals in which the function given by 𝑓(𝑥) = 3 10 𝑥4 − 4 5 𝑥3 − 3𝑥2 + 36 5 𝑥 + 11 is (a) strictly increasing (b) strictly decreasing. [CBSE 2014] 3. MAXIMA AND MINIMA Objective Qs (1 mark) 27. The maximum value of � 1 𝑥 � 𝑥 is: (a) 𝑒1/𝑒 (b) 𝑒 (c) � 1 𝑒 � 1/𝑒 (d) 𝑒𝑒 [CBSE Term-1 2021] 28. The maximum value of [𝑥(𝑥 − 1) + 1]1/3 , 0 ≤ 𝑥 ≤ 1 is: (a) 0 (b) 1 2 (c) 1 (d) � 1 3 � 1/3 [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
29. The least value of the function 𝑓(𝑥) = 2cos 𝑥 + 𝑥 in the closed interval �0, 𝜋 2 � is: (a) 2 (b) 𝜋 6 + √3 (c) 𝜋 2 (d) The least value does not exist. [CBSE Term-1 SQP 2021] 30. The area of a trapezium is defined by function 𝑓 and is given by 𝑓(𝑥) = (10 + 𝑥)�100 − 𝑥2 Then the area when it is maximised is: (a) 75 cm2 (b) 7√3 cm2 (c) 75√3 cm2 (d) 5 cm2 [CBSE Term-1 SQP 2021] 31. The absolute minimum value of the function 𝑓(𝑥) = 𝑥3 − 12𝑥 on the interval [0,3] is: (a) 0 (b) -9 (c) -16 (d) -19 [CBSE Term-1 2021] 32. The maximum value of slope of the curve 𝑦 = −𝑥3 + 3𝑥2 + 12𝑥 − 5 is: (a) 15 (b) 12 (c) 9 (d) 0 [CBSE 2020] 33. Let 𝑓(𝑥) be a polynomial function of degree 6 such that 𝑑 𝑑𝑥 (𝑓(𝑥)) = (𝑥 − 1)3 (𝑥 − 3)3 , then Assertion (𝐴): 𝑓(𝑥) has a minimum at 𝑥 = 1. Reason (R): When 𝑑 𝑑𝑥 (𝑓(𝑥)) < 0, ∀𝑥 ∈ (𝑎 − ℎ, 𝑎), and 𝑑 𝑑𝑥 (𝑓(𝑥)) > 0, ∀𝑥 ∈ (𝑎, 𝑎 + ℎ); where ' ℎ ' is an infinitesimally small positive quantity, then 𝑓(𝑥) has a minimum at 𝑥 = 𝑎, provided 𝑓(𝑥) is continuous at 𝑥 = 𝑎. [CBSE SQP 2023] (a) Both (A) and (R) are true and (R) is the correct explanation of (A). (b) Both (A) and (R) are true but (R) is not the correct explanation of (𝐴). (c) (A) is true but (R) is false. CLICK HERE FOR SOLUTIONS
(d) (A) is false but (𝑅) is true. Case Based Qs (4 - 5 marks) 34. Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore. The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area 75𝜋cm2 . Based on the above information, answer the following questions: (A) If the radius of cylinder is 𝑟 cm and height is ℎ cm, then write the volume 𝑉 of cylinder in terms of radius 𝑟. (B) Find 𝑑𝑉 𝑑𝑟 . (C) Find the radius of cylinder when its volume is maximum. OR For maximum volume, ℎ > 𝑟. State true or false and justify. [CBSE 2023] CLICK HERE FOR SOLUTIONS
35. In an elliptical sport field, the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of 𝑥2 𝑎2 + 𝑦2 𝑏2 = 1. Based on the above information, answer the following questions: (A) If the length and the breadth of the rectangular field be 2𝑥 and 2𝑦 respectively, then find the area function in terms of 𝑥. (B) Find the critical point of the function. (C) Use first derivative test to find the length 2𝑥 and width 2𝑦 of the soccer field (in terms of 𝑎 and 𝑏 ) that maximize its area. [CBSE SQP 2022] Very Short & Short Qs (1 - 3 marks) 36. If 𝑓(𝑥) = 1 4𝑥2+2𝑥+1 ; 𝑥 ∈ 𝑅, then find the maximum value of 𝑓(𝑥). [CBSE SQP 2023] 37. Find the maximum profit that a company can make, if the profit function is given by 𝑃(𝑥) = 72 + 42𝑥 − 𝑥2 , where 𝑥 is the number of units and 𝑃 is the profit in rupees. [CBSE SQP 2023] 38. Find the absolute minimum volume of 𝑓(𝑥) = 2sin 𝑥 in �0, 3𝜋 2 �. [CBSE 2020] 39. Find the least value of the function 𝑓(𝑥) = 𝑎𝑥 + 𝑏 𝑥 , (𝑎 > 0, 𝑏 > 0, 𝑥 > 0). [CBSE 2020] maximum light through the whole opening. [CBSE 2018] 40. Show that the height of the right circular cylinder of greatest volume which can be inscribed in a right circular cone of height ℎ and radius 𝑟 is one-third of the height of the cone and greatest volume of the cylinder is 4 9 times the volume of the cone. [CBSE 2020] 41. Find the minimum value of (𝑎𝑥 + 𝑏𝑦), where 𝑥𝑦 = 𝑐2 . CLICK HERE FOR SOLUTIONS
[CBSE 2020, 15] 42. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3 . If building of tank costs ₹70 per sq. meters for the base and ₹ 45 per square meter for sides. What is the cost of least expensive tank? [CBSE 2019] 43. Find the point on the curve 𝑦2 = 4𝑥, which is nearest to the point (2, −8). [CBSE 2019] 44. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius 𝑅 is 2𝑅 √3 . Also find the volume of the largest cylinder inscribed in a sphere of radius 𝑅. [CBSE 2019] 45. A window is of the form of a semi-circle with a rectangle on its diameter. The total perimeter of the window is 10 m. Find the dimensions of the window to admit 46. An open tank with a square base and vertical sides is to be construted from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. [CBSE 2018] 47. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is cos−1 1 √3 . [CBSE 2016] 48. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius 𝑟 is 4𝑟 3 . Also find maximum volume in terms of volume of the sphere. [CBSE 2016, 14] 49. Prove that the least perimeter of an isosceles triangle in which a circle of radius 𝑟 can be inscribed is 6√3𝑟. [CBSE 2016] 50. Find the local maxima and local minima of the function 𝑓(𝑥) = sin 𝑥 − cos 𝑥, 0 < 𝑥 < 2𝜋. Also find the local maximum and local minimum values. [CBSE 2015] 51. The sum of the perimeters of a circle and square is 𝑘, where 𝑘 is some constant. Prove that the sum of their areas is least when the side of the square is double the radius of the circle. [CBSE 2014] CLICK HERE FOR SOLUTIONS
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
Integrals 1. INDEFINITE INTEGRALS Objective Qs (1 mark) 1. If 𝑑 𝑑𝑥 𝑓(𝑥) = 2𝑥 + 3 𝑥 and 𝑓(1) = 1, then 𝑓(𝑥) is: (a) 𝑥2 + 3log |𝑥| + 1 (c) 2 − 3 𝑥2 (b) 𝑥2 + 3log |𝑥| (d) 𝑥2 + 3log |𝑥| − 4 [CBSE 2023] 2. ∫ 𝑥3 𝑥+1 𝑑𝑥 is: (a) 𝑥 + 𝑥2 2 + 𝑥3 3 − log |1 − 𝑥| + 𝐶 (b) 𝑥 + 𝑥2 2 − 𝑥3 3 − log |1 − 𝑥| + 𝐶 (c) 𝑥 − 𝑥2 2 − 𝑥3 3 − log |1 + 𝑥| + 𝐶 (d) 𝑥 − 𝑥2 2 + 𝑥3 3 − log |1 + 𝑥| + 𝐶 [Delhi Gov. SQP 2022] Very Short & Short Qs (1-3 marks) 3. Find: ∫ � 𝑥 1−𝑥3 𝑑𝑥; 𝑥 ∈ (0,1). [CBSE SQP 2023] 4. Find ∫ sin2 𝑥−cos2 𝑥 sin 𝑥cos 𝑥 𝑑𝑥 [CBSE 2017] 5. Write the anti-derivative of �3√𝑥 + 1 √𝑥 �. [CBSE 2014] 6. Evaluate ∫ cos−1 (sin 𝑥)𝑑𝑥 CH-7 CLICK HERE FOR SOLUTIONS
[CBSE 2014] 2. METHODS OF INTEGRATION Objective Qs (1 mark) 7. Anti-derivative of tan 𝑥−1 tan 𝑥+1 with respect to 𝑥 is: (a) sec2 � 𝜋 4 − 𝑥� + 𝑐 (b) −sec2 � 𝜋 4 − 𝑥� + 𝑐 (c) log �sec � 𝜋 4 − 𝑥�� + 𝑐 (d) −log �sec � 𝜋 4 − 𝑥�� + 𝑐 8. ∫ 𝑥2 𝑒𝑥3 𝑑𝑥 equals: (a) 1 3 𝑒𝑥3 + 𝐶 (b) 1 3 𝑒𝑥4 + 𝐶 (c) 1 2 𝑒𝑥3 + 𝐶 (d) 1 2 𝑒𝑥2 + 𝐶 [Delhi Gov. SQP 2022, CBSE 2020] 9. ∫ 𝑒𝑥 � 𝑥log 𝑥+1 𝑥 � 𝑑𝑥 is equal to (a) log (𝑒𝑥 log 𝑥) + 𝐶 (b) 𝑒𝑥 𝑥 + 𝐶 (c) 𝑥log 𝑥 + 𝑒𝑥 + 𝐶 (d) 𝑒𝑥 log 𝑥 + 𝐶 [CBSE 2020] 10. ∫ 1 sin2 𝑥cos2 𝑥 𝑑𝑥 is: (a) tan 𝑥 + cot 𝑥 + 𝐶 (b) tan 𝑥 − cot 𝑥 + 𝐶 (c) tan 𝑥cot 𝑥 + 𝐶 (d) tan 𝑥 − cot 2𝑥 + 𝐶 [CBSE 2014] Very Short & Short Qs (1 − 3 marks) 11. Find: ∫ 2𝑥2+3 𝑥2(𝑥2+9) 𝑑𝑥; 𝑥 ≠ 0 CLICK HERE FOR SOLUTIONS
[CBSE 2023] 12. Find: ∫ 𝑒𝑥 � 1−sin 𝑥 1−cos 𝑥 � 𝑑𝑥. [CBSE 2023] 13. Find: ∫ sin−1 𝑥 (1−𝑥2)3/2 𝑑𝑥. [CBSE 2023] 14. Find ∫ �𝑥3+𝑥+1� (𝑥2−1) 𝑑𝑥 [CBSE SQP 2022] 15. Evaluate: ∫ 𝑑𝑥 √4𝑥−𝑥2 ⋅ [CBSE Term-2 2022] 16. Find: ∫ sin 2𝑥 √9−cos4 𝑥 𝑑𝑥. [CBSE Term-2 SQP 2022] 17. Find: ∫ 𝑑𝑥 √3−2𝑥−𝑥2 [CBSE SQP 2022] 18. Integrate ∫ 2𝑥 (𝑥2+1)(𝑥2+2) with respect of 𝑥. [CBSE Term-2 2022] 19. Find: ∫ 𝑥+1 (𝑥2+1)𝑥 𝑑𝑥 [CBSE Term-2 SQP 2022] 20. Find: ∫ 𝑑𝑥 √9−4𝑥2 [CBSE 2020] 21. Find: ∫ 𝑥4 log 𝑥𝑑𝑥 [CBSE 2020] 22. Find: ∫ sec3 𝑥𝑑𝑥. [CBSE 2020] 23. Evaluate: ∫ 𝑥+1 (𝑥+2)(𝑥+3) 𝑑𝑥 [CBSE 2020] 24. Integrate: ∫ 2cos 𝑥 (1−sin 𝑥)(1+sin2 𝑥) 𝑑𝑥 [CBSE 2020] CLICK HERE FOR SOLUTIONS
25. Find: ∫ √1 − sin 2𝑥𝑑𝑥, 𝜋 4 < 𝑥 < 𝜋 2 [CBSE 2019] 26. Evaluate: ∫ 𝑑𝑥 √5−4𝑥−2𝑥2 [CBSE 2019] 27. Integrate the function cos (𝑥+𝑎) sin (𝑥+𝑏) with respect to 𝑥. [CBSE 2019] 28. Find: ∫ sec2 𝑥 √tan2 𝑥+4 𝑑𝑥 [CBSE 2019] 29. Find: ∫ log 𝑥 (1+log 𝑥)2 𝑑𝑥. [CBSE Term-2 SQP 2022] 30. Integrate: ∫ 1 𝑒𝑥+1 𝑑𝑥 [CBSE Term-2 2022] Longs Qs (4 - 5 marks) 31. Find: ∫ sin−1 (2𝑥)𝑑𝑥. [CBSE 2019] 32. Find: ∫ 𝑥tan−1 𝑥𝑑𝑥 [CBSE 2019] 33. Find: ∫ 2cos 𝑥 (1−sin 𝑥)(2−cos2 𝑥) 𝑑𝑥. [CBSE 2019] 34. Find: ∫ 2𝑥 (𝑥2+1)(𝑥4+4) 𝑑𝑥. [CBSE 2017] 35. Find: ∫ cos 𝜃 (4+sin2 𝜃)(5−4cos2 𝜃) 𝑑𝜃 𝜃 [CBSE 2017] 36. Find: ∫ 𝑑𝑥 5−8𝑥−𝑥2. [CBSE 2017] 37. Find: ∫ (3sin 𝜃−2)cos 𝜃 5−cos2 𝜃−4sin 𝜃 𝑑𝜃. [CBSE 2016] CLICK HERE FOR SOLUTIONS
38. Evaluate: ∫ √𝑥 √𝑎3−𝑥3 𝑑𝑥 [CBSE 2016] 39. Integrate the following rational function: 𝑥 (𝑥−1)2(𝑥+2) with respect to 𝑥 [CBSE 2015] 40. Find ∫ 𝑥2+𝑥+1𝑑𝑥 (𝑥+2)(𝑥2+1) [CBSE 2015] 41. Find: ∫ sin6 𝑥 cos8 𝑥 𝑑𝑥 [CBSE 2014] 42. Find ∫ 𝑥3 (𝑥4+3𝑥2+2) 𝑑𝑥 [CBSE 2014] 43. Evaluate: ∫ (√cot 𝑥 + √tan 𝑥)𝑑𝑥. [CBSE 2014] 44. Integrate the function √𝑥2 + 1[log (𝑥2 + 1) − 2log 𝑥] 𝑥4 [CBSE 2014] 45. Find: ∫ 𝑥2 (𝑥2+1)(𝑥2+4) 𝑑𝑥 [CBSE 2014] 46. Evaluate: ∫ 1 cos4 𝑥+sin4 𝑥 𝑑𝑥 [CBSE 2014] 47. Evaluate: ∫ 1 sin4 𝑥+sin2 𝑥cos2 𝑥+cos4 𝑥 𝑑𝑥. [CBSE 2014] 3. DEFINITE INTEGRALS Objective Qs (1 mark) 48. For any integer 𝑛, the value of � 𝜋 −𝜋 𝑒cos2 𝑥 sin3 (2𝑛 + 1)𝑥𝑑𝑥 is: CLICK HERE FOR SOLUTIONS
(a) -1 (b) 0 (c) 1 (d) 2 [CBSE SQP 2023] 49. ∫−𝜋/4 𝜋/4 (sec2 𝑥)𝑑𝑥 is equal to: (a) -1 (b) 0 (c) 1 (d) 2 [CBSE 2020] Very Short & Short Qs (1 – 3 marks) 50. Evaluate: ∫−1 1 log𝑒 � 2−𝑥 2+𝑥 � 𝑑𝑥. [CBSE SQP 2023] 51. Evaluate: ∫0 𝜋 4 log (1 + tan 𝑥)𝑑𝑥. [CBSE SQP 2023] 52. Evaluate: ∫−1 1 |𝑥4 − 𝑥|𝑑𝑥. [CBSE 2023] 53. Evaluate: ∫log √2 log √3 1 (𝑒𝑥+𝑒−𝑥)(𝑒𝑥−𝑒−𝑥) 𝑑𝑥. [CBSE 2023] 54. Evaluate: ∫2 3 𝑥 𝑥2+1 𝑑𝑥 [CBSE SQP 2022] 55. Evaluate: ∫0 4 |𝑥 − 1|𝑑𝑥 [CBSE SQP 2022] 56. Evaluate: ∫𝜋/6 𝜋/3 𝑑𝑥 1+√tan 𝑥 [CBSE SQP 2022] 57. Evaluate: ∫1 4 {|𝑥| + |3 − 𝑥|}𝑑𝑥. [CBSE Term-2 2022] 58. Evaluate: ∫1 3 √𝑥 √𝑥+√4−𝑥 𝑑𝑥. [CBSE Term-2 2022] CLICK HERE FOR SOLUTIONS
59. Evaluate: ∫−1 2 |𝑥3 − 3𝑥2 + 2𝑥|𝑑𝑥 [CBSE Term-2 SQP 2022] 60. Evaluate: ∫1 2 � 1 𝑥 − 1 2𝑥2� 𝑒2𝑥 𝑑𝑥. [CBSE 2020] 61. Evaluate: ∫−2 2 |𝑥|𝑑𝑥. [CBSE 2020] 62. Evaluate: ∫1 3 |2𝑥 − 1|𝑑𝑥. [CBSE 2020] 63. Evaluate: ∫−𝜋 𝜋 (1 − 𝑥2)sin 𝑥 ⋅ cos2 𝑥𝑑𝑥. [CBSE 2019] 64. Evaluate: ∫−1 2 |𝑥| 𝑥 𝑑𝑥 [CBSE 2019] 65. Prove that: ∫0 𝑎 𝑓(𝑥)𝑑𝑥 = ∫0 𝑎 𝑓(𝑎 − 𝑥)𝑑𝑥, hence evaluate ∫0 𝜋 𝑥sin 𝑥 1+cos2 𝑥 𝑑𝑥 [CBSE 2019] 66. Prove that: ∫0 𝑎 𝑓(𝑥)𝑑𝑥 = ∫0 𝑎 𝑓(𝑎 − 𝑥)𝑑𝑥 and hence evaluate ∫0 1 𝑥2 (1 − 𝑥)𝑛 𝑑𝑥, [CBSE 2019] 67. Evaluate: ∫0 𝜋/4 sin 𝑥+cos 𝑥 16+9sin 2𝑥 𝑑𝑥. [CBSE 2018] 68. Evaluate: ∫2 3 3𝑥 𝑑𝑥 [CBSE 2017] 69. Evaluate: ∫0 𝜋 𝑥tan 𝑥 sec 𝑥+tan 𝑥 𝑑𝑥 [CBSE 2017] 70. Evaluate: ∫0 𝜋 𝑒2𝑥 ⋅ sin � 𝜋 4 + 𝑥� 𝑑𝑥. [CBSE 2016] 71. Evaluate: ∫−1 2 |𝑥3 − 𝑥|𝑑𝑥 [CBSE 2016] 72. Evaluate: ∫−𝜋 𝜋 (cos 𝑎𝑥 − sin 𝑏𝑥)2 𝑑𝑥. [CBSE 2015] 73. Evaluate: ∫0 𝜋/2 𝑑𝑥 1+√tan 𝑥 CLICK HERE FOR SOLUTIONS
[CBSE 2015] 74. Evaluate: ∫0 𝜋/4 𝑑𝑥 cos3 𝑥√2sin 2𝑥 𝑑𝑥. [CBSE 2015] 75. Evaluate: ∫0 1 𝑥𝑒𝑥2 𝑑𝑥 [CBSE 2014] 76. Evaluate: ∫2 4 𝑥 𝑥2+1 𝑑𝑥. [CBSE 2014] 77. If 𝑓(𝑥) = ∫0 𝑥 𝑡sin 𝑡𝑑𝑡, then write the value of 𝑓′ (𝑥). [CBSE 2014] 78. Evaluate: ∫0 𝜋/2 𝑥2 sin 𝑥𝑑𝑥. [CBSE 2014] 79. Evaluate: ∫𝜋/6 𝜋/3 sin 𝑥+cos 𝑥 √sin 2𝑥 𝑑𝑥. [CBSE 2014] CLICK HERE FOR SOLUTIONS
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
Application of Integrals 1. APPLICATION OF DEFINITE INTEGRALS Very Short & Short Qs (1 -3 marks) 1. Using integration, find the area of the region {(𝑥, 𝑦): 𝑦2 ≤ 𝑥 ≤ 𝑦}. [CBSE Term-2 2022] 2. Find the area bounded by the curves 𝑦 = |𝑥 − 1| and 𝑦 = 1, using integration. [CBSE Term-2 2022] 3. Using integration, find the smaller area enclosed by the circle 𝑥2 + 𝑦2 = 4 and the line 𝑥 + 𝑦 = 2. [CBSE 2020] 4. Find the area enclosed between the parabola 4𝑦 = 3𝑥2 and the straight line 3𝑥 − 2𝑦 + 12 = 0. [CBSE 2015] 5. Using integration, find the area of the region bounded by the line 𝑥 − 𝑦 + 2 = 0, the curve 𝑥 = �𝑦 and 𝑦-axis. [CBSE 2015] 6. Find the area of the smaller region bounded by the ellipse 𝑥2 9 + 𝑦2 4 = 1 and the line 𝑥 3 + 𝑦 2 = 1 [CBSE 2014] Long Qs (4 – 5 marks) 7. Make a rough sketch of the region {(𝑥, 𝑦): 0 ≤ 𝑦 ≤ 𝑥2 + 1,0 ≤ 𝑦 ≤ 𝑥 + 1,0 ≤ 𝑥 ≤ 2} and find the area of the region, using the method of integration. [CBSE SQP 2023] 8. Using Integration, find the area of triangle whose vertices are (−1,1), (0,5) and (3,2). [CBSE 2023] 9. Make a rough sketch of the region {(𝑥, 𝑦) : 0 ≤ 𝑦 < 𝑥2 , 0 ≤ 𝑦 < 𝑥, 0 ≤ 𝑥 ≤ 2} and find the area of the region using integration. [CBSE SQP 2022] 10. Using integration, find the area of the region bounded by the curves 𝑥2 + 𝑦2 = 4, 𝑥 = √3𝑦 and 𝑥- axis lying in the first quadrant. [CBSE Term-2 2022] CH-8 CLICK HERE FOR SOLUTIONS
11. Using integration, find the area of the region in the first quadrant enclosed by the 𝑥-axis, the line 𝑦 = 𝑥 and the circle 𝑥2 + 𝑦2 = 32. [CBSE 2018, 2015, 2014] 12. Find the area bounded by the circle 𝑥2 + 𝑦2 = 16 and the line √3𝑦 = 𝑥 in the first quadrant, using integration. [CBSE 2017] 13. Using integration, find the area bounded by the curves: 𝑦 = |𝑥 + 1| + 1, 𝑥 = −3, 𝑥 = 3 and 𝑦 = 0. [CBSE 2014] 14. Using integration, find the area bounded by the curve 𝑥2 = 4𝑦 and the line 𝑥 = 4𝑦 − 2. [CBSE 2014] 15. Using integration, find the area of the region in the first quadrant enclosed by the 𝑥-axis, the line 𝑦 = 𝑥 and the circle 𝑥2 + 𝑦2 = 18. [CBSE 2014] 16. Using integration, find the area of the region bounded by the triangle whose vertices are (−1,2), (1,5) and (3,4). [CBSE 2014] CLICK HERE FOR SOLUTIONS
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
Differential Equations 1. ORDER AND DEGREE OF DIFFERENTIAL EQUATION Objective Qs (1 mark) 1. The degree of the differential equation �1 + � 𝑑𝑦 𝑑𝑥 � 2 � 3 = � 𝑑2𝑦 𝑑𝑥2� 2 is: (a) 4 (b) 3 2 (c) 2 (d) not defined [CBSE SQP 2023] 2. Degree of the differential equation sin 𝑥 + cos � 𝑑𝑦 𝑑𝑥 � = 𝑦2 is: (a) 2 (b) 1 (c) not defined (d) 0 [CBSE 2023] 3. If 𝑚 and 𝑛 respectively, are the order and the degree of the differential equation 𝑑 𝑑𝑥 �� 𝑑𝑦 𝑑𝑥 �� 4 = 0, then 𝑚 + 𝑛 = (a) 1 (b) 2 (c) 3 (d) 4 [CBSE SQP 2022] Very Short & Short Qs (1 - 3 marks) 4. Write the sum of the order and the degree of the following differential equation: 𝑑 𝑑𝑥 � 𝑑𝑦 𝑑𝑥 � = 5. [CBSE SQP Term-2 2022] 5. The degree of the differential equation 1 + � 𝑑𝑦 𝑑𝑥 � 2 = 𝑥 is [CBSE 2020] 6. Find the order and the degree of the differential equation 𝑥2 𝑑2𝑦 𝑑𝑥2 = �1 + � 𝑑𝑦 𝑑𝑥 � −2 � 4 . [CBSE 2019] CH-9 CLICK HERE FOR SOLUTIONS
7. Write the degree of the differential equation 𝑥3 � 𝑑2𝑦 𝑑𝑥2� 2 + � 𝑑𝑦 𝑑𝑥 � 4 = 0 [CBSE 2019] 8. Find the order and degree (if defined) of the differential equation 𝑑2 𝑦 𝑑𝑥2 + 𝑥 � 𝑑𝑦 𝑑𝑥 � 2 = 2𝑥3 log � 𝑑2 𝑦 𝑑𝑥2 � [CBSE 2019] 9. Write the sum of the order and degree of the differential equation 𝑑 𝑑𝑥 �� 𝑑𝑦 𝑑𝑥 � 3 � = 0. [CBSE 2015] 10. Write the sum of the order and degree of the differential equation � 𝑑2𝑦 𝑑𝑥2� 2 + � 𝑑𝑦 𝑑𝑥 � 3 + 𝑥4 = 0. [CBSE 2015] 11. Write the sum of order and degree of differential equation, 1 + � 𝑑𝑦 𝑑𝑥 � 4 = 7 � 𝑑2 𝑦 𝑑𝑥2 � 3 [CBSE 2015] 2. SOLUTIONS OF FIRST ORDER, FIRST DEGREE DIFFERENTIAL EQUATIONS Objective Qs (1 mark) 12. The general solution of the differential equation 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 0; (Given 𝑥, 𝑦 > 0 ), is of the form: (a) 𝑥𝑦 = 𝑐 (b) 𝑥 = 𝑐𝑦2 (c) 𝑦 = 𝑐𝑥 (d) 𝑦 = 𝑐𝑥2 (Where ' 𝑐 ' is an arbitrary positive constant of integration) [CBSE SQP 2023] 13. The integrating factor of the differential equation (1 − 𝑦2) 𝑑𝑥 𝑑𝑦 + 𝑥𝑦 = 𝑎𝑦, (−1 < 𝑦 < 1) is: (a) 1 𝑦2−1 (b) 1 �𝑦2−1 CLICK HERE FOR SOLUTIONS
(c) 1 1−𝑦2 (d) 1 �1−𝑦2 [CBSE 2023] 14. The integrating factor of the differential equation (𝑥 + 3𝑦2) 𝑑𝑦 𝑑𝑥 = 𝑦 is: (a) 𝑦 (b) −𝑦 (c) 1 𝑦 (d) − 1 𝑦 [CBSE 2020] 15. The number of arbitrary constants in the particular solution of a differential equation of second order is (are): (a) 0 (b) 1 (c) 2 (d) 3 [CBSE 2020] Very Short & Short Qs (1 - 3 marks) 16. Solve the differential equation: 𝑦𝑒 𝑥 𝑦𝑑𝑥 = �𝑥𝑒 𝑥 𝑦 + 𝑦2 � 𝑑𝑦, (𝑦 ≠ 0) [CBSE SQP 2023] 17. Solve the differential equation: (cos2 𝑥) 𝑑𝑦 𝑑𝑥 + 𝑦 = tan 𝑥; �0 ≤ 𝑥 < 𝜋 2 � [CBSE SQP 2023] 18. If (𝑎 + 𝑏𝑥)𝑒 𝑦 𝑥 = 𝑥, then prove that 𝑥 𝑑2 𝑦 𝑑𝑥2 = � 𝑎 𝑎 + 𝑏𝑥 � 2 [CBSE SQP 2023] 19. Find the general solution of the differential equation (𝑥𝑦 − 𝑥2)𝑑𝑦 = 𝑦2 𝑑𝑥 [CBSE 2023] CLICK HERE FOR SOLUTIONS
20. Find the general solution of the differential equation: (𝑥2 + 1) 𝑑𝑦 𝑑𝑥 + 2𝑥𝑦 = �𝑥2 + 4 [CBSE 2023] 21. Solve the differential equation: 𝑦𝑑𝑥 + (𝑥 − 𝑦2)𝑑𝑦 = 0 [CBSE SQP 2022] 22. Find the general solution of the following differential equation: 𝑥 𝑑𝑦 𝑑𝑥 = 𝑦 − 𝑥sin � 𝑦 𝑥 � [CBSE Term-2 SQP 2022] 23. Find the particular solution of the following differential equation, given that 𝑦 = 0 when 𝑥 = 𝜋 4 𝑑𝑦 𝑑𝑥 + 𝑦cot 𝑥 = 2 1 + sin 𝑥 [CBSE Term-2 SQP 2022] 24. Solve the differential equation: 𝑥𝑑𝑦 − 𝑦𝑑𝑥 = �𝑥2 + 𝑦2𝑑𝑥. [CBSE SQP 2022] 25. Find the general solution of the differential equation: 𝑑𝑦 𝑑𝑥 = 𝑒𝑥−𝑦 + 𝑥2 𝑒−𝑦 [CBSE Term-2 2022] 26. Find the general solution of the differential equation: sec2 𝑥tan 𝑦𝑑𝑥 + sec2 𝑦tan 𝑥𝑑𝑦 = 0 [CBSE Term-2 2022] 27. For what value of ' 𝑛 ' is the following a homogeneous differential equation? 𝑑𝑦 𝑑𝑥 = 𝑥3 − 𝑦𝑛 𝑥2𝑦 + 𝑥𝑦2 [CBSE SQP 2020] CLICK HERE FOR SOLUTIONS
28. The integrating factor of the differential equation 𝑥 𝑑𝑦 𝑑𝑥 + 2𝑦 = 𝑥2 is [CBSE 2020] 29. Solve the following differential equation: 𝑑𝑦 𝑑𝑥 = 𝑥3 cosec 𝑦, given that 𝑦(0) = 0. [CBSE SQP 2020] 30. Find the particular solution of the differential equation 𝑥 𝑑𝑦 𝑑𝑥 = 𝑦 − 𝑥tan � 𝑦 𝑥 �, given that 𝑦 = 𝜋 4 at 𝑥 = 1. [CBSE 2020] 31. Solve the differential equation 𝑥𝑑𝑦 𝑑𝑥 + 𝑦 = 𝑥cos 𝑥 + sin 𝑥, given that 𝑦 = 1 when 𝑥 = 𝜋 2 . [CBSE 2019] 32. Solve the following differential equation: 𝑑𝑦 𝑑𝑥 + 𝑦 = cos 𝑥 − sin 𝑥 [CBSE 2019] 33. Solve the differential equation cos � 𝑑𝑦 𝑑𝑥 � = 𝑎, 𝑎 ∈ R [CBSE 2018] 34. Solve the differential equation log � 𝑑𝑦 𝑑𝑥 � = 3𝑥 + 4𝑦 [CBSE 2017] 35. Write the integrating factor of the following differential equation: (1 + 𝑦2) + (2𝑥𝑦 − cot 𝑦) 𝑑𝑦 𝑑𝑥 = 0. [CBSE 2015] 36. Write the solution of the differential equation: 𝑑𝑦 𝑑𝑥 = 2−𝑦 [CBSE 2015] 37. Find the integrating factor of the differential equation � 𝑒−2√𝑥 √𝑥 − 𝑦 √𝑥 � 𝑑𝑥 𝑑𝑦 = 1. [CBSE 2015] 38. Find the solution of the differential equation 𝑑𝑦 𝑑𝑥 = 𝑥3 𝑒−2𝑦 . [CBSE 2015] CLICK HERE FOR SOLUTIONS
39. If 𝑦(𝑥) is a solution of � 2+sin 𝑥 1+𝑦 � 𝑑𝑦 𝑑𝑥 = −cos 𝑥 and 𝑦(0) = 1, then find the value of 𝑦 � 𝜋 2 �. [CBSE 2014] 40. Find the particular solution of the differential equation 𝑒𝑥 �1 − 𝑦2𝑑𝑥 + 𝑦 𝑥 𝑑𝑦 = 0, given that 𝑦 = 1, when 𝑥 = 0. [CBSE 2014] Long Qs (4 – 5 marks) 41. Solve: (1 + 𝑦2)𝑑𝑥 = (tan−1 𝑦 − 𝑥)𝑑𝑦. [CBSE 2015] 42. Solve the following differential equation �𝑦 − 𝑥cos � 𝑦 𝑥 �� 𝑑𝑦 + �𝑦cos � 𝑦 𝑥 � −2𝑥sin � 𝑦 𝑥 �� 𝑑𝑥 = 0 [CBSE 2015] 43. Show that the following differential equation is homogeneous and solve it. �𝑥sin2 � 𝑦 𝑥 � − 𝑦� 𝑑𝑥 + 𝑥𝑑𝑦 = 0 𝑦 = 𝜋 4 when 𝑥 = 1 [CBSE 2015, 14] 44. Find the general solution of the following differential equations: 𝑥log 𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 = 2 𝑥 log 𝑥 [CBSE 2014] CLICK HERE FOR SOLUTIONS
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
Vector Algebra 1. BASIC CONCEPTS OF VECTORS Objective Qs (1 mark) 1. 𝐴𝐵𝐶𝐷 is a rhombus whose diagonals intersect at 𝐸. Then 𝐸𝐴 �����⃗ + 𝐸𝐵 �����⃗ + 𝐸𝐶 �����⃗ + 𝐸𝐷 �����⃗ equals to: (a) 0 �⃗ (b) 𝐴𝐷 �����⃗ (c) 2BD �����⃗ (d) 2𝐴𝐷 �����⃗ [CBSE SQP 2023] 2. If 𝑎 ⃗ = 4𝚤 ˆ + 6𝚥 ˆ and 𝑏 �⃗ = 3𝚥 ˆ + 4𝑘 ˆ, then the vector form of the component of 𝑎 ⃗ along 𝑏 �⃗ is: (a) 18 5 (3𝚤 ˆ + 4𝑘 ˆ) (b) 18 25 (3𝚥 ˆ + 4𝑘 ˆ) (c) 18 5 (3𝚤 ˆ + 4𝑘 ˆ) (d) 18 25 (4𝚤 ˆ + 6𝚥 ˆ) [CBSE SQP 2023] 3. If in △ ABC, BA �����⃗ = 2𝑎 ⃗ and BC �����⃗ = 3𝑏 �⃗, then AC �����⃗ is: (a) 2𝑎 ⃗ + 3𝑏 �⃗ (b) 2𝑎 ⃗ − 3𝑏 �⃗ (c) 3𝑏 �⃗ − 2𝑎 ⃗ (d) −2𝑎 ⃗ − 3𝑏 �⃗ [CBSE 2023] 4. If two vectors 𝑎 ⃗ and 𝑏 �⃗ are such that |𝑎 ⃗| = 2, |𝑏 �⃗| = 3 and 𝑎 ⃗ ⋅ 𝑏 �⃗ = 4, then |𝑎 ⃗ − 2𝑏 �⃗| is equal to: (a) √2 (b) 2√6 (c) 24 (d) 2√2 [CBSE SQP 2022] 5. The value of 𝑝 for which 𝑝(𝚤 ˆ + 𝚥 ˆ + 𝑘 ˆ) is a unit vector is: (a) 0 (b) 1 √3 (c) 1 (d) √3 [CBSE 2020] CH-10 CLICK HERE FOR SOLUTIONS
Very Short & Short Qs (1-3 marks) 6. If 𝑎 ⃗ = 𝚤 ˆ − 𝚥 ˆ + 7𝑘 ˆ and 𝑏 �⃗ = 5𝚤 ˆ − 𝚥 ˆ + 𝜆𝑘 ˆ, then find the value of 𝜆 so that vectors 𝑎 ⃗ + 𝑏 �⃗ and 𝑎 ⃗ − 𝑏 �⃗ are orthogonal. [CBSE SQP 2022] 7. The position vectors of two points 𝐴 and B are 𝑂𝐴 �����⃗ = 2𝚤 ˆ − 𝚥 ˆ − 𝑘 ˆ and 𝑂𝐵 �����⃗ = 2𝚤 ˆ − 𝚥 ˆ + 2𝑘 ˆ respectively. The position vector of a point 𝑃 which divides the line segment joining 𝐴 and 𝐵 in the ratio 2: 1 is [CBSE 2020] 8. 𝑋 and 𝑌 are two points with position vectors 3𝑎 ⃗ + 𝑏 �⃗ and 𝑎 ⃗ − 3𝑏 �⃗ respectively. Write the position vector of a point 𝑍 which divides the line segment 𝑋𝑌 in the ratio 2:1 externally. [CBSE 2019] 9. Find the value of 𝑝 for which the vectors 3𝚤 ˆ + 2𝚥 ˆ + 9𝑘 ˆ and 𝚤 ˆ − 2𝑝𝚥 ˆ + 3𝑘 ˆ are parallel. [CBSE 2017] 10. Write the position vector of the point which divides the join of the points with position vectors 3𝑎 ⃗ − 2𝑏 �⃗ and 2𝑎 ⃗ + 3𝑏 �⃗ in the ratio 2: 1. [CBSE 2016] 11. Find the unit vector in the direction of the sum of the vectors 2𝚤 ˆ + 3𝚥 ˆ − 𝑘 ˆ and 4𝚤 ˆ − 3𝚥 ˆ + 2𝑘 ˆ. [CBSE 2016] 12. Find the position vector of a point which divides the join of points with position vectors 𝑎 ⃗ − 2𝑏 �⃗ and 2𝑎 ⃗ + 𝑏 �⃗ externally in the ratio 2 : 1 . [CBSE 2016] 13. Find a vector in the direction of vector 𝑎 ⃗ = 𝚤 ˆ − 2𝚥 ˆ that has magnitude 7 units. [CBSE 2015] 14. Write the direction ratios of the vector 3𝑎 ⃗ + 2𝑏 �⃗ where 𝑎 ⃗ = 𝚤 ˆ + 𝚥 ˆ − 2𝑘 ˆ and 𝑏 �⃗ = 2𝚤 ˆ − 4𝚥 ˆ + 5𝑘 ˆ. [CBSE 2015] 15. Write the value of cosine of the angle which the vector 𝑎 ⃗ = 𝚤 ˆ + 𝚥 ˆ + 𝑘 ˆ makes with the 𝑦-axis. [CBSE 2014] 16. Find a vector 𝑎 ⃗ of magnitude 5√2, making an angle of 𝜋 4 with 𝑥-axis, 𝜋 2 with 𝑦-axis and an acute angle 𝜃 with 𝑧-axis. [CBSE 2014] CLICK HERE FOR SOLUTIONS
Long Qs (4 – 5 marks) 17. The two vectors 𝚥 ˆ + 𝑘 ˆ and 3𝚤 ˆ − 𝚥 ˆ + 4𝑘 ˆ represent the two sides 𝐴𝐵 and 𝐴𝐶, respectively of a △ 𝐴𝐵𝐶. Find the length of the median through 𝐴. [CBSE 2016, 15] 2. PRODUCT OF TWO VECTORS Objective Qs (1 mark) 18. The value of 𝜆 for which two vectors 2𝚤 ˆ − 𝚥 ˆ + 2𝑘 ˆ and 3𝚤 ˆ + 𝜆𝚥 ˆ + 𝑘 ˆ are perpendicular is: (a) 2 (b) 4 (c) 6 (d) 8 [CBSE SQP 2023] 19. Unit vector along 𝑃𝑄 �����⃗, where coordinates of 𝑃 and 𝑄 respectively are (2,1, −1) and (4,4, −7), is: (a) 2𝚤 ˆ + 3𝚥 ˆ − 6𝑘 ˆ (b) −2𝚤 ˆ − 3𝚥 ˆ + 6𝑘 ˆ (c) − 2 7 𝚤 ˆ − 3 7 𝚥 ˆ + 6 7 𝑘 ˆ (d) 2 7 𝚤 ˆ + 3 7 𝚥 ˆ − 6 7 𝑘 ˆ [CBSE 2023] 20. If |𝑎 ⃗ × 𝑏 �⃗| = √3 and 𝑎 ⃗ ⋅ 𝑏 �⃗ = −3, then angle between 𝑎 ⃗ and 𝑏 �⃗ is: (a) 2𝜋 3 (b) 𝜋 6 (c) 𝜋 3 (d) 5𝜋 6 [CBSE 2023] 21. The scalar projection of the vector 3𝚤 ˆ − 𝚥 ˆ − 2𝑘 ˆ on the vector 𝚤 ˆ + 2𝚥 ˆ − 3𝑘 ˆ is: (a) 7 √14 (b) 7 14 (c) 6 13 (d) 7 2 [CBSE SQP 2022] 22. If the projection of 𝑎 ⃗ = 𝚤 ˆ − 2𝚥 ˆ + 3𝑘 ˆ on 𝑏 �⃗ = 2𝚤 ˆ + 𝜆𝑘 ˆ is zero, then the value of 𝜆 is: (a) 0 (b) 1 CLICK HERE FOR SOLUTIONS
(c) − 2 3 (d) − 3 2 [CBSE 2020] 23. If 𝚤 ˆ, 𝚥 ˆ, 𝑘 ˆ are unit vectors along three mutually perpendicular directions, then: (a) 𝚤 ˆ. 𝚥 ˆ = 1 (b) 𝚤 ˆ × 𝚥 ˆ = 1 (c) 𝚤 ˆ ⋅ 𝑘 ˆ = 0 (d) 𝚤 ˆ × 𝑘 ˆ = 0 [CBSE 2020] Case Based Qs (4 – 5 marks) 24. Teams 𝐴, 𝐵, 𝐶 went for playing a tug of war game. Teams A, B, C have attached a rope to a metal ring and is trying to pull the ring into their own area. Team A pulls with force 𝐹1 = 6𝚤 ˆ + 0𝚥 ˆ𝑁. Team 𝐵 pulls with force 𝐹2 = −4𝚤 ˆ + 4𝚥 ˆ𝑁. Team C pulls with force 𝐹3 = −3𝚤 ˆ − 3𝚥 ˆ𝑁. Based on the above information, answer the following questions: (A) What is the magnitude of the force of Team A? (B) Which team will win the game? (C) Find the magnitude of the resultant force exerted by the teams. OR In what direction is the ring getting pulled? Very Short & Short Qs (1 − 3 marks) 25. If 𝑎 ⃗, 𝑏 �⃗, 𝑐 ⃗ are three non-zero unequal vectors such that 𝑎 ⃗ ⋅ 𝑏 �⃗ = 𝑎 ⃗ ⋅ 𝑐 ⃗, then find the angle between 𝑎 ⃗ and 𝑏 �⃗ − 𝑐 ⃗. CLICK HERE FOR SOLUTIONS
[CBSE 2023] 26. If 𝑎 ⃗ and 𝑏 �⃗ are two vectors of equal magnitude and 𝛼 is the angle between them, then prove that � 𝑎 �⃗+𝑏 �⃗ 𝑎 �⃗−𝑏 �⃗ � = cot � 𝛼 2 �. [CBSE Term-2 2022] 27. If 𝑎 ⃗ and 𝑏 �⃗ are two vectors such that |𝑎 ⃗ + 𝑏 �⃗| = |𝑏 �⃗|, then prove that (𝑎 ⃗ + 2𝑏 �⃗) is perpendicular to 𝑎 ⃗. [CBSE Term-2 2022] 28. If 𝑎 ˆ and 𝑏 ˆ are unit vectors, then prove that |𝑎 ˆ + 𝑏 ˆ| = 2cos 𝜃 2 , where 𝜃 is the angle between them. [CBSE Term-2 SQP 2022] 29. Find |𝑥 ⃗| if (𝑥 ⃗ − 𝑎 ⃗) ⋅ (𝑥 ⃗ + 𝑎 ⃗) = 12, where 𝑎 ⃗ is a unit vector. [CBSE SQP 2022] 30. The area of the parallelogram whose diagonals are 2𝚤 ˆ and −3𝑘 ˆ is ............. square units. [CBSE 2020] 31. The value of 𝜆 for which the vectors 2𝚤 ˆ − 𝜆𝚥 ˆ + 𝑘 ˆ and 𝚤 ˆ + 2𝚥 ˆ − 𝑘 ˆ are orthogonal is [CBSE 2020] 32. Show that the vectors 2𝚤 ˆ − 𝚥 ˆ + 𝑘 ˆ, 3𝚤 ˆ + 7𝚥 ˆ + 𝑘 ˆ and 5𝚤 ˆ + 6𝚥 ˆ + 2𝑘 ˆ form the sides of a right-angled triangle. [CBSE 2020] 33. If the sum of two unit vectors 𝑎 ⃗ and 𝑏 �⃗ is a unit vector, show that the magnitude of their difference is √3. [CBSE 2019] 34. If 𝑎 ⃗ + 𝑏 �⃗ + 𝑐 ⃗ = 0 and |𝑎 ⃗| = 5, |𝑏 �⃗| = 6 and |𝑐 ⃗| = 9, then find the angle between 𝑎 ⃗ and 𝑏 �⃗. [CBSE 2018] 35. If 𝜃 is the angle between two vectors 𝚤 ˆ − 2𝚥 ˆ + 3𝑘 ˆ and 3𝚤 ˆ − 2𝚥 ˆ + 𝑘 ˆ, find sin 𝜃. [CBSE 2018] 36. Find the magnitude of each of the two vectors 𝑎 ⃗ and 𝑏 �⃗, having the same magnitude, such that the angle between them is 60∘ and their scalar product is 9 2 . [CBSE 2018] 37. Write the vectors of unit length perpendicular to both the vectors 𝑎 ⃗ = 2𝚤 ˆ + 𝚥 ˆ + 2𝑘 ˆ and 𝑏 �⃗ = 𝚥 ˆ + 𝑘 ˆ. [CBSE 2016] CLICK HERE FOR SOLUTIONS
38. Write the number of vectors of unit length perpendicular to both the vector 𝑎 ⃗ = 2𝚤 ˆ + 𝚥 ˆ + 2𝑘 ˆ and vector 𝑏 �⃗ = 𝚥 ˆ + 𝑘 ˆ. [CBSE 2016] 39. If 𝑎 ⃗ = 7𝚤 ˆ + 𝚥 ˆ − 4𝑘 ˆ and 𝑏 �⃗ = 2𝚤 ˆ + 6𝚥 ˆ + 3𝑘 ˆ, then find the projection of 𝑎 ⃗ and 𝑏 �⃗. [CBSE 2015] 40. If vector 𝑎 ⃗ and vector 𝑏 �⃗ are two unit vectors such that vector (𝑎 ⃗ + 𝑏 �⃗) is also a unit vector, then find the angle between vector 𝑎 ⃗ and vector 𝑏 �⃗. [CBSE 2014] 41. Vectors 𝑎 ⃗ and 𝑏 �⃗ are such that |𝑎 ⃗| = √3, |𝑏 �⃗| = 2 3 and (𝑎 ⃗ × 𝑏 �⃗) is a unit vector. Write the angle between 𝑎 ⃗ and 𝑏 �⃗. [CBSE 2014] Long Qs 4 - 5 marks 42. If vector 𝑎 ⃗ = 𝚤 ˆ + 2𝚥 ˆ + 3𝑘 ˆ and vector 𝑏 �⃗ = 2𝚤 ˆ + 4𝚥 ˆ − 5𝑘 ˆ repesent two adjacent sides of a parallelogram, find unit vectors parallel to the diagonals of the parallelogram. [CBSE 2020] 43. If 𝚤 ˆ + 𝚥 ˆ + 𝑘 ˆ, 2𝚤 ˆ + 5𝚥 ˆ, 3𝚤 ˆ + 2𝚥 ˆ − 3𝑘 ˆ and 𝚤 ˆ − 6𝚥 ˆ − 𝑘 ˆ are the position vectors of points 𝐴, 𝐵, 𝐶 and 𝐷 respectively, then find the angle between 𝐴𝐵 �����⃗ and 𝐶𝐷 �����⃗. Deduce that 𝐴𝐵 �����⃗ and 𝐶𝐷 �����⃗ are collinear. [CBSE 2019] 44. Show that the points 𝐴, 𝐵, 𝐶 with position vectorts 2𝚤 ˆ − 𝚥 ˆ + 𝑘 ˆ, 𝚤 ˆ − 3𝚥 ˆ − 5𝑘 ˆ, 3𝚤 ˆ − 4𝚥 ˆ − 4𝑘 ˆ, respectively, are the vertices of a right angled triangle. Hence find the area of the triangle. [CBSE 2018] 45. Let vector 𝑎 ⃗ = 4𝚤 ˆ + 5𝚥 ˆ + 𝑘 ˆ, vector 𝑏 �⃗ = 𝚤 ˆ − 4𝚥 ˆ + 5𝑘 ˆ and vector 𝑐 ⃗ = 3𝚤 ˆ + 𝚥 ˆ + −𝑘 ˆ Find 𝑎 ⃗ vector 𝑑 ⃗ which is perpendicular to both vector 𝑐 ⃗ and vector 𝑏 �⃗ and 𝑑 ⃗ ⋅ 𝑎 ⃗ = 21 [CBSE 2018] 46. If 𝑎 ⃗, 𝑏 �⃗ and 𝑐 ⃗ are three mutually perpendicular vectors of the same magnitude, then prove that 𝑎 ⃗ + 𝑏 �⃗ + 𝑐 ⃗ is equally inclined with the vectors 𝑎 ⃗, 𝑏 �⃗ and 𝑐 ⃗. [CBSE 2017] 47. The two adjacent sides of a parallelogram are 2𝚤 ˆ − 4𝚥 ˆ − 5𝑘 ˆ and 2𝚤 ˆ + 2𝚥 ˆ + 3𝚥 ˆ. Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram. [CBSE 2016] CLICK HERE FOR SOLUTIONS
48. The scalar product of the vector 𝑎 ⃗ = 𝚤 ˆ + 2𝚥 ˆ + 3𝑘 ˆ with a unit vector along the sum of vectors 𝑏 �⃗ = 2𝚤 ˆ + 4𝚥 ˆ − 5𝑘 ˆ and 𝑐 ⃗ = 𝜆𝚤 ˆ + 2𝚥 ˆ + 3𝑘 ˆ is equal to one. Find the value of 𝜆 and hence find the unit vector along 𝑏 �⃗ + 𝑐 ⃗. [CBSE 2014] CLICK HERE FOR SOLUTIONS
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
Three Dimensional Geometry 1. BASIC CONCEPTS Objective Qs (1 mark) 1. The lines 𝑟 ⃗ = 𝚤 ˆ + 𝚥 ˆ − 𝑘 ˆ + 𝜆(2𝚤 ˆ + 3𝚥 ˆ − 6𝑘 ˆ) and 𝑟 ⃗ = 2𝚤 ˆ − 𝚥 ˆ − 𝑘 ˆ + 𝜇(6𝚤 ˆ + 9𝚥 ˆ − 18𝑘 ˆ); (where 𝜆 and 𝜇 are scalars) are: (a) coincident (b) skew (c) intersecting (d) parallel [CBSE SQP 2023] 2. If the direction cosines of a line are � 1 𝑐 , 1 𝑐 , 1 𝑐 �, then: (a) 0 < 𝑐 < 1 (b) 𝑐 > 2 (c) 𝑐 = ±√2 (d) 𝑐 = ±√3 [CBSE SQP 2023] 3. Equation of line passing through origin and making 30∘ , 60∘ and 90∘ with 𝑥, 𝑦, 𝑧-axes respectively, is: (a) 2𝑥 √3 = 𝑦 2 = 𝑧 0 (b) 2𝑥 √3 = 2𝑦 1 = 𝑧 0 (c) 2𝑥 = 2𝑦 √3 = 𝑧 1 (d) 2𝑥 √3 = 2𝑦 1 = 𝑧 1 [CBSE 2023] 4. The length of the perpendicular drawn from the point (4, −7,3) on the 𝑦-axis is: (a) 3 units (b) 4 units (c) 5 units (d) 7 units [CBSE 2020] 5. Assertion (A): If a line makes angles 𝛼, 𝛽, 𝛾 with positive direction of the coordinate axes, then sin2 𝛼 + sin2 𝛽 + sin2 𝛾 = 2. Reason (R): The sum of squares of the direction cosines of a line is 1. [CBSE 2023] (a) Both (A) and (R) are true and (R) is the correct explanation of (𝐴). CH-11 CLICK HERE FOR SOLUTIONS
(b) Both (A) and (R) are true, but (R) is not the correct explanation of (A). (c) (A) is true, but (R) is false. (d) (A) is false, but (R) is true. Very Short & Short Qs (1 − 3 marks) 6. If the equation of a line is 𝑥 = 𝑎𝑦 + 𝑏, 𝑧 = 𝑐𝑦 + 𝑑 then find the direction ratios of the line and a point on the line. [CBSE 2023] 7. The cartesian equation of a line 𝐴𝐵 is: 2𝑥 − 1 12 = 𝑦 + 2 2 = 𝑧 − 3 3 Find the direction cosines of a line parallel to line AB. [CBSE Term 2 2022] 8. Find all the possible vector of magnitude 5√3 which are equally inclined to the coordinate axes. [CBSE Term-2 2022] 9. If a line makes 60∘ and 45∘ angles with the positive directions of 𝑥-axis and 𝑧-axis respectively, then find the angle that it makes with the positive direction of 𝑦-axis. Hence, write the direction cosines of the line. [CBSE Term-2 2022] 10. Find the direction cosines of the following line: 3 − 𝑥 −1 = 2𝑦 − 1 2 = 𝑧 4 [CBSE Term-2 SQP 2022] 11. Find the value of 𝑘 so that the lines 𝑥 = −𝑦 = 𝑘𝑧 and 𝑥 − 2 = 2𝑦 − 1 = −𝑧 + 1 are perpendicular to each other. [CBSE 2020] 12. The line of shortest distance between two skew lines is to both the lines. [CBSE 2020] 13. The vector equation of the line through the points (3,4, −7) and (1, −1,6) is ...... . [CBSE 2020] 14. Find the vector equation of a line passing through the point (2,3,2) and parallel to the line 𝑟 ⃗ = (−2𝚤 ˆ + 3𝚥 ˆ) + 𝜆(2𝚤 ˆ − 3𝚥 ˆ + 6𝑘 ˆ). Also, find the distance between these two lines. [CBSE 2019] CLICK HERE FOR SOLUTIONS
15. Find the vector equation of a line which passes through the point (3,4,5) and is parallel to the vector (2𝚤 ˆ + 2𝚥 ˆ − 3𝑘 ˆ). [CBSE 2019] 16. If a line makes angles 90∘ , 135∘ and 45∘ with the 𝑥, 𝑦 and 𝑧 axes respectively, find its direction cosines. [CBSE 2019] 17. Find the shortest distance between the lines vector 𝑟 ⃗ = (4𝚤 ˆ − 𝚥 ˆ) + 𝜆(𝚤 ˆ + 2𝚥 ˆ − 3𝑘 ˆ) 𝑟 ⃗ = (𝚤 ˆ − 𝚥 ˆ + 2𝑘 ˆ) + 𝜇(2𝚤 ˆ + 4𝚥 ˆ − 5𝑘 ˆ) [CBSE 2018] 18. The 𝑥-coordinate of a point on the line joining the points 𝑃(2,2,1) and 𝑄(5,1, −2) is 4. Find its 𝑧- coordinate. [CBSE 2017] 19. Find the vector equation of the line passing through the point 𝐴(1,2, −1) and parallel to the line 5𝑥 − 25 = 14 − 7𝑦 = 35𝑧. [Delhi 2017] 20. If a line makes angles 90∘ , 60∘ and 𝜃 with 𝑥, 𝑦 and 𝑧-axes respectively, where 𝜃 is acute, then find 𝜃. [CBSE 2015] 21. The equations of a line is 5𝑥 − 3 = 15𝑦 + 7 = 3 - 10z. Write the direction cosines of the line. [CBSE 2015] 22. Prove that the line 𝑟 ⃗ = (𝚤 ˆ + 𝚥 ˆ − 𝑘 ˆ) + 𝜆(3𝚤 ˆ − 𝚥 ˆ) and 𝑟 ⃗ = (4𝚤 ˆ − 𝑘 ˆ) + 𝜇(2𝚤 ˆ + 3𝑘 ˆ) intersect and find their point of intersection. [CBSE 2014] 23. Find the vector and cartesian equations of the line passing through the point (2,1,3) and perpendicular to the lines 𝑥−1 1 = 𝑦−2 2 = 𝑧−3 3 and 𝑥 −3 = 𝑦 2 = 𝑧 5 . [CBSE 2014] 24. A line passes through (2, −1,3) and is perpendicular to the lines vector 𝑟 ⃗ = (𝚤 ˆ + 𝚥 ˆ − 𝑘 ˆ) + 𝜆(2𝚤 ˆ − 2𝚥 ˆ + 𝑘 ˆ) and vector 𝑟 ⃗ = (2𝚤 ˆ − 𝚥 ˆ − 3𝑘 ˆ) + 𝜇(𝚤 ˆ + 2𝚥 ˆ + 2𝑘 ˆ). Obtain its equation in vector and cartesian form. [CBSE 2014] 25. Find the direction cosines of the line 𝑥+2 2 = 2𝑦−7 6 = 5−𝑧 6 . Also find the vector equation of the line through the point 𝐴(−1,2,3) and parallel to the given line. [CBSE 2014] CLICK HERE FOR SOLUTIONS
26. Find the value of 𝑝 so that the lines 1−𝑥 3 = 7𝑦−14 2𝑝 = 𝑧−3 2 and 7−7𝑥 3𝑝 = 𝑦−5 1 = 6−𝑧 5 are at right angles. [CBSE 2014] 27. If the cartesian equations of a line are 3−𝑥 5 = 𝑦+4 7 = 2𝑧−6 4 , write the vector equation for the line. [CBSE 2014] 28. Write the equation of the straight line through the point (𝛼, 𝛽, 𝛾) and parallel to z-axis. [CBSE 2014] Long Qs (4 - 5 marks) 29. Find the coordinates of the image of the point (1,6,3) with respect to the line 𝑟 ⃗ = (𝚥 ˆ + 2𝑘 ˆ) + 𝜆(𝚤 ˆ + 2𝚥 ˆ + 3𝑘 ˆ); where ' 𝜆 ' is a scalar. Also, find the distance of the image from the 𝑦-axis. [CBSE SQP 2023] 30. An aeroplane is flying along the line 𝑟 ⃗ = 𝜆(𝚤 ˆ − 𝚥 ˆ + 3𝑘 ˆ); where ' 𝜆 ' is a scalar and another aeroplane is flying along the 𝑟 ⃗ = 𝚤 ˆ − 𝚥 ˆ + 𝜇(−2𝚥 ˆ + 𝑘 ˆ); where ' 𝜇 ' is a scalar. At what points on the lines should they reach, so that the distance between them is the shortest? Find the shortest possible distance between them. [CBSE SQP 2023] 31. A line 𝑙 passes through point (−1,3, −2) and is perpendicular to both the lines 𝑥 1 = 𝑦 2 = 𝑧 3 and 𝑥+2 −3 = 𝑦−1 2 = 𝑧+1 5 . Find the vector equation of the line 𝑙. Hence, obtain its distance from origin. [CBSE 2023] 32. Find the equations of the diagonals of the parallelogram PQRS whose vertices are 𝑃(4,2, −6), 𝑄(5, −3,1), 𝑅(12,4,5) and 𝑆(11,9, −2). Use these equations to find the point of intersection of diagonals. [CBSE 2023] 33. Find the vector and Cartesian equations of a line passing through (1,2, −4) and perpendicular to the two lines 𝑥 − 8 3 = 𝑦 + 9 −16 = 𝑧 − 10 7 and 𝑥 − 15 3 = 𝑦 − 29 8 = 𝑧 − 5 −5 [CBSE 2017] CLICK HERE FOR SOLUTIONS
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
1. BASIC CONCEPTS OF LINEAR PROGRAMMING Objective Qs (1 mark) 1. The corner points of the bounded feasible region determined by a system of linear constraints are (0,3), (1,1) and (3,0). Let 𝑍 = 𝑝𝑥 + 𝑞𝑦, where 𝑝, 𝑞 > 0. The condition on 𝑝 and 𝑞 so that the minimum of 𝑍 occurs at (3,0) and (1,1) is: (a) 𝑝 = 2𝑞 (b) 𝑝 = 𝑞 2 (c) 𝑝 = 3𝑞 (d) 𝑝 = 𝑞 [CBSE SQP 2023] 2. The feasible region corresponding to the linear constraints of a Linear Programming Problem is given below. Which of the following is not a constraint to the given Linear Programming Problem? (a) 𝑥 + 𝑦 ≥ 2 (b) 𝑥 + 2𝑦 ≤ 10 (c) 𝑥 − 𝑦 ≥ 1 (d) 𝑥 − 𝑦 ≤ 1 [CBSE SQP 2023] 3. The objective function 𝑍 = 𝑎𝑥 + 𝑏𝑦 of an LPP has maximum value 42 at (4,6) and minimum value 19 at (3,2). Which of the following is true? (a) 𝑎 = 9, 𝑏 = 1 (b) 𝑎 = 5, 𝑏 = 2 (c) 𝑎 = 3, 𝑏 = 5 (d) 𝑎 = 5, 𝑏 = 3 [CBSE 2023] 4. The corner points of the feasible region of a linear programming problem are (0,4), CH-12 CLICK HERE FOR SOLUTIONS
(8,0) and � 20 3 , 4 3 �. If 𝑍 = 30𝑥 + 24𝑦 is the objective function, then (maximum value of 𝑍 - minimum value of 𝑍 ) is equal to: (a) 40 (b) 96 (c) 120 (d) 136 [CBSE 2023] 5. The corner points of the shaded unbounded feasible region of an LPP are (0,4), (0.6,1.6) and (3,0) as shown in the figure. The minimum value of the objective function 𝑍 = 4𝑥 + 6𝑦 occurs at: (a) (0.6,1.6) only (b) (3,0) only (c) (0.6,1.6) and (3,0) only (d) every point of the line-segment joining the points (0.6,1.6) and (3,0) [CBSE SQP 2022] 6. The solution set of the inequality 3𝑥 + 5𝑦 < 4 is: (a) an open half-plane not containing the origin. (b) an open half-plane containing the origin. (c) the whole 𝑋𝑌-plane not containing the line 3𝑥 + 5𝑦 = 4 (d) a closed half plane containing the origin. [CBSE SQP 2022] 7. In the given graph, the feasible region for a LPP is shaded. CLICK HERE FOR SOLUTIONS
The objective function 𝑍 = 2𝑥 − 3𝑦 will be minimum at: (a) (4,10) (b) (6,8) (c) (0,8) (d) (6,5) [CBSE Term-1 SQP 2021] 8. Based on the given shaded region as the feasible region in the graph, at which point(s) is the objective function Z = 3𝑥 + 9𝑦 maximum? (a) Point B (b) Point 𝐶 (c) Point D (d) Every point on the line segment 𝐶𝐷 [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
9. In a linear programming problem, the constraints on the decision variables 𝑥 and 𝑦 are 𝑥 − 3𝑦 ≥ 0, 𝑦 ≥ 0,0 ≤ 𝑥 ≤ 3. The feasible region: (a) is not in the first quadrant. (b) is bounded in the first quadrant. (c) is unbounded in the first quadrant. (d) does not exist. [CBSE Term-1 SQP 2021] 10. For an objective function 𝑍 = 𝑎𝑥 + 𝑏𝑦, where 𝑎, 𝑏 > 0; the corner points of the feasible region determined by a set of constraints (linear inequalities) are (0,20), (10,10), (30,30) and (0,40). The condition on 𝑎 and 𝑏 such that the maximum 𝑍 occurs at both the points (30,30) and (0,40) is: (a) 𝑏 − 3𝑎 = 0 (b) 𝑎 = 3𝑏 (c) 𝑎 + 2𝑏 = 0 (d) 2𝑎 − 𝑏 = 0 [CBSE Term-1 SQP 2021] 11. The corner points of feasible region for a linear programming problem are P(0,5), 𝑄(1,5), 𝑅(4,2) and 𝑆(12,0). The minimum value of the objective function 𝑍 = 2𝑥 + 5𝑦 is at the point: (a) P (b) Q (c) R (d) 𝑆 [CBSE Term-1 SQP 2021] 12. A linear programming problem is as follows: Minimise 𝑍 = 30𝑥 + 50𝑦 subject to the constraints, 3𝑥 + 5𝑦 ≥ 15 2𝑥 + 3𝑦 ≤ 18 𝑥 ≥ 0, 𝑦 ≥ 0 In the feasible region, the minimum value of 𝑍 occurs at: (a) a unique point (b) no point (c) infinitely many poins (d) two points only [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
Very Short & Short Qs (1-3 marks) 13. Solve the following Linear Programming Problem graphically: Maximise: 𝑍 = −𝑥 + 2𝑦, subject to the constraints: 𝑥 ≥ 3, 𝑥 + 𝑦 ≥ 5, 𝑥 + 2𝑦 ≥ 6, 𝑦 ≥ 0. [CBSE SQP 2023] 14. Solve the following Linear Programming Problem graphically: Minimise: 𝑍 = 𝑥 + 2𝑦, subject to the constraints: 𝑥 + 2𝑦 ≥ 100, 2𝑥 − 𝑦 ≤ 0,2𝑥 + 𝑦 ≤ 200, 𝑥, 𝑦 ≥ 0. [CBSE SQP 2023] 15. Solve the following linear programming problem graphically: Maximise: Z = x + 2y Subject to constraints: 𝑥 + 2𝑦 ≥ 100, 2𝑥 − 𝑦 ≤ 0 2𝑥 + 𝑦 ≤ 200 𝑥 ≥ 0, 𝑦 ≥ 0 [CBSE 2023] 16. Solve the following Linear Programming Problem graphically: Maximise 𝑍 = 400𝑥 + 300𝑦 Subject to 𝑥 + 𝑦 ≤ 200, 𝑥 ≤ 40, 𝑥 ≥ 20, 𝑦 ≥ 0 [CBSE SQP 2022] 17. The corner points of the feasible region determined by the system of linear constraints are as shown below: CLICK HERE FOR SOLUTIONS
(A) Let 𝑍 = 3𝑥 − 4𝑦 be the objective function. Find the maximum and minimum value of Z and also the corresponding point at which the maximum and minimum value occurs. (B) Let X = 𝑝x + 𝑞𝑦, where 𝑝, 𝑞 > 0 be the objective function. Find the condition on 𝑝 and 𝑞 so that the maximum value of 𝑍 occurs at 𝐵(4,10) and 𝐶(6,8). Also mention the number of optimal solutions in this case. [CBSE SQP 2020] 18. Solve the following LPP graphically: Minimise, 𝑍 = 5𝑥 + 7𝑦 Subject to the constraints 2𝑥 + 𝑦 ≥ 8 𝑥 + 2𝑦 ≥ 10 𝑥, 𝑦 ≥ 0 [CBSE 2020] 19. A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large van is ₹ 400 and each small van is ₹ 200. Not more than ₹ 3000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimise cost. [CBSE 2017] 20. Two tailors, A and B, earn ₹ 300 and ₹ 400 per day, respectively. A can stitch 6 shirts and 4 pairs of trousers while 𝐵 can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP. [CBSE 2017] CLICK HERE FOR SOLUTIONS
21. A small firm manufactures necklace and bracelets. The total number of necklace and bracelet that it can handle per day is at most 24. It takes 1 hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is ₹ 100 and that on a bracelet is ₹ 300, how many of each should be produced daily to maximise the profit? It is being given that at least one of each must be produced. [CBSE 2017] 22. Solve the following linear programming problem (LPP) graphically: Maximise and Minimise 𝑍 = 5𝑥 + 10𝑦 subject to the constraints 𝑥 + 2𝑦 ≤ 120, 𝑥 + 𝑦 ≥ 60, 𝑥 − 2𝑦 ≥ 0, 𝑥 ≥ 0, 𝑦 ≥ 0 [CBSE 2017] 23. If a 20 year old girl drives her car at 25 km/h, she has to spend ₹4/km on petrol. If she drives her car at 40 km/h, the petrol cost increases to ₹ 5/km. She has ₹ 200 to spend on petrol and wishes to find the maximum distance she can travel within one hour. Express the above problem as Linear Programming Problem. Write any one value reflected in the problem. [CBSE SQP 2016] Long Qs (4 - 5 marks) 24. A manufacturer has three machine I, II, and III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for atleast 5 hours a day. She produces only two items M and N each requiring the use of all the three machines. The number of hours required for producing 1 unit each of M and N on the three machines are given in the following table: Items Number of hours required on machines I II III M 1 2 1 N 2 1 1.25 She makes a profit of ₹ 600 and ₹ 400 on items 𝑀 and 𝑁 respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produce? What will be the maximum profit? [CBSE 2020] 25. A small manufacturer has employed 5 skilled men and 10 semi-skilled men and makes an article in two qualities deluxe model and an ordinary model. The making of a deluxe model requires 2hrs. worked by a skilled man and 2 hrs. worked by a semiskilled man. The ordinary model CLICK HERE FOR SOLUTIONS
requires 1hr by a skilled man and 3hrs. by a semi-skilled man. By union rules no man may work more than 8hrs. per day. The manufacturers clear profit on deluxe model is ₹ 15 and on an ordinary model is ₹ 10. How many of each type should be made in order to maximise his total daily profit. [CBSE 2019] 26. Find graphically, the maximum value of 𝑧 = 2𝑥 + 5𝑦, subject to constraints given below: 2𝑥 + 4𝑦 ≤ 8,3𝑥 + 𝑦 ≤ 6, 𝑥 + 𝑦 ≤ 4; 𝑥 ≥ 0, 𝑦 ≥ 0 [CBSE 2015] 27. Maximise 𝑍 = 8𝑥 + 9𝑦 subject to the constraints given below: 2𝑥 + 3𝑦 ≤ 6,3𝑥 − 2𝑦 ≤ 6, 𝑦 ≤ 1; 𝑥, 𝑦 ≥ 0 [CBSE 2015] CLICK HERE FOR SOLUTIONS
CLASS 12 MATHEMATICS CHAPTER WISE , TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
Probability 1. BASIC CONCEPTS AND CONDITIONAL PROBABILITY Objective Qs (1 mark) 1. If 𝐴 and 𝐵 are two events such that 𝑃(𝐴/𝐵) = 2 × 𝑃(𝐵/𝐴) and 𝑃(𝐴) + 𝑃(𝐵) = 2 3 , then 𝑃(𝐵) is equal to: (a) 2 9 (b) 7 9 (c) 4 9 (d) 5 9 [CBSE 2023] 2. A problem in Mathematics is given to three students whose chances of solving it are 1 2 , 1 3 , 1 4 , respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is: (a) 1 4 (b) 1 3 (c) 1 2 (d) 3 4 [CBSE SQP 2023] 3. Given two independent events 𝐴 and 𝐵 such that 𝑃(𝐴) = 0.3, 𝑃(𝐵) = 0.6 and 𝑃(𝐴′ ∩ 𝐵′) is: (a) 0.9 (b) 0.18 (c) 0.28 (d) 0.1 [CBSE SQP 2022] 4. If 𝐴 and 𝐵 are two independent events with 𝑃(𝐴) = 1 3 and 𝑃(𝐵) = 1 4 , then 𝑃(𝐵′ /𝐴) is: (a) 1 4 (b) 1 8 (c) 3 4 (d) 1 [CBSE 2020] CH-13 CLICK HERE FOR SOLUTIONS
5. A die is thrown once. Let 𝐴 be the event that the number obtained is greater than 3 . Let 𝐵 be the event that the number obtained is less than 5 . Then 𝑃(𝐴 ∪ 𝐵) is: (a) 2 5 (b) 3 5 (c) 0 (d) 1 [CBSE 2020] 6. A card is picked at random from a pack of 𝟓𝟐 playing cards. Given that the picked card is a queen, the probability of this card to be a card of spade is: (a) 1 3 (b) 4 13 (c) 1 4 (d) 1 2 [CBSE 2020] Very Short & Short Qs (1 -3 marks) 7. 𝐴 and 𝐵 throw a die alternately till one of them gets 𝑎 ' 6 ' and wins the game. Find their respective probabilities of winning, if 𝐴 starts the game first. [CBSE 2023] 8. Three friends go for coffee. They decide who will pay the bill, by each tossing a coin and then letting the "odd person" pay. There is no odd person if all three tosses produce the same result. It there is no odd person in the first round, they make a second round of tosses and they continue to do so until there is an odd person. What is the probability that exactly three rounds of tosses are made? [CBSE SQP 2022] 9. Events 𝐴 and 𝐵 are such that 𝑃(𝐴) = 1 2 , 𝑃(𝐵) = 7 12 and 𝑃( not 𝐴 or not 𝐵) = 1 4 . State whether A and B are independent? [CBSE Term-2 2022] 10. A box 𝐵1 contains 1 white ball and 3 red balls. Another box 𝐵2 contains 2 white balls and 3 red balls. If one ball is drawn at random from each of the boxes 𝐵1 and 𝐵2, then find the probability that the two balls drawn are of the same colour. [CBSE Term-2 2022] 11. Two cards are drawn at random from a pack of 52 cards one-by-one without replacement. What is the probability of getting first card red and second card jack? [CBSE Term-2 SQP 2022] 12. Two cards are drawn at random and oneby-one without replacement from a well shuffled pack of 52 playing cards. Find the probability that one card is red and other is black. CLICK HERE FOR SOLUTIONS
[CBSE 2020] 13. A bag contains 3 black, 4 red and 2 green balls. If three balls are drawn simultaneously at random, then the probability that the balls are of different colours is . [CBSE 2020] 14. The probability of finding a green signal on a busy crossing 𝑋 is 30%. What is the probability of finding a green signal on 𝑋 on two consecutive days out of three? [CBSE 2020] 15. A speaks truth in 80% cases and B speak truth in 90% cases. In what percentage of cases are they likely to agree with each other in stating the same fact? [CBSE SQP 2020] 16. If 𝐴 and 𝐵 are two events such that 𝑃(𝐴) = 0.4, 𝑃(𝐵) = 0.3 and 𝑃(𝐴 ∪ 𝐵) = 0.6, then find P(B′ ∩ 𝐴). [CBSE 2020] 17. Out of 8 outstanding students of a school, in which there are 3 boys and 5 girls, a team of 4 students is to be selected for a quiz competition. Find the probability that 2 boys and 2 girls are selected. [CBSE 2019] 18. Prove that if 𝐸 and 𝐹 are independent events, then the events 𝐸′ and 𝐹′ are also independent. [CBSE 2019] 19. 12 cards numbered 1 to 12 (one number on one card), are placed in a box and mixed up thoroughly. Then a card is drawn at random from the box. If it is known that the number on the drawn card is greater than 5, find the probability that the card bears an odd number. [CBSE 2019] 20. A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let 𝐴 be the event, 'the number is even', and 𝐵 be the event, 'the number is red'. Are A and B independent? [CBSE 2019] 21. A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. [CBSE 2018] 22. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Given that (A) the youngest is a girl. (B) at least one is a girl. [CBSE 2014] CLICK HERE FOR SOLUTIONS
23. A bag contains 3 red and 7 black balls. Two balls are drawn one by one at a time at random without replacement. If second drawn ball is red then what is the probability the first drawn ball is also red? [CBSE 2014] 24. A couple has two children. Find the probability that both are boys, it is known that: (A) one of the children is a boy (B) older child is a boy. [CBSE 2014] Long Qs (4 – 5 marks) 25. In a game of Archery, each ring of the Archery target is valued. The centre most ring is worth 10 points and rest of the rings are allotted points 9 to 1 in sequential order moving outwards. Archer 𝐴 is likely to earn 10 points with a probability of 0.8 and Archer B is likely to earn 10 points with a probability of 0.9. Based on the above information, answer the following questions: If both of them hit the Archery target, then find the probability that (A) Exactly one of them earns 10 points. (B) both of them earn 10 points. [CBSE Term-2 2022] 26. Probability of solving a specific problem independently by 𝐴 and 𝐵 are 1 2 and 1 3 respectively. If both try to solve the problem independently, find the probability that: (A) the problem is solved. CLICK HERE FOR SOLUTIONS
(B) exactly one of them solves the problem. [CBSE 2018] 27. A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and 𝐵 wins the game if he gets a total of 10. If 𝐴 starts the game, then find the probability that 𝐵 wins. [CBSE 2016] 28. If 𝐴 and 𝐵 are two independent events such that 𝑃(𝐴 ‾ ∩ 𝐵) = 2 15 and 𝑃(𝐴 ∩ 𝐵 ‾) = 1 6 , then find 𝑃(𝐴) and 𝑃(𝐵). [CBSE 2015] 29. Consider the experiment of tossing a coin. If the coin shows head, toss it again, but if it shows tail, then throw a die. Find the conditional probability of the event that 'the die shows a number greater than 4' given that 'there is atleast one tail'. [CBSE 2014] 2. BAYES' THEOREM, RANDOM VARIABLE AND ITS PROBABILITY DISTRIBUTION Case Based Qs 30. In an Office three employees Jayant, S onia and Oliver process incoming copies of a certain form. Jayant processes 50% of the forms, Sonia processes 20% and Oliver the remaining 30% of the forms. Jayant has an error rate of 0.06, Sonia has an error rate of 0.04 and Oliver has an error rate of 0.03. Based on the above information, answer the following questions: (A) Find the probability that Sonia processed the form and committed an error. (B) Find the total probability of committing an error in processing the form. (C) The manager of the Company wants to do a quality check. During inspection, he selects a form at random from the days output of processed form. If the form selected at random has an error, find the probability that the form is not processed by Jayant. OR CLICK HERE FOR SOLUTIONS
Let E be the event of committing an error in processing the form and let 𝐸1, 𝐸2 and 𝐸3, be the events that Jayant, Sonia and Oliver processed the form. Find the value of ∑𝑖=1 3 𝑃(𝐸𝑖 ∣ 𝐸). [CBSE SQP 2023] 31. An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company's statistics show that an accident-prone person will have an accident at sometime within a fixed one-year period with probability 0.6, whereas this probability is 0.2 for a person who is not accident prone. The company knows that 20 percent of the population is accident prone. (A) What is the probability that a new policyholder will have an accident within a year of purchasing a policy? (B) Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone? [CBSE Term-2 SQP 2022] 32. There are two antiaircraft guns, named as 𝐴 and 𝐵. The probabilities that the shell fired from them hits an airplane are 0.3 and 0.2 respectively. Both of them fired one shell at an airplane at the same time. (A) What is the probability that the shell fired from exactly one of them hit the plane? (B) If it is known that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from B? CLICK HERE FOR SOLUTIONS
[CBSE SQP 2022] 33. Recent studies suggest that roughly 12% of the world population is left handed. Depending upon the parents, the chances of having a left handed child are as follows: 𝐴 : When both father and mother are left handed: Chances of left handed child is 24%. 𝐵 : When father is right handed and mother is left handed: Chances of left handed child is 22%. C : When father is left handed and mother is right handed: Chances of left handed child is 17%. 𝐷 : When both father and mother are right handed: Chances of left handed child is 9%. Assuming that 𝑃(𝐴) = 𝑃(𝐵) = 𝑃(𝐶) = 𝑃(𝐷) = 1 4 and 𝐿 denotes the event that child is left handed. (A) Find 𝑃(𝐿/𝐶) (B) Find 𝑃(𝐿 ‾/𝐴) (C) Find 𝑃(𝐴/𝐿) Find the probability that a randomly selected child is left handed given that exactly one of the parents is left handed. [CBSE 2023] Very Short & Short Qs (1 - 3 marks) 34. Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, find the mean of random variable. CLICK HERE FOR SOLUTIONS
[CBSE 2023] 35. The random variable 𝑋 has a probability distribution 𝑃(𝑋) of the following form, where ' 𝑘 ' is some real number: 𝑃(𝑋) = � 𝑘, if 𝑥 = 0 2𝑘, if 𝑥 = 1 3𝑘, if 𝑥 = 2 0, otherwise (A) Determine the value of 𝑘. (B) Find 𝑃(𝑋 < 2). (C) Find 𝑃(𝑋 > 2). [CBSE 2023] 36. A bag contains 1 red and 3 white balls. Find the probability distribution of the number of red balls if 2 balls are drawn at random from the bag one-by-one without replacement. [CBSE Term-2 SQP 2022] 37. There are two bags. Bag I contain 1 red and 3 white balls, and Bag II contains 3 red and 5 white balls. A bag is selected at random and a ball is drawn from it. Find the probability that the ball so drawn red in colour. [CBSE Term-2 2022] 38. A coin is tossed twice. The following table shows the probability distribution of number of tails: 𝑋 0 1 2 𝑃(𝑋) 𝐾 6𝐾 9𝐾 (A) Find the value of 𝐾. (B) Is the coin tossed biased or unbiased? Justify your answer. [CBSE Term-2 2022] 39. Three rotten apples are mixed with seven fresh apples. Find the probability distribution of the number of rotten apples, if three apples are drawn one by one with replacement. Find the mean of the number of rotten apples. [CBSE 2020] 40. In a shop 𝑋, 30 tins of ghee of type 𝐴 and 40 tins of ghee of type B which look alike, are kept for sale. While in shop 𝑌, similar 50 tins of ghee of type 𝐴 and 60 tins of ghee of type 𝐵 are there. One tin of ghee is purchased from one of the randomly selected shop and is found to be of type B. Find the probability that it is purchased from shop Y. [CBSE 2020] 41. Find the probability distribution of 𝑋, the number of heads in a simultaneous toss of two coins. [CBSE 2019] CLICK HERE FOR SOLUTIONS
42. Of the students in a school, it is know that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance? Is regularity required only school? Justify your answer. [CBSE 2017] 43. Three persons 𝐴, 𝐵 and 𝐶 apply for a job of manager in a private company. Chances of their selection (A, B, C) are in the ratio 1: 2: 4. The probabilities that 𝐴, 𝐵 and 𝐶 can introduce changes to improve profits of the company are 0.8,0.5 and 0.3 respectively. If the changes does not take place, find the probability that it is due to the appointment of C. [CBSE 2016] 44. Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spades. Hence, find the mean of the distribution. [CBSE 2015] 45. A bag A contains 4 black and 6 red balls and bag 𝐵 contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag 𝐴 is choosen, otherwise bag B. If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black. [CBSE 2015] Long Qs (4 – 5 marks) 46. There are two boxes, namely box-I and box-II. Box-I contains 3 red and 6 black balls. Box-II contains 5 red and 5 black balls. One of the two boxes is selected at random and a ball is drawn at random. The ball drawn is found to be red. Find the probability that this red ball comes out from box-II. [CBSE Term-2 2022] 47. An insurance company insured 3000 cyclists, 6000 scooter drivers, and 9000 car drivers. The probability of an accident involving a cyclist, a scooter driver, and a car driver are 0.3,0.05 and 0.02 respectively. One of the insured persons meets with an accident. What is the probability that he is a cyclist? [CBSE 2019] 48. A manufacturer has three machine operators A, B and C. The first operator A produces 1% of defective items, whereas the other two operators 𝐵 and 𝐶 produces 5% and 7% defective items respectively. 𝐴 is on the job for 50% of the time, B is on the job 30% of time and 𝐶 on the job for 20% of the time. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by 𝐴 ? [CBSE 2019] 49. Often it is taken that a truthful person commands more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six. Find the probability that it is actually a six. Do you also agree that the value of truthfulness leads to more respect in the society? CLICK HERE FOR SOLUTIONS
[CBSE 2017] 50. A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white? [CBSE 2016] 51. A class has 15 students whose ages are 14,17,15,14,21,17,19,20,16,18,20,17, 16,19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age 𝑋 of the selected student is recorded. What is the probability distribution of the random variable X? Find the mean of X. [CBSE 2014] 52. In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins loses. [CBSE 2014] 53. Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable X and hence find the mean of the distribution. [CBSE 2014] CLICK HERE FOR SOLUTIONS
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Class 12 MATHS CBSE PYQ Chapter Wise Topic Wise.pdf

  • 1.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 2.
    Relations And Functions PreviousYear Questions (Topic-wise) 1. RELATIONS AND ITS TYPES Objective Qs (1 mark) 1. Let R be a relation in the set N given by 𝑅 = {(𝑎, 𝑏): 𝑎 = 𝑏 − 2, 𝑏 > 6}, then: (a) (2,4) ∈ 𝑅 (b) (3,8) ∈ 𝑅 (c) (6,8) ∈ 𝑅 (d) (8,7) ∈ 𝑅 [CBSE Term-1 SQP 2021] 2. 𝐴 = {1,2,3,4}. A relation 𝑅 in the set 𝐴 is given by 𝑅 = {(1,1), (2,3), (3,2), (4,3), (3,4)}, then relation 𝑅 is: (a) reflexive (b) symmetric (c) transitive (d) equivalence [Delhi Gov. Term-1 SQP 2021] 3. Let set 𝑋 = {1,2,3} and a relation 𝑅 is defined in 𝑋 as 𝑅 = {(1,3), (2,2), (3,2)}, then minimum ordered pairs which should be added in relation R to make it reflexive and symmetric are: (a) {(1,1), (2,3), (1,4)} (b) {(5,3), (3,1), (1,2)} (c) {(1,1), (3,3), (3,1), (2,3)} (d) {(1,1), (3,3), (3,1), (1,2)} [CBSE Term-1 2021] 4. If 𝑅 = {(𝑥, 𝑦): 𝑥, 𝑦 ∈ 𝑍, 𝑥2 + 𝑦2 ≤ 4} is a relation in set 𝑍. Then domain of 𝑅 is: (a) {0,1,2} (b) {−2, −1,0,1,2} (c) [0, −1, −2] (d) {−1,0,1} [CBSE Term-1 2021] 5. A relation 𝑅 is defined on 𝑁. Which of the following is the reflexive relation? CH-1
  • 3.
    (a) R ={(𝑥, 𝑦): 𝑥 > 𝑦; 𝑥, 𝑦 ∈ N} (b) R = {(𝑥, 𝑦): 𝑥 + 𝑦 = 10; 𝑥, 𝑦 ∈ N} (c) R = {(𝑥, 𝑦): 𝑥𝑦 is the square number; 𝑥, 𝒚 ∈ N} (d) 𝑅 = {(𝑥, 𝑦): 𝑥 + 4𝑦 = 10; 𝑥, 𝑦 ∈ 𝑁} [CBSE Term-1 2021] 6. The number of equivalence relations in the set {1,2,3} containing the elements (1,2) and (2,1) is: (a) 0 (b) 1 (c) 2 (d) 3 [CBSE Term-1 2021] 7. A relation 𝑅 is defined on 𝑍 as: 𝑎R𝑏 if and only if 𝑎2 − 7𝑎𝑏 + 6𝑏2 = 0 Then, 𝑅 is: (a) reflexive and symmetric (b) symmetric but not reflexive (c) transitive but not reflexive (d) reflexive but not symmetric [CBSE Term-1 2021] Very Short & Short Qs (𝟏 - 3 marks) 8. Let 𝑅 be the equivalence relation on the set 𝑍 of integers given by 𝑅 = {(𝑎, 𝑏): 2 divides 𝑎 − 𝑏}. Write the equivalence class {0}. [CBSE 2021] 9. Check if the relation 𝑅 in the set 𝑅 of real numbers defined as 𝑅 = {(𝑎, 𝑏): 𝑎 < 𝑏} is: (A) symmetric; (B) transitive [CBSE 2020] 10. Let 𝐴 = {𝑥 ∈ 𝑍: 0 ≤ 𝑥 ≤ 12}. Show that R = {(𝑎, 𝑏): 𝑎, 𝑏 ∈ A, |𝑎 − 𝑏| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also, write the equivalence class [2]. [CBSE 2018] 11. If 𝑅 = {(𝑥, 𝑦): 2𝑥 + 𝑦 = 8} is a relation on 𝑁, write the range of 𝑅.
  • 4.
    [CBSE 2014] 12. Let𝑅 = {(𝑎, 𝑎3): 𝑎 is a prime number less than 5} be a relation. Find the range of 𝑅. [CBSE 2014] 13. Let 𝑅 be a relation defined on the set of natural numbers 𝑁 as 𝑅 = {(𝑥, 𝑦): 𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 and 2𝑥 + 𝑦 = 24}. Then, find the domain and range of the relation 𝑅. Also, find whether 𝑅 is an equivalence relation or not. [CBSE 2014] Long Qs (𝟒 - 5 marks) 14. Let N be the set of all natural numbers and R be a relation on N × N defined by (𝑎, 𝑏)R(𝑐, 𝑑) ⇔ 𝑎𝑑 = 𝑏𝑐 for all (𝑎, 𝑏), (𝑐, 𝑑) ∈ N × N. Show that R is an equivalence relation on N × N. Also, find the equivalence class of (2,6), i.e., [(2,6)]. [CBSE SQP 2023] 15. Given a non-empty set 𝑋, define the relation 𝑅 in 𝑃(𝑋) as follows: For 𝐴, 𝐵 ∈ 𝑃(𝑋), (𝐴, 𝐵) ∈ 𝑅 if 𝐴 ⊂ 𝐵. Prove that 𝑅 is reflexive, transitive and not symmetric. [CBSE SQP 2022] 16. Define the relation 𝑅 in the set 𝑁 × 𝑁 as follows: For (𝑎, 𝑏), (𝑐, 𝑑) ∈ N × N, (𝑎, 𝑏)R(𝑐, 𝑑) if 𝑎𝑑 = 𝑏𝑐. Prove that 𝑅 is an equivalence relation in N × N. [CBSE SQP 2022] 17. Show that the relation 𝑆 in the set 𝐴 = [𝑥 ∈ 𝑍 : 0 ≤ 𝐗 ≤ 12] given by 𝑆 = [(𝑎, 𝑏): 𝑎, 𝑏 ∈ 𝑍, |𝑎 − 𝑏| is divisible by 3 ] is an equivalence relation. [CBSE 2019] 18. Let 𝐍 denote the set of all natural numbers and R be a relation on N × N defined by (𝑎, 𝑏)R(𝑐, 𝑑) if 𝑎𝑑(𝑏 + 𝑐) = 𝑏𝑐(𝑎 + 𝑑). Prove that 𝑅 is an equivalence relation. [CBSE 2015] 19. Show that the relation 𝑅 in the set 𝐴 = {1,2,3,4,5} given by 𝑅 = {(𝑎, 𝑏): |𝑎 − 𝑏| is divisible by 2} is an equivalence relation. Write all the equivalence classes of 𝑅. [CBSE 2015]
  • 5.
    2. FUNCTIONS ANDITS TYPES Objective Qs (1 mark) 20. Let A = {1,2,3, … , 𝑛} and B = {𝑎, 𝑏}. Then the number of surjections from 𝐴 to 𝐵 is: (a) 𝑛 P2 (b) 2𝑛 − 2 (c) 2𝑛 − 1 (d) None of these [Delhi Gov. SQP 2022] 21. Let 𝑋 = {𝑥2 : 𝑥 ∈ 𝑁} and the function 𝑓: 𝑁 → 𝑋 is defined by 𝑓(𝑥) = 𝑥2 , 𝑥 ∈ 𝑁. Then this function is: (a) injective only (b) not bijective (c) surjective (d) bijective [CBSE Term-1 2021] 22. A function 𝑓: 𝑅 → 𝑅 defined by 𝑓(𝑥) = 2 + 𝑥2 is: (a) not one-one (b) one-one (c) not onto (d) neither one-one nor onto [CBSE Term-1 2021] 23. The function 𝑓: 𝑅 → 𝑅 defined by 𝑓(𝑥) = 4 + 3cos 𝑥 is: (a) bijective (b) one-one but not onto (c) onto but not one-one (d) neither one-one nor onto [CBSE Term-1 2021] 24. The number of functions defined from {1,2, 3,4,5} → {𝑎, 𝑏} which are one-one is: (a) 5 (b) 3 (c) 2 (d) 0 [CBSE Term-1 2021]
  • 6.
    25. Let 𝑓:𝑅 → 𝑅 be defined by 𝑓(𝑥) = 1 𝑥 , 𝑥 ∈ 𝑅. Then 𝑓 is: (a) one-one (b) onto (c) bijective (d) f is not defined [CBSE Term-1 2021] 26. Assertion (A): The relation 𝑓: {1,2,3,4} → {𝑥, 𝑦, 𝑧, 𝑝} defined by 𝑓 = {(1, 𝑥), (2, 𝑦), (3, 𝑧)} is a bijective function. Reason (R): The function 𝑓: {1,2,3} → {𝑥, 𝑦, 𝑧, 𝑝} such that 𝑓 = {(1, 𝑥), (2, 𝑦), (3, 𝑧)} is one-one. [CBSE SQP 2023] Very Short & Short Qs (1 - 3 marks) 27. Let 𝑓: N → N be defined as: 𝑓(𝑛) = � 𝑛+1 2 , if 𝑛 is odd 𝑛 2 , if 𝑛 is even for all 𝑛 ∈ 𝑁. Find whether the function 𝑓 is bijective or not. [Delhi Gov. SQP 2022] 28. Let 𝑓: 𝑋 → 𝑌 be a function. Define a relation 𝑅 on 𝑋 given by 𝑅 = {(𝑎, 𝑏): 𝑓(𝑎) = 𝑓(𝑏)}. Show that 𝑅 is an equivalence relation on 𝑋. [Delhi Gov. SQP 2022] 29. Prove that the function 𝑓 is surjective, where 𝑓: N → N such that 𝑓(𝑛) = � 𝑛 + 1 2 , if 𝑛 is 𝑛2 , if 𝑛 is even Is the function injective? Justify your answer. [CBSE SQP 2022] 30. Let 𝐴 = {1,2,3}, 𝐵 = {4,5,6,7} and 𝑓 = {(1,4), (2,5), (3,6)} be a function from A to 𝐵. State whether ' 𝑓 ' is one-one or not. [CBSE 2014]
  • 7.
    Long Qs (4- 5 marks) 31. Show that the function 𝑓: 𝑅 → {𝑥 ∈ 𝑅 : −1 < 𝑥 < 1} defined by 𝑓(𝑥) = 𝑥 1+|𝑥| , 𝑥 ∈ 𝑅 is one-one and onto function. [CBSE SQP 2023] 32. A function 𝑓: [−4,4] → [0,4] is given by 𝑓(𝑥) = √16 − 𝑥2. Show that 𝑓 is an onto function but not a one-one function. Further, find all possible values of ' 𝑎 ' for which 𝑓(𝑎) = √7. [CBSE 2023] 33. Show that the function 𝑓: (−∞, 0) → (−1,0) and 𝑓(𝑥) = 𝑥 1+|𝑥| , 𝑥 ∈ (−∞, 0) is one-one and onto. [CBSE 2020]
  • 8.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLUTIONS (FROM 2014 TO 2023) PYQ
  • 9.
    SOLUTIONS 1. RELATIONS ANDITS TYPES 1. (c) (6,8) ∈ 𝑅 Explanation: Given, 𝑎 = 𝑏 − 2 and 𝑏 > 6 ⇒ (6,8) ∈ R 2. (b) symmetric Explanation: We have, 𝑅 = {(1,1), (2,3), (3,2), (4,3), (3,4)} ∵ (2,2), (3,3), (4,4) ∉ 𝑅 ∴ 𝑅 is not reflexive. For (2,3) ∈ 𝑅, we have (3,2) ∈ 𝑅 Similarly for (4,3) ∈ 𝑅, we have (3,4) ∈ 𝑅. ∴ 𝑅 is symmetric. For (2,3) ∈ 𝑅 and (3,2) ∈ 𝑅, we have (2,2) ∉ 𝑅 ∴ 𝑅 is not transitive ∵ 𝑅 is not reflexive and transitive so it is not an equivalence relation. NOTE: A relation 𝑅 in a set 𝐴 is called (i) reflexive, if (𝑎, 𝑎) ∈ 𝑅, for every 𝑎 ∈ 𝐴, (ii) symmetric, if (𝑎1, 𝑎2) ∈ 𝑅 implies that (𝑎2, 𝑎1) ∈ 𝑅, For all 𝑎1, 𝑎2 ∈ 𝐴. (iii) transitive, if (𝑎1, 𝑎2) ∈ 𝑅 and (𝑎2, 𝑎3) ∈ 𝑅 implies that (𝑎1, 𝑎3) ∈ 𝑅 for all 𝑎1, 𝑎2, 𝑎3 ∈ 𝑅. 3. (c) {(1,1), (3,3), (3,1), (2,3)} Explanation: The ordered pairs to be added in R are: (1,1), (3,3) {needed to make R reflexive} CH-1
  • 10.
    (3,1), (2,3) {neededto make 𝑅 symmetric} So, {(1,1)(3,3), (3,1), (2,3)} are required. 4. (b) {−2, −1,0,1,2} Explanation: Given, 𝑥2 + 𝑦2 ≤ 4 ⇒ 𝑦2 ≤ 4 − 𝑥2 ⇒ 𝑦 ≤ �4 − 𝑥2 For domain: 4 − 𝑥2 ≥ 0 ⇒ 𝑥2 ≤ 4 ∴ −2 ≤ 𝑥 ≤ 2 So, Domain = {−2, −1,0,1,2} 5. (c) 𝑅 = {(𝑥, 𝑦) : 𝑥𝑦 is the square number; 𝑥, 𝑦 ∈ 𝑁} Explanation: When 𝑥 ∈ 𝑁, 𝑥2 is a square number So, (𝑥, 𝑥) ∈ R for all 𝑥 ∈ N. Therefore, R is reflexive. 6. (c) 2 Explanation: Total possible pairs = (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) Reflexive means (𝑎, 𝑎) should be in relation. So, (1,1), (2,2), (3,3) should be in relation. Symmetric means if (𝑎, 𝑏) is in relation, then (𝑏, 𝑎) should be in relation. So, since (1,2) is in relation, (2,1) should also be in relation. Transitive means if (𝑎, 𝑏) is in relation and (𝑏, 𝑐) is in relation, then (𝑎, 𝑐) is in relation. So, if (1,2) is in relation and (2,1) is in relation, then (1,1) should be in relation. Relation R1 = {(1,2), (2,1), (1,1), (2,2), (3,3)} Total possible pairs = (1,1), (1,2), (1,3), (2,1), (2, 2), (2,3), (3,1), (3,2), (3,3) So, smallest relation is 𝑅1 = {(1,2), (2,1), (1,1), (2,2), (3,3)}. If we add (2,3) then we have to add (3,2) also, as it is symmetric but, as (1,2) and (2,3) are there, we need to add (1,3) also, as it is transitive.
  • 11.
    As we areadding (1,3) we should add (3,1) also, as it is symmetric. Relation 𝑅2 = {(1,2), (2,1), (1,1), (2,2), (3,3), (2,3), (3,2), (1,3), (3,1)} Hence, only two possible relations are there which are equivalence. 7. (d) reflexive but not symmetric Explanation: We have, 𝑎2 − 7𝑎𝑎 + 6𝑎2 = 7𝑎2 − 7𝑎2 = 0 Therefore, 𝑎 Ra for all 𝑎 in Z 𝑆𝑜, 𝑅 is reflexive Let 𝑎𝑅𝑏, then 𝑎2 − 7𝑎𝑏 + 6𝑏2 = 0 Consider 𝑏2 − 7𝑏𝑎 + 6𝑎2 = 𝑏2 − (𝑎2 + 6𝑏2) + 6𝑎2 = 5𝑎2 − 5𝑏2 So, 𝑅 is not symmetric. 8. 𝑅 = {(𝑎, 𝑏): 2 divides (𝑎 − 𝑏)} ⇒ (𝑎 − 𝑏) is a multiple of 2. To find equivalence class 0, put 𝑏 = 0 So, (𝑎 − 0) is a multiple of 2 ⇒ 𝑎 is a multiple of 2 So, in set 𝑍 of integers, all the multiple of 2 will come in equivalence class {0} Hence, equivalence class {0} = {2𝑥} where 𝑥 = integer (Z). NOTE → An equivalence class of 𝑎 is denoted as [𝑎] = {𝑥 ∈ 𝐴 : (𝑎, 𝑥) ∈ 𝑅}. This comprises all of 𝐴 's elements related to letter ' 𝑎 '. 9. 𝑅 = {(𝑎, 𝑏): 𝑎 < 𝑏} (A) Checking for symmetric, if (𝑎, 𝑏) ∈ R such that 𝑎 < 𝑏 then, (𝑏, 𝑎) ∈ 𝑅 not possible (2,3) ∈ R but, (3,2) ∈ R ∴ Relation 𝑅 is not symmetric. (B) Checking for transitive,
  • 12.
    if (𝑎, 𝑏)∈ 𝑅 and (𝑏, 𝑐) ∈ 𝑅 such that 𝑎 < 𝑏 and 𝑏 < 𝑐 then clearly, 𝑎 < 𝑐 i.e., (𝑎, 𝑐) ∈ R ∴ Relation R is transitive. 10. For reflexive: Let 𝑎 ∈ 𝑍, then |𝑎 − 𝑎| = 0, which is divisible by 4. So, (𝑎, 𝑎) ∈ 𝑅. Thus, 𝑅 is a reflexive relation. For symmetric: Let 𝑎, 𝑏 ∈ 𝑍 such that |𝑎 − 𝑏| is divisible by 4. Then |𝑏 − 𝑎| is also divisible by 4 [∵ |𝑏 − 𝑎| = |𝑎 − 𝑏|] So, (𝑎, 𝑏) ∈ 𝑅 ⇒ (𝑏, 𝑎) ∈ 𝑅. Thus, 𝑅 is a symmetric relation. For transitive: Let 𝑎, 𝑏, 𝑐 ∈ 𝑍 and (𝑎, 𝑏) ∈ 𝑅 and (𝑏, 𝑐) ∈ 𝑅. Since (𝑎, 𝑏) ∈ 𝐑 and (𝑏, 𝑐) ∈ 𝑅 Therefore, |𝑎 − 𝑏| = 4𝑘 for some 𝑘 ∈ 𝑍 ⇒ (𝑎 − 𝑏) = ±4𝑘 and |𝑏 − 𝑐| = 4𝑙 for some 𝑙 ∈ 𝑍 ⇒ (𝑏 − 𝑐) = ±4𝑙 Now, (𝑎 − 𝑐) = (𝑎 − 𝑏) + (𝑏 − 𝑐) = ±4𝑘 ± 4𝑙 = 4(±𝑘 ± 𝑙), which is divisible by 4. So, (𝑎, 𝑏) ∈ R and (𝑏, 𝑐) ∈ R ⇒ (𝑎, 𝑐) ∈ 𝑅. Thus, 𝑅 is transitive. Since, 𝑅 is reflexive, symmetric and transitive, hence, it is an equivalence relation. Set of elements related to 1 is {1,5,9} Set of elements related to 2 is {2,6,10} So, equivalence class of [2] is {2,6,10}. 11. 𝑅 = {(𝑥, 𝑦): 2𝑥 + 𝑦 = 8} is a relation on 𝑁.
  • 13.
    Therefore, 𝑅 ={(3,2), (2,4), (1,6)} So, Range = {2,4,6}. 12. Relation 𝑅 = {(𝑎, 𝑎3): 𝑎 is a prime number less than 5}. Therefore, 𝑅 = {(2,8), (3,27)}, ⇒ Range = {8,27} 13. We have, 2𝑥 + 𝑦 = 24 ⇒ 𝑦 = 24 − 2𝑥 For 𝑥 = 1, 𝑦 = 24 − 2 × 1 = 24 − 2 = 22 For 𝑥 = 2, 𝑦 = 24 − 2 × 2 = 24 − 4 = 20 For 𝑥 = 3, 𝑦 = 24 − 2 × 3 = 24 − 6 = 18 For 𝑥 = 4, 𝑦 = 24 − 2 × 4 = 24 − 8 = 16 For 𝑥 = 5, 𝑦 = 24 − 2 × 5 = 24 − 10 = 14 For 𝑥 = 6, 𝑦 = 24 − 2 × 6 = 24 − 12 = 12 For 𝑥 = 7, 𝑦 = 24 − 2 × 7 = 24 − 14 = 10 For 𝑥 = 8, 𝑦 = 24 − 2 × 8 = 24 − 16 = 8 For 𝑥 = 9, 𝑦 = 24 − 2 × 9 = 24 − 18 = 6 For 𝑥 = 10, 𝑦 = 24 − 2 × 10 = 24 − 20 = 4 For 𝑥 = 11, 𝑦 = 24 − 2 × 11 = 24 − 22 = 2 In ordered pair, 𝑅 = {(1,22), (2,20), (3,18), (4,16), (5,14), (6,12), (7,10), (8,8), (9,6), (10,4), (11,2)} ∴ Domain of 𝑅 = {1,2,3,4,5,6,7,8,9,10,11} Range of 𝑅 = {22,20,18,16,14,12,10,8,6,4,2} Here, 1 ∈ 𝑁 but (1,1) ∉ 𝑅, hence 𝑅 is not reflexive. Hence, R is not an equivalence relation.
  • 14.
    NOTE: A relation issaid to be equivalence if it is reflexive, symmetric and transitive. 14. Let (𝑎, 𝑏) be an arbitrary element of 𝑁 × 𝑁. Then, (𝑎, 𝑏) ∈ 𝑁 × 𝑁 and 𝑎, 𝑏 ∈ 𝑁 We have, 𝑎𝑏 = 𝑏𝑎; ( As 𝑎, 𝑏 ∈ 𝑁 and multiplication is commutative on N ) ⇒ (𝑎, 𝑏)R(𝑎, 𝑏), according to the definition of the relation R on N × N. Thus, (𝑎, 𝑏)𝑅(𝑎, 𝑏), ∀(𝑎, 𝑏) ∈ N × N. So, 𝑅 is reflexive relation on N × N. Let (𝑎, 𝑏), (𝑐, 𝑑) be arbitrary elements of N × N such that (𝑎, 𝑏)𝑅(𝑐, 𝑑). Then, (𝑎, 𝑏)R(𝑐, 𝑑) ⇒ 𝑎𝑑 = 𝑏𝑐 ⇒ 𝑏𝑐 = 𝑎𝑑 (changing LHS and RHS) ⇒ 𝑐𝑏 = 𝑑𝑎; (As 𝑎, 𝑏, 𝑐, 𝑑 ∈ N and multiplication is commutative on N ) ⇒ (𝑐, 𝑑)𝑅(𝑎, 𝑏); according to the definition of the relation R on N × N Thus, (𝑎, 𝑏)R(𝑐, 𝑑) ⇒ (𝑐, 𝑑)𝑅(𝑎, 𝑏) So, R is symmetric relation on N × N. Let (𝑎, 𝑏), (𝑐, 𝑑), (𝑒, 𝑓) be arbitrary elements of N × N such that (𝑎, 𝑏)R(𝑐, 𝑑) and (𝑐, 𝑑)R(𝑒, 𝑓). Then (𝑎, 𝑏)R(𝑐, 𝑑) ⇒ 𝑎𝑑 = 𝑏𝑐 and (𝑐, 𝑑)𝑅(𝑒, 𝑓) ⇒ 𝑐𝑓 = 𝑑𝑒 ⇒ (𝑎𝑑)(𝑐𝑓) = (𝑏𝑐)(𝑑𝑒) ⇒ 𝑎𝑓 = 𝑏𝑒 ⇒ (𝑎, 𝑏)R(𝑒, 𝑓); (according to the definition of the relation R on N × N ) Thus, (𝑎, 𝑏)R(𝑐, 𝑑) and (𝑐, 𝑑)R(𝑒, 𝑓)
  • 15.
    ⇒ (𝑎, 𝑏)R(𝑒,𝑓) So, 𝑅 is transitive relation on N × N. As the relation 𝑅 is reflexive, symmetric and transitive so, it is equivalence relation on N × N. [(2,6)] = {(𝑥, 𝑦) ∈ N × N: (𝑥, 𝑦)𝑅(2,6)} = {(𝑥, 𝑦) ∈ N × N: 3𝑥 = 𝑦} = {(𝑥, 3𝑥): 𝑥 ∈ N} = {(1,3), (2,6), (3,9), … … . . } 15. Let 𝐴 ∈ 𝑃(𝑋). Then 𝐴 ⊂ 𝐴 ⇒ (𝐴, 𝐴) ∈ 𝑅 Hence, 𝑅 is reflexive. Let 𝐴, 𝐵, 𝐶, ∈ 𝑃(𝑋) such that (𝐴, 𝐵), (𝐵, 𝐶) ∈ 𝑅 ⇒ 𝐴 ⊂ 𝐵, 𝐵, ⊂ 𝐶 ⇒ 𝐴 ⊂ 𝐶 ⇒ (𝐴, 𝐶) ∈ 𝑅 Hence, 𝑅 is transitive. 𝜙, 𝑋 ∈ 𝑃(𝑋) such that 𝜙 ⊂ 𝑋. Hence, (𝜙, 𝑋) ∈ 𝑅. But X ⊄ 𝜙 which implies that (𝑋, 𝜙) ∉ 𝑅. Thus, 𝑅 is not symmetric. 16. Let (𝑎, 𝑏) ∈ 𝑁 × 𝑁. Then we have, 𝑎𝑏 = 𝑏𝑎 (by commutative property of multiplication of natural numbers) ⇒ (𝑎, 𝑏)R(𝑎, 𝑏) Hence, 𝑅 is reflexive. Let (𝑎, 𝑏), (𝑐, 𝑑) ∈ 𝑁 × 𝑁 such that (𝑎, 𝑏)𝑅(𝑐, 𝑑). Then 𝑎𝑑 = 𝑏𝑐 ⇒ 𝑐𝑏 = 𝑑𝑎 (by commutative property of multiplication of natural numbers
  • 16.
    ⇒ (𝑐, 𝑑)R(𝑎,𝑏) Hence, 𝑅 is symmetric. Let (𝑎, 𝑏), (𝑐, 𝑑), (𝑒, 𝑓) ∈ 𝑁 × 𝑁 such that (𝑎, 𝑏)R(𝑐, 𝑑) and (𝑐, 𝑑)R(𝑒, 𝑓). Then, 𝑎𝑑 = 𝑏𝑐, 𝑐𝑓 = 𝑑𝑒 ⇒ 𝑎𝑑𝑐𝑓 = 𝑏𝑐𝑑𝑒 ⇒ 𝑎𝑓 = 𝑏𝑒 ⇒ (𝑎, 𝑏)R(𝑒, 𝑓) Hence, R is transitive. Since, 𝑅 is reflexive, symmetric and transitive, 𝑅 is an equivalence relation on 𝑁 × 𝑁. 17. A = {𝑥 ∈ 𝑍: 0 ≤ 𝑥 ≤ 12} = {1,2,3,4,5,6,7,8,9, 10,11,12} 𝑅 = {(𝑎, 𝑏): |𝑎 − 𝑏| is divisible by 3} For any element 𝑎 ∈ A, we have (𝑎, 𝑎) ∈ R as |𝑎 − 𝑎| = 0 is divisible by 3. ∴ R is reflexive. Now, let (𝑎, 𝑏) ∈ 𝑅 ⇒ |𝑎 − 𝑏| is divisible 3. ⇒∣ 𝑏 − 𝑎)| = |𝑎 − 𝑏 ∣ is divisible by 3 ⇒ (𝑏, 𝑎) ∈ R ∴ R is symmetric. Now, let (𝑎, 𝑏), (𝑏, 𝑐) ∈ 𝑅. ⇒ |𝑎 − 𝑏| is divisible by 3 and |𝑏 − 𝑐| is divisible by 3. ⇒ ±(𝑎 − 𝑏) is divisible by 3 and ±(𝑏 − 𝑐) is divisible by 3. ⇒ (𝑎 − 𝑐) = (𝑎 − 𝑏) + (𝑏 − 𝑐) is divisible by 3. and −(𝑎 − 𝑐) = −{(𝑎 − 𝑏) + (𝑏 − 𝑐)} is divisible by 3. ⇒ |𝑎 − 𝑐| is divisible by 3. ⇒ (𝑎, 𝑐) ∈ 𝑅
  • 17.
    ∴ R istransitive. Hence, R is an equivalence relation. 18. Reflexive: Let (𝑎, 𝑏) ∈ 𝑁 × 𝑁. ∵ 𝑎𝑏(𝑏 + 𝑎) = 𝑏𝑎(𝑎 + 𝑏) ⇒ (𝑎, 𝑏)R(𝑎, 𝑏) ⇒ 𝑅 is reflexive. Symmetric: For (𝑎, 𝑏), (𝑐, 𝑑) ∈ 𝑁 × 𝑁 such that (𝑎, 𝑏)𝑅(𝑐, 𝑑). ∴ 𝑎𝑑(𝑏 + 𝑐) = 𝑏𝑐(𝑎 + 𝑑) or, 𝑏𝑐(𝑎 + 𝑑) = 𝑎𝑑(𝑏 + 𝑐) or, 𝑐𝑏(𝑑 + 𝑎) = 𝑑𝑎(𝑐 + 𝑏) ⇒ (𝑐, 𝑑)𝑅(𝑎, 𝑏) ⇒ 𝑅 is symmetric. Transitive: Let (𝑎, 𝑏), (𝑐, 𝑑), (𝑒, 𝑓) ∈ 𝑁 × 𝑁 such that (𝑎, 𝑏)R(𝑐, 𝑑) and (𝑐, 𝑑)R(𝑒, 𝑓). ∴ 𝑎𝑑(𝑏 + 𝑐) = 𝑏𝑐(𝑎 + 𝑑) and 𝑐𝑓(𝑑 + 𝑒) = 𝑑𝑒(𝑐 + 𝑓) ⇒ 𝑎𝑑𝑏 + 𝑎𝑑𝑐 = 𝑏𝑐𝑎 + 𝑏𝑐𝑑 ….(i) and 𝑐𝑓𝑑 + 𝑐𝑓𝑒 = 𝑑𝑒𝑐 + 𝑑𝑒𝑓 ….(ii) Multiplying (i) by 𝑒𝑓 and (ii) by 𝑎𝑏 and then adding them, we get 𝑎𝑑𝑏𝑒𝑓 + 𝑎𝑑𝑐𝑒𝑓 + 𝑐𝑓𝑑𝑎𝑏 + 𝑐𝑓𝑒𝑎𝑏 = 𝑏𝑐𝑎𝑒𝑓 + 𝑏𝑐𝑑𝑒𝑓 + 𝑑𝑒𝑐𝑎𝑏 + 𝑑𝑒𝑓𝑎𝑏 ⇒ 𝑎𝑑𝑐𝑒𝑓 + 𝑎𝑑𝑐𝑓𝑏 = 𝑏𝑐𝑑𝑒𝑎 + 𝑏𝑐𝑑𝑒𝑓 ⇒ adcf (𝑒 + 𝑏) = 𝑏𝑐𝑑𝑒(𝑎 + 𝑓) ⇒ 𝑎𝑓(𝑏 + 𝑒) = 𝑏𝑒(𝑎 + 𝑓) ⇒ (𝑎, 𝑏)R(𝑒, 𝑓) ⇒ 𝑅 is transitive. Hence, 𝑅 is an equivalence relation. 19. Given 𝑅 = {(𝑎, 𝑏): |𝑎 − 𝑏| is divisible by 2}
  • 18.
    and 𝐴 ={1,2,3,4,5} 𝑅 = {(1,1), (2,2), (3,3), (4,4), (5,5)(1,3), (1,5), (2,4), (3,5), (3,1), (5,1), (4,2), (5,3)} Reflexive: ∀𝑎 ∈ A, (𝑎, 𝑎) ∈ R, ∴ R is reflexive ∴ [As {(1,1), (2,2), (3,3), (5,5)} ∈ 𝑅] Symmetric ∀(𝑎, 𝑏) ∈ R, (𝑏, 𝑎) ∈ R, ∴ R is symmetric. [ As {(1,3), (1,5), (2,4), (3,5), (3,1), (5,1), (4,2), (5,3)} ∈ 𝑅] Transitive ∀(𝑎, 𝑏), (𝑏, 𝑐) ∈ 𝑅, (𝑎, 𝑐) ∈ R ∴ R is transitive. [As (1,3), (3,1) ∈ 𝑅 ⇒ (1,1) ∈ 𝑅 and similarly others] ∴ R is an equivalence relation. Equivalence classes are and [1] = {1,3,5} [2] = {2,4} [3] = {1,3,5} [4] = {2,4} [5] = {1,3,5} 2. FUNCTIONS AND ITS TYPES 20. (b) 2𝑛 − 2 Explanation: Given that, 𝐴 = {1,2,3, … 𝑛} and 𝐵 = {𝑎, 𝑏} If function is surjective then its range must be set 𝐵 = {𝑎, 𝑏}
  • 19.
    Now number ofonto functions = Number of ways ' 𝑛 ' distinct objects can be distributed in two boxes ' 𝑎 ' and ' 𝑏 ' in such a way that no box remains empty. Now for each object there are two options, either it is put in box 'a' or in box 'b' So total number of ways of ' 𝑛 ' different objects put in two boxes = 2 × 2 × 2 … 𝑛 times = 2𝑛 But in one case all the objects are put in box ' 𝑎 ' and in one case all the objects are put in box 'b'. So, number of surjective functions = 2𝑛 − 2 21. (d) bijective Explanation: Given, 𝑓(𝑥) = 𝑥2 For injective: Let 𝑓(𝑥1) = 𝑓(𝑥2) ⇒ 𝑥1 2 = 𝑥2 2 ⇒ 𝑥1 = 𝑥2 ( ∵ - ve rejected) So, it is injective. For subjective: Range = 𝑋 Co-domain = 𝑋 So, range = co-domain So, it is subjective. Hence, it is bijective. 22. (d) neither one-one nor onto Explanation: For one-one: 𝑓(𝑥1) = 𝑓(𝑥2) ⇒ 2 + 𝑥1 2 = 2 + 𝑥2 2 ⇒ 𝑥1 2 = 𝑥2 2 ⇒ 𝑥1 = ±𝑥2 [∵ 𝑥1, 𝑥2 ∈ 𝑅] So, it is not one-one. For onto:
  • 20.
    Range = Positivereal numbers Co-domain = R ∴ Range ≠ co-domain So, it is not onto. Hence, 𝑓(𝑥) is neither one-one nor onto. 23. (d) neither one-one nor onto Explanation: Given: 𝑓(𝑥) = 4 + 3cos 𝑥 Since, cos 𝜋 2 = cos �− 𝜋 2 � ⇒ 4 + 3cos 𝜋 2 = 4 + cos �− 𝜋 2 � ⇒ 𝑓 � 𝜋 2 � = 𝑓 �− 𝜋 2 � But 𝜋 2 ≠ − 𝜋 2 So, 𝑓 is not one-one. Range of cos 𝑥 is [−1,1] ⇒ −1 ≤ cos 𝑥 ≤ 1 ⇒ −3 ≤ 3cos 𝑥 ≤ 3 ⇒ 1 ≤ 4 + 3cos 𝑥 ≤ 7 ⇒ 1 ≤ 𝑓(𝑥) ≤ 7 So, the range of 𝑓 is [1,7] Thus, 𝑓 is not onto. Hence, 𝑓 is neither one-one nor onto. 24. (d) 0 Explanation: If we consider one-one function, only two elements of the set {1,2,3,4,5} can have images. Therefore, there can't be a one-one function from {1,2,3,4,5} → {𝑎, 𝑏} Hence, the number of one-one functions is 0. 25. (d) fis not defined
  • 21.
    Explanation: We have,𝑓(𝑥) = 1 𝑥 , ∀𝑥 ∈ 𝑅 For 𝑥 = 0, 𝑓(𝑥) is not defined. Hence, 𝑓(𝑥) is not a defined function. 26. (d) (A) is false but (R) is true. Explanation: Assertion is false. As element 4 has no image under 𝑓, so relation 𝑓 is not a function. Reason is true. The given function 𝑓: {1, 2,3} → {𝑥, 𝑦, 𝑧, 𝑝} is one - one, as for each 𝑎 ∈ {1,2,3}, there is a different image in {𝑥, 𝑦, 𝑧, 𝑝} under 𝑓. 27. The given function is 𝑓: 𝑁 → 𝑁 such that 𝑓(𝑛) = � 𝑛 + 1 2 , if 𝑛 is odd 𝑛 2 , if 𝑛 is even for all 𝑛 ∈ N We shall verify whether 𝑓(𝑥) is one-one and onto. One-One: From the definition of 𝑓(𝑛) 𝑓(1) = 1+1 2 = 1 and 𝑓(2) = 2 2 = 1 𝑓(𝑛) is not an one-one function because at two distinct values from domain (N), 𝑓(𝑛) has same image. Onto: For onto function, we check whether Range of 𝑓(𝑛) = Co-domain of 𝑓(𝑛) Now, if 𝑛 is an odd natural number, then (2𝑛 − 1) is also an odd natural number. Now, 𝑓(2𝑛 − 1) = 2𝑛−1+1 2 = 𝑛 … . (𝑖) Again, if 𝑛 is an even natural number, then 2𝑛 is also an even natural number. Then, 𝑓(2𝑛) = 2𝑛 2 = 𝑛 … . (𝑖𝑖)#(𝑖𝑖) From equations, (i) and (ii), we observe that for each 𝑛 (whether even or odd), there exists its pre- image in N. i.e., Range of 𝑓(𝑛) = Co-domain of 𝑓(𝑛).
  • 22.
    Hence, 𝑓 isonto. Since, 𝑓(𝑛) is onto but not one-one, it is not a bijective function. 28. The given function is 𝑓: 𝑋 → 𝑌 and relation on 𝑋 is 𝑅 = {(𝑎, 𝑏): 𝑓(𝑎) = 𝑓(𝑏)} Reflexive: Since, for every 𝑥 ∈ 𝑋, we have 𝑓(𝑥) = 𝑓(𝑥) ⇒ (𝑥, 𝑥) ∈ 𝑅, ∀𝑥 ∈ 𝑋 Therefore, 𝑅 is reflexive. Symmetric: Let (𝑥, 𝑦) ∈ 𝑅 Then, 𝑓(𝑥) = 𝑓(𝑦) ⇒ 𝑓(𝑦) = 𝑓(𝑥) ⇒ (𝑦, 𝑥) ∈ 𝑅 Thus, (𝑥, 𝑦) ∈ 𝑅 ⇒ (𝑦, 𝑥) ∈ 𝑅, ∀𝑥, 𝑦 ∈ 𝑋 Therefore, 𝑅 is symmetric. Transitive: Let 𝑥, 𝑦, 𝑧 ∈ 𝑋 such that (𝑥, 𝑦) ∈ 𝑅 and (𝑦, 𝑧) ∈ 𝑅 Then, 𝑓(𝑥) = 𝑓(𝑦) … (𝑖) And 𝑓(𝑦) = 𝑓(𝑧) … . (𝑖𝑖) From eqs. (i) and (ii), we get 𝑓(𝑥) = 𝑓(𝑧) ⇒ (𝑥, 𝑧) ∈ 𝑅 Thus, (𝑥, 𝑦) ∈ 𝑅 and (𝑦, 𝑧) ∈ 𝑅 ⇒ (𝑥, 𝑧) ∈ 𝑅, ∀𝑥, 𝑦, 𝑧 ∈ 𝑅 Therefore, 𝑅 is transitive.
  • 23.
    Since, 𝑅 isreflexive symmetric and transitive, so it is an equivalence relation. 29. Let 𝑦 ∈ 𝑁 (co-domain). Then ∃2𝑦 ∈ 𝑁 (domain) such that 𝑓(2𝑦) = 2𝑦 2 = 𝑦. Hence, 𝑓 is surjective. 1,2 ∈ 𝑁 (domain) such that 𝑓(1) = 1 = 𝑓(2) Hence, 𝑓 is not injective. 30. Given that 𝐴 = {1,2,3}, 𝐵 = {4,5,6,7} Now, 𝑓: 𝐴 → 𝐵 is defined as 𝑓 = {(1,4), (2,5), (3,6)}. 𝑓(1) = 4, 𝑓(2) = 5, 𝑓(3) = 6, so 𝑓 is one-one. 31. We have, 𝑓(𝑥) = � 𝑥 1+𝑥 , if 𝑥 ≥ 0 𝑥 1−𝑥 , if 𝑥 < 0 Now, we consider the following cases Case 1: when 𝑥 ≥ 0, we have 𝑓(𝑥) = 𝑥 1+𝑥 Injectivity: let 𝑥, 𝑦 ∈ 𝑅+ ∪ {0} such that 𝑓(𝑥) = 𝑓(𝑦), then ⇒ 𝑥 1 + 𝑥 = 𝑦 1 + 𝑦 ⇒ 𝑥 + 𝑥𝑦 = 𝑦 + 𝑥𝑦 ⇒ 𝑥 = 𝑦 So, 𝑓 is injective function. Surjectivity: when 𝑥 ≥ 0, we have 𝑓(𝑥) = 𝑥 1+𝑥 ≥ 0 and 𝑓(𝑥) = 1 − 1 1+𝑥 < 1, as 𝑥 ≥ 0 Let 𝑦 ∈ [0,1), thus for each 𝑦 ∈ [0,1) there exists 𝑥 = 𝑦 1−𝑦 ≥ 0 such that 𝑓(𝑥) = 𝑦 1−𝑦 1+ 𝑦 1−𝑦 = 𝑦. So, 𝑓 is onto function on [0, ∞) to [0,1). Case 2: when 𝑥 < 0, we have 𝑓(𝑥) = 𝑥 1−𝑥 Injectivity: Let 𝑥, 𝑦 ∈ 𝑅− i.e., 𝑥, 𝑦 < 0, such that 𝑓(𝑥) = 𝑓(𝑦), then ⇒ 𝑥 1 − 𝑥 = 𝑦 1 − 𝑦 ⇒ 𝑥 − 𝑥𝑦 = 𝑦 − 𝑥𝑦 ⇒ 𝑥 = 𝑦 So, 𝑓 is injective function. Surjectivity: 𝑥 < 0,
  • 24.
    we have 𝑓(𝑥)= 𝑥 1−𝑥 < 0 also, 𝑓(𝑥) = 𝑥 1 − 𝑥 = −1 + 1 1 − 𝑥 > −1 −1 < 𝑓(𝑥) < 0. Let 𝑦 ∈ (−1,0) be an arbitrary real number and there exists 𝑥 = 𝑦 1+𝑦 < 0 such that, 𝑓(𝑥) = 𝑓 � 𝑦 1 + 𝑦 � = 𝑦 1 + 𝑦 1 − 𝑦 1 + 𝑦 = 𝑦 So, for 𝑦 ∈ (−1,0), there exists 𝑥 = 𝑦 1+𝑦 < 0 such that 𝑓(𝑥) = 𝑦. Hence, 𝑓 is onto function on (−∞, 0) to (−1,0). Case 3: (Injectivity): Let 𝑥 > 0 and 𝑦 < 0 such that 𝑓(𝑥) = 𝑓(𝑦) ⇒ 𝑥 1 + 𝑥 = 𝑦 1 − 𝑦 ⇒ 𝑥 − 𝑥𝑦 = 𝑦 + 𝑥𝑦 ⇒ 𝑥 − 𝑦 = 2𝑥𝑦, Here LHS > 0 but RHS < 0, which is inadmissible. Hence, 𝑓(𝑥) ≠ 𝑓(𝑦) when 𝑥 ≠ 𝑦. Hence, 𝑓 is one-one and onto function. 32. 𝑓(𝑥) = �16 − 𝑥2 for 𝑥 = 2, 𝑓(𝑥) = √12 for 𝑥 = −2, 𝑓(𝑥) = √12 Since, for 𝑥 = 2 and -2, the function has same image
  • 25.
    ∴ The givenfunction is not one-one. Let 𝑦 ∈ [0,4] ∴ 𝑦 ≥ 0 𝑦 = �16 − 𝑥2 𝑦2 = 16 − 𝑥2 𝑥 = �16 − 𝑦2 For 𝑥 ∈ 𝑅, 16 − 𝑦2 ≥ 0 (4 − 𝑦)(4 + 𝑦) ≥ 0 ⇒ Either (4 − 𝑦) ≥ 0 and (4 + 𝑦) ≥ 0 or (4 − 𝑦) < 0 and (4 + 𝑦) < 0. ∴ For every 𝑦 ∈ [0,4]∃𝑥 ∈ [−4,4] such that 𝑦 = 𝑓(𝑥) ∴ The given function is onto 𝑓(𝑎) = √7 �16 − 𝑎2 = √7 Squaring on both sides: 16 − 𝑎2 = 7 𝑎2 = 9 𝑎 = ±3 33. Given, 𝑓(𝑥) = 𝑥 1 + |𝑥| , 𝑥 ∈ (−∞, 0) = 𝑥 1 − 𝑥 [∵ 𝑥 ∈ (−∞, 0), |𝑥| = −𝑥] For one-one: Let 𝑓(𝑥1) = 𝑓(𝑥2), 𝑥1, 𝑥2 ∈ (−∞, 0)
  • 26.
    ⇒ 𝑥1 1 − 𝑥1 = 𝑥2 1− 𝑥2 ⇒ 𝑥1(1 − 𝑥2) = 𝑥2(1 − 𝑥1) ⇒ 𝑥1 − 𝑥1𝑥2 = 𝑥2 − 𝑥1𝑥2 ⇒ 𝑥1 = 𝑥2 Hence, if 𝑓(𝑥1) = 𝑓(𝑥2), then 𝑥1 = 𝑥2 ∴ f is one-one For onto: Let 𝑓(𝑥) = 𝑦 ⇒ 𝑦 = 𝑥 1 − 𝑥 ⇒ 𝑦(1 − 𝑥) = 𝑥 ⇒ 𝑦 − 𝑥𝑦 = 𝑥 ⇒ 𝑥 + 𝑥𝑦 = 𝑦 ⇒ 𝑥(1 + 𝑦) = 𝑦 ⇒ 𝑥 = 𝑦 1 + 𝑦 So, x is defined for all values of y. ∴ 𝑓 is onto.
  • 27.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 28.
    Inverse Trigonometric Functions 1.INVERSE TRIGONOMETRIC FUNCTIONS Objective Qs (1 mark) 1. The value of sin �sin−1 � 1 2 � + cos−1 � 1 2 �� is: (a) 1 (b) 2 (c) 1 2 (d) 1 4 [Delhi Gov. SQP 2022] 2. The value of sin−1 �cos 13𝜋 5 � is: (a) − 3𝜋 5 (b) − 𝜋 10 (c) 3𝜋 10 (d) 𝜋 10 [CBSE Term-1 2021] 3. The value of the expression sec−1 (2) + sin−1 � 1 2 � + tan−1 (−√3) is: (a) 5𝜋 3 (b) 𝜋 3 (c) −𝜋 3 (d) 𝜋 6 [Delhi Gov. Term-1 2021] 4. If 𝑎 ≤ 2sin−1 𝑥 + cos−1 𝑥 ≤ 𝑏, then: (a) 𝑎 = 0, 𝑏 = 𝜋 (b) 𝑎 = 𝜋, 𝑏 = 2𝜋 (c) 𝑎 = −𝜋 2 , 𝑏 = 𝜋 2 (d) 𝑎 = 0, 𝑏 = 𝜋 2 [Delhi Gov. Term-1 SQP 2021] 5. If tan−1 𝑥 = 𝑦, then: (a) −1 < 𝑦 < 1 (b) −𝜋 2 ≤ 𝑦 ≤ 𝜋 2 CH-2 CLICK HERE FOR SOLUTIONS
  • 29.
    (c) −𝜋 2 < 𝑦 < 𝜋 2 (d)𝑦 ∈ � −𝜋 2 , 𝜋 2 � [CBSE Term-1 SQP 2021] 6. If sin−1 𝑥 + sin−1 𝑦 = 2𝜋 3 , then cos−1 𝑥 + cos−1 𝑦 is: (a) −𝜋 3 (b) 𝜋 3 (c) 𝜋 (d) 𝜋 2 [Delhi Gov. Term - 1 SQP 2021] 7. The principal value of �tan−1 √3 − cot−1 (−√3)� is: (a) 𝜋 (b) − 𝜋 2 (c) 0 (d) 2√3 [CBSE Term-1 2021] In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices. (a) Both (A) and (R) are true and (R) is the correct explanation of (A). (b) Both (A) and (R) are true but (R) is not the correct explanation of (𝐴). (c) (A) is true but (R) is false. (d) (A) is false but (R) is true. 8. Assertion (A): Maximum value of (cos−1 𝑥)2 is 𝜋2 . Reason (R): Range of the principal value branch of cos−1 𝑥 is � −𝜋 2 , 𝜋 2 �. [CBSE 2023] 9. Assertion (A): The domain of the function sec−1 2𝑥 is �−∞, − 1 2 � ∪ � 1 2 , ∞�. Reason (R): sec−1 (−2) = − 𝜋 4 [CBSE SQP 2022] 10. Assertion (A): sin−1 �sin � 2𝜋 3 �� = 2𝜋 3 Reason (R): sin−1 (sin 𝜃) = 𝜃 if 𝜃 ∈ �� −𝜋 2 � , 𝜋 2 � CLICK HERE FOR SOLUTIONS
  • 30.
    [Delhi Gov. SQP2022] Very Short & Short Qs (1 - 3 marks) 11. Find the value of sin−1 �cos � 33𝜋 5 ��. [CBSE SQP 2023] 12. Find the domain of sin−1 (𝑥2 − 4). [CBSE SQP 2023] 13. Evaluate sin−1 �sin 3𝜋 4 � + cos−1 (cos 𝜋) + tan−1 (1). [CBSE 2023] 14. Draw the graph of cos−1 𝑥, where 𝑥 ∈ [−1,0]. Also, write its range. [CBSE 2023] 15. Find the value of sin−1 �sin � 13𝜋 7 ��. 16. Prove that: sin−1 �2𝑥�1 − 𝑥2� = 2cos−1 𝑥, 1 √2 ≤ 𝑥 ≤ 1 [CBSE 2020] 17. Prove that 3sin−1 𝑥 = sin−1 (3𝑥 − 4𝑥3), 𝑥 ∈ �− 1 2 , 1 2 � [CBSE 2018] 18. Find the value of tan−1 √3 − sec−1 (−2). [CBSE 2018] 19. Prove that: 3cos−1 𝑥 = cos−1 (4𝑥3 − 3𝑥), 𝑥 ∈ � 1 2 , 1� [CBSE 2018] 20. Solve for 𝑥: tan−1 (𝑥 − 1) + tan−1 𝑥 + tan−1 (𝑥 + 1) = tan−1 3𝑥. [CBSE 2014] 21. If (tan−1 𝑥)2 + (cot−1 𝑥)2 = 5𝜋2 8 , find 𝑥. [CBSE 2015] 22. Prove that: 2tan−1 � 1 5 � + sec−1 � 5√2 7 � + 2 CLICK HERE FOR SOLUTIONS
  • 31.
    tan−1 � 1 8 � = 𝜋 4 [CBSE 2014] 23.If tan−1 𝑥tan−1 𝑦 = 𝜋 4 , 𝑥𝑦 < 1, then write the value of 𝑥 + 𝑦 + 𝑥𝑦. [CBSE 2014] 24. Write the principal value of tan−1 �sin � −𝜋 2 �� [CBSE 2014] 25. Find the value of cot � 𝜋 2 − 2cot−1 √3�. [CBSE 2014] 26. Write the value of cos−1 �− 1 2 � + 2sin−1 � 1 2 � [CBSE 2014] 27. Solve for 𝑥: tan−1 𝑥 + 2cot−1 𝑥 = 2𝜋 3 [CBSE 2014] 28. Prove that: cot−1 � √1+sin 𝑥+√1−sin 𝑥 √1+sin 𝑥−√1−sin 𝑥 � = 𝑥 2 , 𝑥 ∈ �0, 𝜋 4 � [CBSE 2014] 29. If sin �sin−1 1 5 + cos−1 𝑥� = 1, then find the value of 𝑥. [CBSE 2014] 30. Prove that: sin−1 � 8 17 � + sin−1 � 3 5 � = cos−1 � 36 85 �. Long Qs (4 - 5 marks) 31. Solve tan−1 2𝑥 + tan−1 3𝑥 = 𝜋 4 . [CBSE 2019, 15] 32. Find the value of sin �cos−1 4 5 + tan−1 2 3 �. [CBSE 2014] 33. Prove that: cos−1 12 13 + sin−1 3 5 = sin−1 56 65 CLICK HERE FOR SOLUTIONS
  • 32.
    [CBSE 2019] 34. Solve:tan−1 (𝑥 + 1) + tan−1 (𝑥 − 1) = tan−1 8 31 [CBSE 2019, 15] 35. Prove that: 2tan−1 � 1 2 � + tan−1 � 1 7 � = sin−1 � 31 25√2 � [CBSE 2019] CLICK HERE FOR SOLUTIONS
  • 33.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 34.
    Matrices 1. MATRIX ANDITS TYPES Objective Qs (1 mark) 1. If A = �𝑎𝑖𝑗� is a square matrix of order 2 such that 𝑎𝑖𝑗 = � 1, when 𝑖 ≠ 𝑗 0, when 𝑖 = 𝑗 , then 𝐴2 is: (a) � 1 0 1 0 � 2×2 (b) � 1 1 0 0 � 2×2 (c) � 1 1 1 0 � 2×2 (d) � 1 0 0 1 � 2×2 [CBSE SQP 2023] 2. If 𝐴 is a 2 × 3 matrix such that 𝐴𝐵 and 𝐴𝐵′ both are defined, then order of the matrix 𝐵 is: (a) 2 × 2 (b) 2 × 1 (c) 3 × 2 (d) 3 × 3 [CBSE 2023] 3. If � 2𝑎 + 𝑏 𝑎 − 2𝑏 5𝑐 − 𝑑 4𝑐 + 3𝑑 � = � 4 −3 11 24 �, then the value of 𝑎 + 𝑏 − 𝑐 + 2𝑑 is: (a) 8 (b) 10 (c) 4 (d) -8 [CBSE Term-1 SQP 2021] Very Short & Short Qs (1 - 3 marks) 4. Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3. [CBSE 2016] 5. Write the element 𝑎23 of a 3 × 3 matrix A = �𝑎𝑖𝑗� whose elements 𝑎𝑖𝑗 are given by 𝑎𝑖𝑗 = |𝑖−𝑗| 2 [CBSE 2015] 6. Find the element 𝑎32 of a 3 × 3 matrix: if 𝑎𝑖𝑗 is given by 𝑎𝑖𝑗 = 1 2 | − 3𝑖 + 𝑗|. [CBSE 2014] CH-3 CLICK HERE FOR SOLUTIONS
  • 35.
    7. If � 𝑥𝑦4 𝑧 + 6 𝑥 + 𝑦 � = � 8 𝑤 0 6 �, then write the value of (𝑥 + 𝑦 + 𝑧). [CBSE 2014] 8. If � 𝑥 − 𝑦 𝑧 2𝑥 − 𝑦 𝑤� = � −1 4 0 5 �, find the value of 𝑥 + 𝑦. [CBSE 2014] 9. If � 𝑎 + 4 3𝑏 8 −6 � = � 2𝑎 + 2 𝑏 + 2 8 𝑎 − 8𝑏 �, then find the value of (𝑎 − 2𝑏). [CBSE 2014] 2. OPERATIONS ON MATRICES Objective Qs (1 mark) 10. If � 1 −1 2 3 � + 𝑋 = � 3 4 5 6 �, where X = � 𝑎 𝑏 𝑐 𝑑 � , then 𝑎 + 𝑐 − 𝑏 − 𝑑 = (a) 13 (b) 5 (c) -8 (d) -3 [Delhi Gov. Term-1 SQP 2021] 11. Given that matrices 𝐴 and 𝐵 are of order 3 × 𝑛 and 𝑚 × 5 respectively, then the order of matrix 𝐶 = 5𝐴 + 3𝐵 is: (a) 3 × 5 and 𝑚 = 𝑛 (b) 3 × 5 (c) 3 × 3 (d) 5 × 5 [CBSE Term-1 SQP 2021] 12. If 𝐴 = �𝑎𝑖𝑗� is a square matrix of order 2 such that 𝑎𝑖𝑗 = � 1, when 𝑖 ≠ 𝑗 0, when 𝑖 = 𝑗 , then 𝐴2 is: (a) � 1 0 1 0 � (b) � 1 1 0 0 � (c) � 1 1 1 0 � (d) � 1 0 0 1 � [CBSE Term-1 SQP 2021] 13. If 𝐴 is square matrix such that 𝐴2 = 𝐴, then (𝐼 + 𝐴)3 − 7𝐴 is equal to: (a) 𝐴 (b) I + A CLICK HERE FOR SOLUTIONS
  • 36.
    (c) I- A (d)I [CBSE Term-1 SQP 2021] 14. If 𝐴 = � 0 2 3 −4 � and 𝑘𝐴 = � 0 3𝑎 2𝑏 24 �, then the values of 𝑘, 𝑎 and 𝑏 respectively are: (a) −6, −12, −18 (b) −6, −4, −9 (c) −6,4,9 (d) −6,12,18 [CBSE Term-1 SQP 2021] 15. If [𝑥 − 25 + 𝑦] � 0 1 1 0 � = 0, then 𝑥 + 𝑦 is: (a) 0 (b) -2 (c) -1 (d) -3 [Delhi Gov. Term-1 SQP 2021] 16. For the matrix 𝑋 = � 0 1 1 1 0 1 1 1 0 � , (𝑋2 − 𝑋) is: (a) 21 (b) 31 (c) 1 (d) 51 [CBSE Term-1 2021] 17. If 𝐴 is a diagonal matrix of order 3 × 3 such that 𝐴2 = 𝐴, then number of possible matrices 𝐴 are: (a) 4 (b) 8 (c) 16 (d) 32 [Delhi Gov. Term-1 2021] Very Short & Short Qs (1 - 3 marks) 18. If 3𝐴 − 𝐵 = � 5 0 1 1 � and 𝐵 = � 4 3 2 5 � and find the value of matrix 𝐴. [CBSE 2019] 19. If 𝐴 = � 2 0 1 2 1 3 1 −1 0 �, then find the value of (𝐴2 − 5𝐴). [CBSE 2019] CLICK HERE FOR SOLUTIONS
  • 37.
    20. If 𝐴= � 4 2 −1 1 �, show that (𝐴 − 2𝐼)(𝐴 − 3𝐼) = 0. [CBSE 2019] 21. Let 𝐴 = � 2 −1 3 4 � , 𝐵 = � 5 2 7 4 � , 𝐶 = � 2 5 3 8 �. Find a matrix 𝐷 such that 𝐶𝐷 − 𝐴𝐵 = 0. [CBSE 2017] Long Qs (4 - 5 marks) 22. If 𝐴 = � 1 0 2 0 2 1 2 0 3 � and 𝐴3 − 6𝐴2 + 7𝐴 + 𝑘|3 = 0, find the value of 𝑘. [CBSE 2016] 23. If A = � 1 −1 2 −1 � , B = � 𝑎 1 𝑏 −1 � and (𝐴 + 𝐵)2 = 𝐴2 + 𝐵2 , then find the values of 𝑎 and 𝑏. [CBSE 2015] 24. If [2𝑥 3] � 1 2 −3 0 � � 𝑥 3 � = 0, find 𝑥. [CBSE 2015] 25. A trust fund has ₹35,000 is to be invested in two different types of bonds. The first bond pays 8% interest per annum which will be given to orphanage and second bond pays 10% interest per annum which will be given to an N.G.O. (Cancer Aid Society). Using matrix multiplication, determine how to divide ₹35,000 among two types of bonds if the trust fund obtains an annual total interest of ₹3,200. What are the values reflected in this question? [CBSE 2015] 26. In a parliament election, a political party hired a public relations firm to promote its candidates in three ways - telephone, house calls and letters. The cost per contact (in paisa) is given in matrix 𝐴 as 𝐴 = � 140 200 150 � Telephone House calls Letters The number of contacts of each type made in two cities 𝑋 and 𝑌 is given in the matrix 𝐵 as Telephone Housecalls Letters 𝐵 = � 1000 500 5000 3000 1000 10000 � City 𝑋 City 𝑌 Find the total amount spent by the party in the two cities. What should one consider before casting his/her vote - party's promotional activity or their social activities? [CBSE 2015] CLICK HERE FOR SOLUTIONS
  • 38.
    27. To promotemaking of toilets for women, an organisation tried to generate awareness through (i) house calls, (ii) letters, and (iii) announcements. The cost for each mode per attempt is given below: (i) ₹ 50 (ii) ₹ 20 (iii) ₹ 40 The number of attempts made in three villages 𝑋, 𝑌 and 𝑍 are given below: (i) (ii) (iii) 𝑋 400 300 100 𝑌 300 250 75 𝑍 500 400 150 Find the total cost incurred by the organisation for three villages separately, using matrices. [CBSE 2015] 28. Solve the matrix equation for 𝑥. [𝑥 1] � 1 0 −2 0 � = 0 [CBSE 2014] 3. TRANSPOSE OF A MATRIX, SYMMETRIC AND SKEW-SYMMETRIC MATRICES Objective Qs (1 mark) 29. If � 2 0 5 4 � = 𝑃 + 𝑄, where 𝑃 is a symmetric and 𝑄 is a skew symmetric matrix, then 𝑄 is equal to: (a) � 2 5 2 5 2 4 � (b) � 0 − 5 2 5 2 0 � (c) � 0 5 2 − 5 2 0 � (d) � 2 − 5 2 5 2 4 � [CBSE 2023] CLICK HERE FOR SOLUTIONS
  • 39.
    30. If 𝐴= �𝑎𝑖𝑗� is a skew-symmetric matrix of order 𝑛, then (a) 𝑎𝑖𝑗 = 1 𝑎𝑗𝑖 ∀𝑖, 𝑗 (b) 𝑎𝑖𝑗 ≠ 0∀𝑖, 𝑗 (c) 𝑎𝑖𝑗 = 0, where 𝑖 = 𝑗 (d) 𝑎𝑖𝑗 ≠ 0 where 𝑖 = 𝑗 [CBSE SQP 2022] 31. For two matrices 𝑃 = � 3 4 −1 2 0 1 � and 𝑄⊤ = � −1 2 1 1 2 3 � 𝑃 − 𝑄 is: (a) � 2 3 −3 0 0 −3 � (b) � 4 3 −3 0 −1 −2 � (c) � 4 3 0 −3 −1 −2 � (d) � 4 3 0 −3 0 −3 � [CBSE Term-1 2021] 32. If 𝐴 = � cos 𝛼 −sin 𝛼 sin 𝛼 cos 𝛼 � and 𝐴 + 𝐴′ = 1, then the value of 𝛼 is: (a) 𝜋 6 (b) 𝜋 3 (c) 𝜋 (d) 3𝜋 2 [CBSE Term-1 2021] 33. If a matrix 𝐴 is both symmetric and skew symmetric, then 𝐴 is necessarily a/an: (a) diagonal matrix (b) zero square matrix (c) square matrix (d) identity matrix [CBSE Term-1 2021] Very Short & Short Qs (1 - 3 marks) 34. If 𝑃 is a 3 × 3 matrix such that 𝑃′ = 2𝑃 + 𝐼, where 𝑃′ is the transpose of 𝑃, then: (a) P = I (b) 𝑃 = −1 CLICK HERE FOR SOLUTIONS
  • 40.
    (c) 𝑃 =21 (d) 𝑃 = −21 [CBSE Term-1 2021] 35. If is a matrix of order 3 × 2, then find the order of the matrix 𝐴′ . [CBSE 2020] 36. If 𝐴 and 𝐵 are symmetric matrices, such that 𝐴𝐵 and 𝐵𝐴 are both defined, then prove that 𝐴𝐵 − 𝐵𝐴 is a skew symmetric matrix. [CBSE 2019] 37. If 𝐴 = � 1 2 2 2 1 𝑥 −2 2 −1 � is a matrix satisfying 𝐴𝐴′ = 91, find 𝑥. [CBSE 2018] 38. If the matrix 𝐴 = � 0 𝑎 −3 2 0 −1 𝑏 1 0 � is skew symmetric, find the values of 𝑎 and 𝑏. [CBSE 2018] 39. Show that all the diagonal elements of a skew-symmetric matrix are zero. [CBSE 2017] 40. If 𝐴 = � 3 5 7 9 � is written as 𝐴 = 𝑃 + 𝑄, where 𝑃 is a symmetric matrix and 𝑄 is a skewsymmetric matrix, then write matrix 𝑃. [CBSE 2016] 41. Matrix 𝐴 = � 0 2𝑏 −2 3 1 3 3𝑎 3 −1 � is given to be symmetric, then find the values of 𝑎 and 𝑏. [CBSE 2016] 42. If 𝐴 = � cos 𝛼 sin 𝛼 −sin 𝛼 cos 𝛼 �, find 𝛼 satisfying 0 < 𝛼 < 𝜋 2 when 𝐴 + 𝐴⊤ = √2𝐼2; where 𝐴⊤ is transpose of 𝐴. [CBSE 2016] 43. Express the matrix 𝐴 = � 2 4 −6 7 3 5 1 −2 4 � as the sum of a symmetric and skew-symmetric matrix. [CBSE 2015] 4. INVERTIBLE MATRICES Objective Qs (1 mark) CLICK HERE FOR SOLUTIONS
  • 41.
    44. If 𝐴= � 1 −1 0 2 3 4 0 1 2 � and 𝐵 = � 2 2 −4 −4 2 −4 2 −1 5 �, then: (a) A−1 = B (b) A−1 = 6 B (c) B−1 = B (d) 𝐵−1 = 1 6 𝐴 45 If 𝐴 = � 3 1 −1 2 �, show that 𝐴2 − 5𝐴 + 7𝐼 = 0. Hence find 𝐴−1 . [CBSE 2020] CLICK HERE FOR SOLUTIONS
  • 42.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 43.
    Determinants 1. DETERMINANTS OFA SQUARE MATRIX AND AREA OF A TRIANGLE Objective Qs (1 mark) 1. If |𝐴| = |𝑘𝐴|, where 𝐴 is a square matrix of order 2, then sum of all possible values of 𝑘 is: (a) 1 (b) -1 (c) 2 (d) 0 [CBSE 2023] 2. If � 1 2 1 2 3 1 3 𝑎 1 � is non-singular matrix and 𝑎 ∈ 𝐴, then the set 𝐴 is: (a) R (b) {0} (c) {4} (d) R − {4} [CBSE 2023] 3. If (𝑎, 𝑏), (𝑐, 𝑑) and (𝑒, 𝑓) are the vertices of △ 𝐴𝐵𝐶 and △ denotes the area of △ 𝐴𝐵𝐶, then � 𝑎 𝑐 𝑒 𝑏 𝑑 𝑓 1 1 1 � 2 is equal to: (a) 2Δ2 (b) 4Δ2 (c) 2Δ (d) 4Δ [CBSE 2023] 4. If 𝐴 and 𝐵 are invertible square matrices of the same order, then which of the following is not correct? (a) |𝐴𝐵−1| = |𝐴| |𝐵| (b) |(𝐴𝐵)−1| = 1 |𝐴||𝐵| (c) (𝐴𝐵)−1 = 𝐵−1 𝐴−1 (d) (𝐴 + 𝐵)−1 = 𝐵−1 + 𝐴−1 [CBSE SQP 2023] CH-4 CLICK HERE FOR SOLUTIONS
  • 44.
    5. If thearea of the triangle with vertices (−3,0), (3,0) and (0, 𝑘) is 9 sq. units, then the value/s of 𝑘 will be: (a) 9 (b) pm 3 (c) -9 (d) 6 [CBSE SQP 2023] 6. The value of |𝐴|, if 𝐴 = � 0 2𝑥 − 1 √𝑥 1 − 2𝑥 0 2√𝑥 −√𝑥 −2√𝑥 0 �, where 𝑥 ∈ R+ , is: (a) (2𝑥 + 1)2 (b) 0 (c) (2𝑥 + 1)3 (d) (2𝑥 − 1)2 [CBSE SQP 2023] 7. Given that 𝐴 is a square matrix of order 3 and |𝐴| = −2, then |adj (2𝐴)| is equal to: (a) −26 (b) +4 (c) −28 (d) 28 [CBSE SQP 2023] 8. If 𝐴 is a square matrix of order 3, |𝐴′| = −3, then |𝐴𝐴′| = (a) 9 (b) -9 (c) 3 (d) -3 [CBSE SQP 2022] 9. The area of a triangle with vertices 𝐴, 𝐵, 𝐶 is given by: (a) |𝐴𝐵 �����⃗ × 𝐴𝐶 �����⃗| (b) 1 2 |𝐴𝐵 �����⃗ × 𝐴𝐶 �����⃗| (c) 1 4 |𝐴𝐶 �����⃗ × 𝐴𝐵 �����⃗| (d) 1 8 |𝐴𝐶 �����⃗ × 𝐴𝐵 �����⃗| [CBSE SQP 2022] 10. Value of 𝑘, for which 𝐴 = � 𝑘 8 4 2𝑘 � is a singular matrix, is: (a) 4 (b) -4 (c) ± 4 (d) 0 [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
  • 45.
    11. Given that𝐴 is a non-singular matrix of order 3, such that 𝐴2 = 2𝐴, then value of |2𝐴| is: (a) 4 (b) 8 (c) 64 (d) 16 [CBSE Term-1 SQP 2021] 12. Let, 𝐴 = � 1 sin 𝛼 1 −sin 𝛼 1 sin 𝛼 −1 −sin 𝛼 1 � , where 0 ≤ 𝛼 ≤ 2𝜋, then: (a) |A| = 0 (b) |A| ∈ (2, ∞) (c) |A| ∈ (2,4) (d) |𝐴| ∈ [2,4] [CBSE Term-1 SQP 2021] 13. Three points 𝑃(2𝑥, 𝑥 + 3), 𝑄(0, 𝑥) and R(𝑥 + 3, 𝑥 + 6) are collinear, then 𝑥 is: (a) 0 (b) 2 (c) 3 (d) 1 [CBSE Term-1 2021] 14. If, 𝐴 is a skew-symmetric matrix of order 3 , then the value of |A| is: (a) 3 (b) 0 (c) 9 (d) 27 [CBSE 2020] 15. If 𝐴 is a 3 × 3 matrix such that |𝐴| = 8, then |3 A| equals: (a) 8 (b) 24 (c) 72 (d) 216 [CBSE 2020] Very Short & Short Qs (1 - 3 marks) 16. If 𝐴 is a square matrix satisfying 𝐴′ 𝐴 = 𝐼, write the value of |𝐴|. [CBSE 2019] 17. If 𝐴 = � 𝑝 2 2 𝑝 � and |𝐴3| = 125, then find the value of 𝑝. [CBSE 2019] CLICK HERE FOR SOLUTIONS
  • 46.
    18. If � 𝑥sin 𝜃 cos 𝜃 −sin 𝜃 −𝑥 1 cos 𝜃 1 𝑥 � = 8, write the value of 𝑥. [CBSE 2016] 19. Find the maximum value of � 1 1 1 1 1 + sin 𝜃 1 1 1 1 + cos 𝜃 � [CBSE 2016] 20. Write the value of the determinant � 𝑝 𝑝 + 1 𝑝 − 1 𝑝 � [CBSE 2014] 21. Write the value of � 2 7 65 3 8 75 5 9 86 �. [CBSE 2014] 2. MINORS AND COFACTORS Objective Qs (1 mark) 22. If 𝐴 is any square matrix of order 3 × 3 such that |adj 𝐴| = 256, then the sum of all possible values of |A| is: (a) 256 (b) 16 (c) -16 (d) 0 [Delhi Gov. Term-1 SQP 2021] 23. Given that 𝐴 = �𝑎𝑖𝑗� is a square matrix of order 3 × 3 and |𝐴| = −7, then the value of ∑𝑖=1 3 𝑎𝑖2𝐴𝑖2, where 𝐴𝑖𝑗 denotes the cofactor of element 𝑎𝑖𝑗 is: (a) 7 (b) -7 (c) 0 (d) 49 [CBSE Term-1 SQP 2021] 24. Given that 𝐴 is a square matrix of order 3 and |𝐴| = −4, then |adj 𝐴| is: (a) -4 (b) 4 (c) -16 (d) 16 [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
  • 47.
    25. For matrix𝐴 = � 2 5 −11 7 � , (adj 𝐴)⊤ is: (a) � −2 −5 11 −7 � (b) � 7 5 11 2 � (c) � 7 11 −5 2 � (d) � 7 −5 11 2 � [CBSE Term-1 SQP 2021] 26. For 𝐴 = � 3 1 −1 2 �, then 14𝐴−1 is given by: (a) 14 � 2 −1 1 3 � (b) � 4 −2 2 6 � (c) 2 � 2 −1 1 −3 � (d) 2 � −3 −1 1 −2 � [CBSE Term-1 SQP 2021] 27. If 𝐴 = � 1 −2 4 2 −1 3 4 2 0 � is the adjoint of a square matrix 𝐵, then 𝐵−1 is equal to: (a) ±A (b) ±√2 A (c) ± 1 √2 𝐵 (d) ± 1 √2 A [CBSE Term-1 2021] 28. If � 1 −tan 𝜃 tan 𝜃 1 � � 1 tan 𝜃 −tan 𝜃 1 � −1 = � 𝑎 −𝑏 𝑏 𝑎 �, then (a) 𝑎 = 1 = 𝑏 (b) 𝑎 = cos 2𝜃, 𝑏 = sin 2𝜃 (c) 𝑎 = sin 2𝜃, 𝑏 = cos 𝜃 (d) 𝑎 = cos 𝜃, 𝑏 = sin 𝜃 [CBSE Term-1 2021] Very Short & Short Qs (1 - 3 marks) 29. If 𝐴 = � 2 3 5 −2 � be such that 𝐴−1 = 𝑘𝐴, then find the value of 𝑘. [CBSE 2018] CLICK HERE FOR SOLUTIONS
  • 48.
    30. Given 𝐴= � 2 −3 −4 7 �, compute 𝐴−1 and show that 2𝐴−1 = 91 − 𝐴. [CBSE 2018] 31. Find the inverse of the matrix 𝐴 = � 1 3 3 1 4 3 1 3 4 �. [CBSE 2018] 32. If for any 2 × 2 square matrix 𝐴, 𝐴(adj 𝐴) = � 8 0 0 8 �, then write the value of |𝐴|. 33. If 𝐴 = � 5 6 −3 −4 3 2 −4 −7 3 �, then write the cofactor of the element 𝑎21 of its 2nd row. [CBSE 2015] 34. In the interval 𝜋 2 < 𝑥 < 𝜋, find the value of 𝑥 for which the matrix, � 2sin 𝑥 3 1 2sin 𝑥 � is singular. [CBSE 2015] 35. If 𝐴 = � 1 −2 3 0 −1 4 −2 2 1 �, then find (𝐴⊤)−1 . [CBSE 2015] 36. Find the adjoint of the matrix 𝐴 = � −1 −2 −2 2 1 −2 2 −2 1 � and hence show that 𝐴(adj 𝐴) = |𝐴|𝐼3. [CBSE 2015] 37. If 𝐴 = � 2 3 1 −4 � and 𝐵 = � 1 −2 −1 3 �, then verify that (𝐴𝐵)−1 = 𝐵−1 𝐴−1 [CBSE 2015] Long Qs (4 - 5 marks) 38. Show that for the matrix 𝐴 = � 1 1 1 1 2 −3 2 −1 3 �, 𝐴3 − 6𝐴2 + 5𝐴 + 11𝐼 = 0. Hence, find 𝐴−1 . [CBSE 2019] 39. If 𝐴 = � cos 𝛼 −sin 𝛼 0 sin 𝛼 cos 𝛼 0 0 0 1 �, find adj 𝐴 and verify that 𝐴(adj 𝐴) = (adj 𝐴)𝐴 = |𝐴|𝐼3. [CBSE 2016] 3. APPLICATIONS OF DETERMINANTS AND MATRICES CLICK HERE FOR SOLUTIONS
  • 49.
    Objective Qs (1mark) 40. The system of linear equations 5𝑥 + 𝑘𝑦 = 5, 3𝑥 + 3𝑦 = 5; will be consistent if: (a) 𝑘 ≠ −3 (b) 𝑘 = −5 (c) 𝑘 = 5 (d) 𝑘 ≠ 5 [CBSE Term-1 2021] 41. If 𝐴 = � 1 −1 0 2 3 4 0 1 2 � and 𝐵 = � 2 2 −4 −4 2 −4 2 −1 5 �, then: (a) A−1 = B (b) A−1 = 6 B (c) B−1 = B (d) B−1 = 1 6 A [CBSE Term-1 SQP 2021] Very Short & Short Qs (1-3 marks) 42. For what values of 𝑘, the system of linear equations 𝑥 + 𝑦 + 𝑧 = 2 2𝑥 + 𝑦 − 𝑧 = 3 3𝑥 + 2𝑦 + 𝑘𝑧 = 4 has a unique solution? [CBSE 2016] 43. The monthly incomes of Aryan and Babban are in the ratio 3: 4 and their monthly expenditures are in the ratio 5: 7. If each saves ₹ 15,000 per month, then find their monthly incomes using matrices. [CBSE 2016] Longs Qs (4 - 5 marks) 44. Using the matrix method, solve the following system of linear equations: 2 𝑥 + 3 𝑦 + 10 𝑧 = 4, 4 𝑥 − 6 𝑦 + 5 𝑧 = 1, 6 𝑥 + 9 𝑦 + 20 𝑧 = 2. [CBSE SQP 2023] 45. If 𝐴 = � −3 −2 −4 2 1 2 2 1 3 � and 𝐵 = � 1 2 0 −2 −1 −2 0 −1 1 �, then find 𝐴𝐵 and use it to solve the following system of equations: CLICK HERE FOR SOLUTIONS
  • 50.
    𝑥 − 2𝑦= 3 2𝑥 − 𝑦 − 𝑧 = 2 −2𝑦 + 𝑧 = 3 [CBSE 2023] 46. If 𝑓(𝛼) = � cos 𝛼 −sin 𝛼 0 sin 𝛼 cos 𝛼 0 0 0 1 �, then prove that 𝑓(𝛼) ⋅ 𝑓(−𝛽) = 𝑓(𝛼 − 𝛽). [CBSE 2023] 47. If 𝐴 = � 1 2 0 −2 −1 −2 0 −1 1 �, find 𝐴−1 . Using 𝐴−1 , solve the system of linear equations 𝑥 − 2𝑦 = 10,2𝑥 − 𝑦 − 𝑧 = 8, −2𝑦 + 𝑧 = 7. [CBSE 2021] 48. Evaluate the product 𝐴𝐵, where 𝐴 = � 1 −1 0 2 3 4 0 1 2 � and 𝐵 = � 2 2 −4 −4 2 −4 2 −1 5 � Hence the solve system of linear equations 𝑥 − 𝑦 = 3 2𝑥 + 3𝑦 + 4𝑧 = 17 𝑦 + 2𝑧 = 7 [CBSE 2021] 49. If 𝐴 = � 1 2 −3 3 2 −2 2 −1 1 �, then find 𝐴−1 and use it to solve the following systems of the equations: 𝑥 + 2𝑦 − 3𝑧 = 6 3𝑥 + 2𝑦 − 2𝑧 = 3 3𝑥 + 2𝑦 − 𝑧 = 5 [CBSE 2020] 50. If 𝐴 = � 2 3 4 1 −1 0 0 1 2 �, find 𝐴−1 . Hence, solve the system of equations 2𝑥 + 3𝑦 + 4𝑧 = 17, 𝑥 − 𝑦 = 3, 𝑦 + 2𝑧 = 7. [CBSE 2020] CLICK HERE FOR SOLUTIONS
  • 51.
    51. Determine theproduct of � −4 4 4 −7 1 3 5 −3 −1 �. and � 1 −1 1 1 −2 −2 2 1 3 �, and then use this to solve the system of equations 𝑥 − 𝑦 + 𝑧 = 4 𝑥 − 2𝑦 − 2𝑧 = 9 2𝑥 − 𝑦 + 𝑧 = 2 [CBSE 2017] 52. A shopkeeper has 3 varieties of pens ' 𝐴 ', ' 𝐵 ' and ' 𝐶 '. Meena purchased 1 pen of each variety for a total of ₹ 21. Jeevan purchased 4 pens of 'A' variety, 3 pens of 'B' variety and 2 pens of ' 𝐶 ' variety for ₹60, while Shikha purchased 6 pens of 'A' variety 2 pens of ' 𝐵 ' variety and 3 pens of ' 𝐶 ' variety for ₹70. Using matrix method, find the cost of each variety of pen. [CBSE 2016] 53. A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received ₹2,800 as interest, However, if trust had interchanged money in bonds, they would have got ₹ 100 less as interest, Using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Helpage India as donation. Which value is reflected in this question? [CBSE 2016] 54. A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is ₹ 9,000 whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is ₹ 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society? [CBSE 2016] 55. Two schools 𝐴 and 𝐵 decided to award prizes to their students for three values, team spirit truthfulness and tolerance at the rate of ₹𝑥, ₹ 𝑦 and ₹𝑧 per student respectively. School A, decided to award a total of ₹ 1,100 for the three values to 3,1 and 2 students respectively while school 𝐵 decided to award ₹1,400 for the three values to 1,2 and 3 students respectively. If one prize for all the three values together amount to ₹ 600 then (A) Represent the above situation by a matrix equation after forming linear equations. (B) Is it possible to sove the system of equations so obtained using matrices? (C) Which value you prefer to be rewarded most and why? [CBSE 2015] CLICK HERE FOR SOLUTIONS
  • 52.
    56. A totalamount of ₹ 7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and 8 1 2 % respectively. The total annual interest from these three accounts is ₹ 550. Equal amounts have been deposited in the 5% and 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices. [CBSE 2014] CLICK HERE FOR SOLUTIONS
  • 53.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 54.
    Continuity And Differentiability 1.LIMITS AND CONTINUITY OF A FUNCTION Objective Qs (1 mark) 1. If 𝑓(𝑥) = � 𝑘𝑥 |𝑥| , if 𝑥 < 0 3, if 𝑥 ≥ 0 is continuous at 𝑥 = 0, then the value of 𝑘 is: (a) -3 (b) 0 (c) 3 (d) any real number [CBSE SQP 2023] 2. The function 𝑓(𝑥) = [𝑥], where [𝑥] denotes the greatest integer function, is continuous at: (a) 4 (b) 1.5 (c) 1 (d) -2 [CBSE Term-1 2021] 3. The value of 𝑘(𝑘 < 0) for which the function 𝑓 defined as: 𝑓(𝑥) = � 1 − cos 𝑘𝑥 𝑥sin 𝑥 , 𝑥 ≠ 0 1 2 , 𝑥 = 0 is continuous at 𝑥 = 0 is: (a) ± 1 (b) -1 (c) ± 1 2 (d) 1 2 [CBSE Term-1 SQP 2021] 4. The point(s), at which the function 𝑓 given by 𝑓(𝑥) = � 𝑥 |𝑥| , 𝑥 < 0 −1, 𝑥 ≥ 0 is continuous, is/are: (a) 𝑥 ∈ 𝑅 (b) 𝑥 = 0 (c) 𝑥 ∈ R − {0} (d) 𝑥 = −1 and 1 [CBSE Term-1 SQP 2021] CH-5 CLICK HERE FOR SOLUTIONS
  • 55.
    5. 𝑓(𝑥) =� 3𝑥 − 8 if 𝑥 ≤ 5 2𝑘 if 𝑥 > 5 is continuous, find 𝑘. (a) 2 7 (b) 3 7 (c) 4 7 (d) 7 2 [CBSE Term-1 2021] Very Short & Short Qs (1 - 3 marks) 6. Find the value of 𝜆 so that the function 𝑓 defined by 𝑓(𝑥) = � 𝜆𝑥, if 𝑥 ≤ 𝜋 cos 𝑥, if 𝑥 > 𝜋 is continuous at 𝑥 = 𝜋. [CBSE 2020] 7. Determine the value of ' 𝑘 ' for which the following function is continuous at 𝑥 = 3. 𝑓(𝑥) = � (𝑥 + 3)2 − 36 𝑥 − 3 , 𝑥 ≠ 3 𝑘 , 𝑥 = 3 [CBSE 2017] 8. Find the values of 𝑝 and 𝑞, for which 𝑓(𝑥) = ⎩ ⎪ ⎨ ⎪ ⎧ 1 − sin3 𝑥 3cos2 𝑥 , if 𝑥 < 𝜋 2 𝑝 , if 𝑥 = 𝜋 2 𝑞(1 − sin 𝑥) (𝜋 − 2𝑥)2 , if 𝑥 > 𝜋 2 is continuous at 𝑥 = 𝜋 2 . [CBSE 2016] 9. Find the value of the constant 𝑘 so that the function 𝑓, defined below, is continuous at 𝑥 = 0, where 𝑓(𝑥) = � 1 − cos 4𝑥 8𝑥2 , if 𝑥 ≠ 0 𝑘 , if 𝑥 = 0 [CBSE SQP 2014] CLICK HERE FOR SOLUTIONS
  • 56.
    Long Qs (4- 5 marks) 10. If 𝑓(𝑥) = ⎩ ⎪ ⎨ ⎪ ⎧ sin (𝑎 + 1)𝑥 + 2sin 𝑥 𝑥 𝑥 < 0 2 √1 + 𝑏𝑥 − 1 𝑥 𝑥 > 0 is continuous at 𝑥 = 0 then find the values of 𝑎 and 𝑏. [CBSE 2016] 2. DIFFERENTIABILITY Objective Qs (1 mark) 11. The set of all points where the function 𝑓(𝑥) = 𝑥 + |𝑥| is differentiable, is: (a) (0, ∞) (b) (−∞, 0) (c) (−∞, 0) ∪ (0, ∞) (d) (−∞, ∞) [CBSE SQP 2023] 12. The function 𝑓(𝑥) = 𝑥|𝑥| is: (a) continuous-and differentiable at 𝑥 = 0. (b) continuous but not differentiable at 𝑥 = 0. (c) differentiable but not continuous at 𝑥 = 0. (d) neither differentiable nor continuous at 𝑥 = 0. [CBSE 2023] 13. The function given below at 𝑥 = 4 is: 𝑓(𝑥) = � 2𝑥 + 3, 𝑥 ≤ 4 𝑥2 − 5, 𝑥 > 4 (a) continuous but not differentiable (b) differentiable but not continuous (c) continuous as well as differentiable (d) neither continuous nor differentiable CLICK HERE FOR SOLUTIONS
  • 57.
    [Delhi. Gov. Term-1SQP 2021] 14. The function 𝑓(𝑥) = � 𝑥2 for 𝑥 ≤ 1 2 − 𝑥 for 𝑥 ≥ 1 is: (a) Not differentiable at 𝑥 = 1 (b) Differentiable at 𝑥 = 1 (c) Not continuous at 𝑥 = 1 (d) Neither continuous nor differentiable at 𝑥 = 1 [CBSE Term-1 2021] Very Short & Short Qs (1-3 marks) 15. Prove that the greatest integer function defined by 𝑓(𝑥) = [𝑥],0 < 𝑥 < 3 is not differentiable at 𝑥 = 1. [CBSE 2020] 16. Let, 𝑓(𝑥) = 𝑥|𝑥|, for all 𝑥 ∈ 𝑅. Check its differentiability at 𝑥 = 0. [CBSE 2020] 17. Prove that the greatest integer function defined by 𝑓(𝑥) = [𝑥],0 < 𝑥 < 2 is not differentiable at 𝑥 = 1 [CBSE 2020] 18. Find the values of 𝑎 and 𝑏, if the function 𝑓 defined by 𝑓(𝑥) = � 𝑥2 + 3𝑥 + 𝑎 , 𝑥 ≤ 1 𝑏𝑥 + 2 , 𝑥 > 1 is differentiable at 𝑥 = 1. [CBSE 2016] 19. For what value of 𝜆, the function defined by 𝑓(𝑥) = �𝜆(𝑥2 + 2) , if 𝑥 ≤ 0 4𝑥 + 6,𝑥 > 0 is continuous at 𝑥 = 0 ? Hence, check the differentiability of 𝑓(𝑥) at 𝑥 = 0. [CBSE 2015] 20. Find whether the following function is differentiable at 𝑥 = 1 and 𝑥 = 2 or not. 𝑓(𝑥) = � 𝑥, 𝑥 < 1 2 − 𝑥, 1 < 𝑥 ≤ 2 −2 + 3𝑥 − 𝑥2 , 𝑥 > 2 [CBSE 2015] CLICK HERE FOR SOLUTIONS
  • 58.
    3. DERIVATIVES Objective Qs(1 mark) 21. If tan � 𝑥+𝑦 𝑥−𝑦 � = 𝑘, then 𝑑𝑦 𝑑𝑥 is equal to: (a) −𝑦 𝑥 (b) 𝑦 𝑥 (c) sec2 � 𝑦 𝑥 � (d) −sec2 � 𝑦 𝑥 � [CBSE 2023] 22. If 𝑒𝑥 + 𝑒𝑦 = 𝑒𝑥+𝑦 , then 𝑑𝑦 𝑑𝑥 is: (a) 𝑒𝑦−𝑥 (b) 𝑒𝑥+𝑦 (c) −𝑒𝑦−𝑥 (d) 2𝑒𝑥−𝑦 [Delhi Gov. SQP 2022, CBSE Term-1 SQP 2021] 23. The derivative of sin−1 �2𝑥√1 − 𝑥2� w.r.t. sin−1 𝑥, 1 √2 < 𝑥 < 1, is: (a) 2 (b) 𝜋 2 − 2 (c) 𝜋 2 (d) -2 [CBSE Term-1 SQP 2021] 24. If (𝑥2 + 𝑦2)2 = 𝑥𝑦, then 𝑑𝑦 𝑑𝑥 is: (a) 𝑥+4𝑥�𝑥2+𝑦2� 4𝑦(𝑥2+𝑦2)−𝑥 (b) 𝑦−4𝑥�𝑥2+𝑦2� 𝑥+4(𝑥2+𝑦2) (c) 𝑦−4𝑥�𝑥2+𝑦2� 4𝑦(𝑥2+𝑦2)−𝑥 (d) 4𝑦�𝑥2+𝑦2�−𝑥 𝑦−4𝑥(𝑥2+𝑦2) [CBSE Term- 1 2021] 25. If 𝑦 = log (cos 𝑒𝑥), then 𝑑𝑦 𝑑𝑥 is: (a) cos 𝑒𝑥−1 (c) 𝑒𝑥 sin 𝑒𝑥 (b) e−𝑥 cos 𝑒𝑥 (d) −𝑒𝑥 tan 𝑒𝑥 [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
  • 59.
    26. If 𝑦2 (2− 𝑥) = 𝑥3 , then � 𝑑𝑦 𝑑𝑥 � (1,1) is equal to: (a) 2 (b) -2 (c) 3 (d) − 3 2 [CBSE Term-1 2021] Very Short & Short Qs (1 - 3 marks) 27. If 𝑓(𝑥) = � 𝑎𝑥 + 𝑏; 0 < 𝑥 ≤ 1 2𝑥2 − 𝑥; 1 < 𝑥 < 2 is a differentiable function in (0,2), then find the values of 𝑎 and b. [CBSE 2023] 28. If 𝑦√1 − 𝑥2 + 𝑥�1 − 𝑦2 = 1, then prove that 𝑑𝑦 𝑑𝑥 = −� 1−𝑦2 1−𝑥2. [CBSE SQP 2022] 29. If 𝑦 = tan−1 𝑥 + cot−1 𝑥, 𝑥 ∈ 𝑅, then find 𝑑𝑦 𝑑𝑥 . [CBSE 2020] 30. If cos (𝑥𝑦) = 𝑘, where 𝑘 is a constant and 𝑥𝑦 ≠ 𝑛𝜋, 𝑛 ∈ 𝑍, then 𝑑𝑦 𝑑𝑥 is equal to ....... [CBSE 2020] 31. If 𝑥 = 𝑎sec 𝜃, 𝑦 = 𝑏 tan 𝜃, then find 𝑑𝑦 𝑑𝑥 at 𝜃 = 𝜋 3 . [CBSE 2020] 32. If 𝑦 = 𝑒𝑥2cos 𝑥 + (cos 𝑥)𝑥 , then find 𝑑𝑦 𝑑𝑥 . [CBSE 2020] 33. Differentiate tan−1 � cos 𝑥−sin 𝑥 cos 𝑥+sin 𝑥 � with respect to 𝑥. [CBSE 2018] 34. If 𝑥 = 𝑎(2𝜃 − sin 2𝜃) and 𝑦 = 𝑎(1 − cos 2𝜃), find 𝑑𝑦 𝑑𝑥 when 𝜃 = 𝜋 3 . [CBSE 2018] 35. Find 𝑑𝑦 𝑑𝑥 at 𝑥 = 1, 𝑦 = 𝜋 4 , if sin2 𝑦 + cos 𝑥𝑦 = 𝑘. [CBSE 2017] 36. If 𝑦 = sin−1 �6𝑥√1 − 9𝑥2�, − 1 3√2 < 𝑥 < 1 3√2 , then find 𝑑𝑦 𝑑𝑥 . [CBSE 2017] CLICK HERE FOR SOLUTIONS
  • 60.
    37. If 𝑦= tan−1 � 𝑎 𝑥 � + log � (𝑥−𝑎) (𝑥+𝑎) , prove that 𝑑𝑦 𝑑𝑥 = 2𝑎3 (𝑥4−𝑎4) [CBSE 2014] Long Qs 38. If log (𝑥2 + 𝑦2) = 2tan−1 � 𝑦 𝑥 � Show that 𝑑𝑦 𝑑𝑥 = 𝑥 + 𝑦 𝑥 − 𝑦 . [CBSE 2019] 39. If 𝑥 = 𝑎cos 𝜃 + 𝑏sin 𝜃 and 𝑦 = 𝑎sin 𝜃 −𝑏cos 𝜃, then show that 𝑦2 𝑑2 𝑦 𝑑𝑥2 − 𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 = 0. [CBSE 2015] 40. If 𝑦 = 𝑥cos−1 𝑥 �(1−𝑥2) − log �(1 − 𝑥2), then prove that 𝑑𝑦 𝑑𝑥 = cos−1 𝑥 (1−𝑥2)3/2. [CBSE 2015] 41. If 𝑥 = 𝑎sin 2𝑡(1 + cos 2𝑡) and 𝑦 = 𝑏cos 2𝑡(1 −cos 2𝑡), show that at 𝑡 = 𝜋 4 , � 𝑑𝑦 𝑑𝑥 � = 𝑏 𝑎 . [CBSE 2015] 42. If 𝑥 = 𝑎𝑒𝑡 (sin 𝑡 + cos 𝑡) and 𝑦 = 𝑎𝑒𝑡 (sin 𝑡 − cos 𝑡), then prove that : 𝑑𝑦 𝑑𝑥 = (𝑥+𝑦) (𝑥−𝑦) . [CBSE 2015] 43. If 𝑒𝑥 + 𝑒𝑦 = 𝑒𝑥+𝑦 , then show that 𝑑𝑦 𝑑𝑥 = −𝑒𝑦−𝑥 . [CBSE 2014] 44. Differentiate tan−1 � 𝑥 √1−𝑥2 � with respect to sin−1 �2𝑥√1 − 𝑥2�. [CBSE 2014] 45. Find the value of 𝑑𝑦 𝑑𝑥 at 𝜃 = 𝑥 4 , if 𝑥 = 𝑎𝑒𝜃 (sin 𝜃 − cos 𝜃) and 𝑦 = 𝑎𝑒𝜃 (sin 𝜃 + cos 𝜃). [CBSE 2014] 4. SECOND ORDER DERIVATIVE 46. If 𝑦 = 5cos 𝑥 − 3sin 𝑥, then 𝑑2𝑦 𝑑𝑥2 is equal to: (a) −𝑦 (b) 𝑦 (c) 25𝑦 (d) 9𝑦 [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
  • 61.
    47. If 𝑥= 𝑎sec 𝜃, 𝑦 = 𝑏tan 𝜃, then 𝑑2𝑦 𝑑𝑥2 at 𝜃 = 𝜋 6 is: (a) −3√3𝑏 𝑎2 (b) −2√3𝑏 𝑎 (c) −3√3𝑏 𝑎 (d) −𝑏 3√3𝑎2 [CBSE Term-1 SQP 2021] 48. If 𝑦 = log𝑒 � 𝑥2 𝑒2�, then 𝑑2𝑦 𝑑𝑥2 is equal to: (a) −1 𝑥 (b) − 1 𝑥2 (c) 2 𝑥2 (d) − 2 𝑥2 [CBSE 2020] Very Short & Short Qs (1 - 3 marks) 49. If 𝑦 = √𝑎𝑥 + 𝑏, prove that 𝑦 � 𝑑2𝑦 𝑑𝑥2� + � 𝑑𝑦 𝑑𝑥 � 2 = 0. [CBSE 2023] 50. If 𝑥 = 𝑎cos 𝜃, 𝑦 = 𝑏sin 𝜃, then find 𝑑2𝑦 𝑑𝑥2. [CBSE 2020] 51. If 𝑥 = 𝑎𝑡2 , 𝑦 = 2𝑎𝑡 then find 𝑑2𝑦 𝑑𝑥2. [CBSE 2020] 52. If 𝑒𝑦 (𝑥 + 1) = 1, then show that 𝑑2𝑦 𝑑𝑥2 = � 𝑑𝑦 𝑑𝑥 � 2 . [CBSE 2017] 53. If 𝑥𝑚 ⋅ 𝑦𝑛 = (𝑥 + 𝑦)𝑚+𝑛 , prove that 𝑑2𝑦 𝑑𝑥2 = 0. [CBSE 2017] 54. If 𝑦 = 2cos (log 𝑥) + 3sin (log 𝑥), prove that 𝑥2 𝑑2 𝑦 𝑑𝑥2 + 𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 = 0 [CBSE 2016] 55. If 𝑦 = 𝐴𝑒𝑚𝑥 + 𝐵𝑒𝑛𝑥 , show that 𝑑2𝑦 𝑑𝑥2 − (𝑚 + 𝑛) 𝑑𝑦 𝑑𝑥 + 𝑚𝑛𝑦 = 0. [CBSE 2015, 2014] CLICK HERE FOR SOLUTIONS
  • 62.
    Long Qs (4- 5 marks) 56. If 𝑥 = sin 𝑡 and 𝑦 = sin 𝑝𝑡, prove that (1 − 𝑥2) 𝑑2 𝑦 𝑑𝑥2 − 𝑥 𝑑𝑦 𝑑𝑥 + 𝑝2 𝑦 = 0 [CBSE 2019] 57. If 𝑦 = (sin−1 𝑥)2 , then prove that: (1 − 𝑥2) 𝑑2 𝑦 𝑑𝑥2 − 𝑥 � 𝑑𝑦 𝑑𝑥 � − 2 = 0 [CBSE 2019] 58. If 𝑦 = 𝑥𝑥 , prove that 𝑑2𝑦 𝑑𝑥2 − 1 𝑦 � 𝑑𝑦 𝑑𝑥 � 2 − 𝑦 𝑥 = 0. [CBSE 2016, 2014] 59. If 𝑦 = 2cos (log 𝑥) + 3sin (log 𝑥), prove that 𝑥2 𝑑2𝑦 𝑑𝑥2 + 𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 = 0 [CBSE 2016] 60. If 𝑥 = 𝑎 �cos 𝑡 + log tan 𝑡 2 � and 𝑦 = 𝑎sin 𝑡, then find 𝑑2𝑦 𝑑𝑥2 at 𝑡 = 𝜋 3 . [CBSE 2014] CLICK HERE FOR SOLUTIONS
  • 63.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 64.
    Application of Derivatives 1.RATE OF CHANGE OF QUANTITIES Case Based Qs (4- 5 marks) 1. The relation between the height of the plant (' 𝑦 ' in cm ) with respect to its exposure to the sunlight is governed by the following equation 𝑦 = 4𝑥 − 1 2 𝑥2 , where ' 𝑥 ' is the number of days exposed to the sunlight, for 𝑥 ≤ 3. Based on the above information, answer the following questions: (A) Find the rate of growth of the plant with respect to the number of days exposed to the sunlight. (B) Does the rate of growth of the plant increase or decrease in the first three days? What will be the height of the plant after 2 days? [CBSE SQP 2023] Very Short & Short Qs (1 − 3 marks) 2. If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius. [CBSE 2023] 3. A man 1.6 m tall walks at the rate of 0.3 m/s away from a street light that is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening? [CBSE SQP 2022] 4. The radius of a circle is increasing at the uniform rate of 3 cm/sec. At the instant when the radius of the circle is 2 cm, find its area increases at the rate of cm2 /sec. [CBSE 2020] CH-6 CLICK HERE FOR SOLUTIONS
  • 65.
    5. A ladder13 m long is leaning against a vertical wall. The bottom of the ladder is dragged away from the wall along the ground at the rate of 2 cm/sec. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall? [CBSE 2019] 6. The volume of a cube is increasing at the rate of 8 cm3 /s. How fast is the surface area increasing when the length of its edge is 12 cm ? [CBSE 2019] 7. The total cost 𝐶(𝑥) in Rupees, associated with the production of 𝑥 units of an item is given by 𝐶(𝑥) = 0.005𝑥3 − 0.02𝑥2 + 30𝑥 + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. [CBSE 2018] 8. The length 𝑥 of a rectangle is decreasing at the rate of 5 cm/ minute and the width 𝑦 is increasing at the rate of 4 cm/ minute. When 𝑥 = 8 cm and 𝑦 = 6 cm, find the rate of change of: (A) the perimeter. (B) area of rectangle. [CBSE 2017] 9. The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm. [CBSE 2017] Long Qs (4 - 5 marks) 10. The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing, when the side of the triangle is 20 cm. [CBSE 2015] 2. INCREASING AND DECREASING FUNCTIONS Objective Qs (1 mark) 11. The real function 𝑓(𝑥) = 2𝑥3 − 3𝑥2 − 36𝑥 + 7 is: (a) strictly increasing in (−∞, −2) and strictly decreasing in (−2, ∞). (b) strictly decreasing in (−2,3). (c) strictly decreasing in (−∞, 3) and strictly increasing in (3, ∞). (d) strictly decreasing in (−∞, −2) ∪ (3, ∞). CLICK HERE FOR SOLUTIONS
  • 66.
    [CBSE Term-1 SQP2021] 12. The value of 𝑏 for which the function 𝑓(𝑥) = 𝑥 + cos 𝑥 + 𝑏 is strictly decreasing over 𝑅 is : (a) 𝑏 < 1 (b) no value of 𝑏 exists (c) 𝑏 ≤ 1 (d) 𝑏 ≥ 1 [CBSE Term-1 SQP 2021] 13. Find the intervals in which the function 𝑓 given by 𝑓(𝑥) = 𝑥2 − 4𝑥 + 6 is strictly increasing. (a) (−∞, 2) ∪ (2, ∞) (b) (2, ∞) (c) (−∞, 2) (d) (−∞, 2] ∪ (2, ∞) [CBSE Term-1 SQP 2021] 14. The interval in which 𝑦 = 𝑥2 𝑒−𝑥 is increasing, is: (a) (−∞, ∞) (b) (−2,0) (c) (2, ∞) (d) (0,2) [CBSE Term-1 2021] Case Based Qs ( 4 - 5 marks) 15. The use of electric vehicles will curb air pollution in the long run. The use of electric vehicles is increasing every year and estimated electric vehicles in use at any time 𝑡 is given by the function 𝑉. 𝑉(𝑡) = 1 5 𝑡3 − 5 2 𝑡2 + 25𝑡 − 2 where 𝑡 represents the time and 𝑡 = 1,2,3 … corresponds to year 2001, 2002, 2003, ....... respectively. Based on the above information, answer the following questions: (A) Can the above function be used to estimate number of vehicles in the year 2000? Justify. (B) Prove that the function 𝑉(𝑡) is an increasing function. CLICK HERE FOR SOLUTIONS
  • 67.
    [CBSE 2023] 16. Thetemperature of a person during an intestinal illness is given by 𝑓(𝑥) = −0.1𝑥2 + 𝑚𝑥 + 98.6, 0 ≤ 𝑥 ≤ 12, 𝑚 being a constant, where 𝑓(𝑥) is the temperature in ∘ F at 𝑥 days. Based on the above information, answer the following questions: (A) Is the function differentiable in the interval (0,12) ? Justify your answer. (B) If 6 is the critical point of the function, then find the value of the constant 𝑚. (C) Find the intervals in which the function is strictly increasing/strictly decreasing. [CBSE SQP 2022] Very Short & Short Qs (1 − 3 marks) 17. Find the interval/s in which the function 𝑓: 𝑅 → 𝑅 defined by 𝑓(𝑥) = 𝑥𝑒𝑥 , is increasing. [CBSE SQP 2023] 18. Check whether the function 𝑓: 𝑅 → 𝑅 defined by 𝑓(𝑥) = 𝑥3 + 𝑥, has any critical point/s or not? If yes, then find the point/s. [CBSE SQP 2023] 19. Find the interval in which the function 𝑓 given by 𝑓(𝑥) = 7 − 4𝑥 − 𝑥2 is strictly increasing. [CBSE 2020] 20. Find the intervals in which the function 𝑓 given by 𝑓(𝑥) = tan 𝑥 − 4𝑥, 𝑥 ∈ �0, 𝜋 2 � is : (A) Strictly increasing (B) Strictly decreasing [CBSE 2020] 21. Find the interval in which the function 𝑓(𝑥) = 𝑥4 4 − 𝑥3 − 5𝑥2 + 24𝑥 + 12 is: (A) strictly increasing. (B) strictly decreasing. CLICK HERE FOR SOLUTIONS
  • 68.
    [CBSE 2018] 22. Showthat the function 𝑓(𝑥) = 4𝑥3 − 18𝑥2 + 27𝑥 − 7 is always increasing on 𝑅. [CBSE 2017] 23. Find the interval in which 𝑓(𝑥) = sin 3𝑥 − cos 3𝑥, 0 < 𝑥 < 𝜋, is strictly increasing or strictly decreasing. [CBSE 2016] 24. Find the value (𝑠) of 𝑥 for which 𝑦 = [𝑥(𝑥 − 2)]2 is an increasing function. [CBSE 2014] 25. Prove that the function 𝑓 defined by 𝑓(𝑥) = 𝑥2 − 𝑥 + 1 is neither increasing nor decreasing in (−1,1). Hence, find the intervals in which 𝑓(𝑥) is: (A) strictly increasing. (B) strictly decreasing. [CBSE 2014] 26. Find intervals in which the function given by 𝑓(𝑥) = 3 10 𝑥4 − 4 5 𝑥3 − 3𝑥2 + 36 5 𝑥 + 11 is (a) strictly increasing (b) strictly decreasing. [CBSE 2014] 3. MAXIMA AND MINIMA Objective Qs (1 mark) 27. The maximum value of � 1 𝑥 � 𝑥 is: (a) 𝑒1/𝑒 (b) 𝑒 (c) � 1 𝑒 � 1/𝑒 (d) 𝑒𝑒 [CBSE Term-1 2021] 28. The maximum value of [𝑥(𝑥 − 1) + 1]1/3 , 0 ≤ 𝑥 ≤ 1 is: (a) 0 (b) 1 2 (c) 1 (d) � 1 3 � 1/3 [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
  • 69.
    29. The leastvalue of the function 𝑓(𝑥) = 2cos 𝑥 + 𝑥 in the closed interval �0, 𝜋 2 � is: (a) 2 (b) 𝜋 6 + √3 (c) 𝜋 2 (d) The least value does not exist. [CBSE Term-1 SQP 2021] 30. The area of a trapezium is defined by function 𝑓 and is given by 𝑓(𝑥) = (10 + 𝑥)�100 − 𝑥2 Then the area when it is maximised is: (a) 75 cm2 (b) 7√3 cm2 (c) 75√3 cm2 (d) 5 cm2 [CBSE Term-1 SQP 2021] 31. The absolute minimum value of the function 𝑓(𝑥) = 𝑥3 − 12𝑥 on the interval [0,3] is: (a) 0 (b) -9 (c) -16 (d) -19 [CBSE Term-1 2021] 32. The maximum value of slope of the curve 𝑦 = −𝑥3 + 3𝑥2 + 12𝑥 − 5 is: (a) 15 (b) 12 (c) 9 (d) 0 [CBSE 2020] 33. Let 𝑓(𝑥) be a polynomial function of degree 6 such that 𝑑 𝑑𝑥 (𝑓(𝑥)) = (𝑥 − 1)3 (𝑥 − 3)3 , then Assertion (𝐴): 𝑓(𝑥) has a minimum at 𝑥 = 1. Reason (R): When 𝑑 𝑑𝑥 (𝑓(𝑥)) < 0, ∀𝑥 ∈ (𝑎 − ℎ, 𝑎), and 𝑑 𝑑𝑥 (𝑓(𝑥)) > 0, ∀𝑥 ∈ (𝑎, 𝑎 + ℎ); where ' ℎ ' is an infinitesimally small positive quantity, then 𝑓(𝑥) has a minimum at 𝑥 = 𝑎, provided 𝑓(𝑥) is continuous at 𝑥 = 𝑎. [CBSE SQP 2023] (a) Both (A) and (R) are true and (R) is the correct explanation of (A). (b) Both (A) and (R) are true but (R) is not the correct explanation of (𝐴). (c) (A) is true but (R) is false. CLICK HERE FOR SOLUTIONS
  • 70.
    (d) (A) isfalse but (𝑅) is true. Case Based Qs (4 - 5 marks) 34. Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore. The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area 75𝜋cm2 . Based on the above information, answer the following questions: (A) If the radius of cylinder is 𝑟 cm and height is ℎ cm, then write the volume 𝑉 of cylinder in terms of radius 𝑟. (B) Find 𝑑𝑉 𝑑𝑟 . (C) Find the radius of cylinder when its volume is maximum. OR For maximum volume, ℎ > 𝑟. State true or false and justify. [CBSE 2023] CLICK HERE FOR SOLUTIONS
  • 71.
    35. In anelliptical sport field, the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of 𝑥2 𝑎2 + 𝑦2 𝑏2 = 1. Based on the above information, answer the following questions: (A) If the length and the breadth of the rectangular field be 2𝑥 and 2𝑦 respectively, then find the area function in terms of 𝑥. (B) Find the critical point of the function. (C) Use first derivative test to find the length 2𝑥 and width 2𝑦 of the soccer field (in terms of 𝑎 and 𝑏 ) that maximize its area. [CBSE SQP 2022] Very Short & Short Qs (1 - 3 marks) 36. If 𝑓(𝑥) = 1 4𝑥2+2𝑥+1 ; 𝑥 ∈ 𝑅, then find the maximum value of 𝑓(𝑥). [CBSE SQP 2023] 37. Find the maximum profit that a company can make, if the profit function is given by 𝑃(𝑥) = 72 + 42𝑥 − 𝑥2 , where 𝑥 is the number of units and 𝑃 is the profit in rupees. [CBSE SQP 2023] 38. Find the absolute minimum volume of 𝑓(𝑥) = 2sin 𝑥 in �0, 3𝜋 2 �. [CBSE 2020] 39. Find the least value of the function 𝑓(𝑥) = 𝑎𝑥 + 𝑏 𝑥 , (𝑎 > 0, 𝑏 > 0, 𝑥 > 0). [CBSE 2020] maximum light through the whole opening. [CBSE 2018] 40. Show that the height of the right circular cylinder of greatest volume which can be inscribed in a right circular cone of height ℎ and radius 𝑟 is one-third of the height of the cone and greatest volume of the cylinder is 4 9 times the volume of the cone. [CBSE 2020] 41. Find the minimum value of (𝑎𝑥 + 𝑏𝑦), where 𝑥𝑦 = 𝑐2 . CLICK HERE FOR SOLUTIONS
  • 72.
    [CBSE 2020, 15] 42.A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3 . If building of tank costs ₹70 per sq. meters for the base and ₹ 45 per square meter for sides. What is the cost of least expensive tank? [CBSE 2019] 43. Find the point on the curve 𝑦2 = 4𝑥, which is nearest to the point (2, −8). [CBSE 2019] 44. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius 𝑅 is 2𝑅 √3 . Also find the volume of the largest cylinder inscribed in a sphere of radius 𝑅. [CBSE 2019] 45. A window is of the form of a semi-circle with a rectangle on its diameter. The total perimeter of the window is 10 m. Find the dimensions of the window to admit 46. An open tank with a square base and vertical sides is to be construted from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. [CBSE 2018] 47. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is cos−1 1 √3 . [CBSE 2016] 48. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius 𝑟 is 4𝑟 3 . Also find maximum volume in terms of volume of the sphere. [CBSE 2016, 14] 49. Prove that the least perimeter of an isosceles triangle in which a circle of radius 𝑟 can be inscribed is 6√3𝑟. [CBSE 2016] 50. Find the local maxima and local minima of the function 𝑓(𝑥) = sin 𝑥 − cos 𝑥, 0 < 𝑥 < 2𝜋. Also find the local maximum and local minimum values. [CBSE 2015] 51. The sum of the perimeters of a circle and square is 𝑘, where 𝑘 is some constant. Prove that the sum of their areas is least when the side of the square is double the radius of the circle. [CBSE 2014] CLICK HERE FOR SOLUTIONS
  • 73.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 74.
    Integrals 1. INDEFINITE INTEGRALS ObjectiveQs (1 mark) 1. If 𝑑 𝑑𝑥 𝑓(𝑥) = 2𝑥 + 3 𝑥 and 𝑓(1) = 1, then 𝑓(𝑥) is: (a) 𝑥2 + 3log |𝑥| + 1 (c) 2 − 3 𝑥2 (b) 𝑥2 + 3log |𝑥| (d) 𝑥2 + 3log |𝑥| − 4 [CBSE 2023] 2. ∫ 𝑥3 𝑥+1 𝑑𝑥 is: (a) 𝑥 + 𝑥2 2 + 𝑥3 3 − log |1 − 𝑥| + 𝐶 (b) 𝑥 + 𝑥2 2 − 𝑥3 3 − log |1 − 𝑥| + 𝐶 (c) 𝑥 − 𝑥2 2 − 𝑥3 3 − log |1 + 𝑥| + 𝐶 (d) 𝑥 − 𝑥2 2 + 𝑥3 3 − log |1 + 𝑥| + 𝐶 [Delhi Gov. SQP 2022] Very Short & Short Qs (1-3 marks) 3. Find: ∫ � 𝑥 1−𝑥3 𝑑𝑥; 𝑥 ∈ (0,1). [CBSE SQP 2023] 4. Find ∫ sin2 𝑥−cos2 𝑥 sin 𝑥cos 𝑥 𝑑𝑥 [CBSE 2017] 5. Write the anti-derivative of �3√𝑥 + 1 √𝑥 �. [CBSE 2014] 6. Evaluate ∫ cos−1 (sin 𝑥)𝑑𝑥 CH-7 CLICK HERE FOR SOLUTIONS
  • 75.
    [CBSE 2014] 2. METHODSOF INTEGRATION Objective Qs (1 mark) 7. Anti-derivative of tan 𝑥−1 tan 𝑥+1 with respect to 𝑥 is: (a) sec2 � 𝜋 4 − 𝑥� + 𝑐 (b) −sec2 � 𝜋 4 − 𝑥� + 𝑐 (c) log �sec � 𝜋 4 − 𝑥�� + 𝑐 (d) −log �sec � 𝜋 4 − 𝑥�� + 𝑐 8. ∫ 𝑥2 𝑒𝑥3 𝑑𝑥 equals: (a) 1 3 𝑒𝑥3 + 𝐶 (b) 1 3 𝑒𝑥4 + 𝐶 (c) 1 2 𝑒𝑥3 + 𝐶 (d) 1 2 𝑒𝑥2 + 𝐶 [Delhi Gov. SQP 2022, CBSE 2020] 9. ∫ 𝑒𝑥 � 𝑥log 𝑥+1 𝑥 � 𝑑𝑥 is equal to (a) log (𝑒𝑥 log 𝑥) + 𝐶 (b) 𝑒𝑥 𝑥 + 𝐶 (c) 𝑥log 𝑥 + 𝑒𝑥 + 𝐶 (d) 𝑒𝑥 log 𝑥 + 𝐶 [CBSE 2020] 10. ∫ 1 sin2 𝑥cos2 𝑥 𝑑𝑥 is: (a) tan 𝑥 + cot 𝑥 + 𝐶 (b) tan 𝑥 − cot 𝑥 + 𝐶 (c) tan 𝑥cot 𝑥 + 𝐶 (d) tan 𝑥 − cot 2𝑥 + 𝐶 [CBSE 2014] Very Short & Short Qs (1 − 3 marks) 11. Find: ∫ 2𝑥2+3 𝑥2(𝑥2+9) 𝑑𝑥; 𝑥 ≠ 0 CLICK HERE FOR SOLUTIONS
  • 76.
    [CBSE 2023] 12. Find:∫ 𝑒𝑥 � 1−sin 𝑥 1−cos 𝑥 � 𝑑𝑥. [CBSE 2023] 13. Find: ∫ sin−1 𝑥 (1−𝑥2)3/2 𝑑𝑥. [CBSE 2023] 14. Find ∫ �𝑥3+𝑥+1� (𝑥2−1) 𝑑𝑥 [CBSE SQP 2022] 15. Evaluate: ∫ 𝑑𝑥 √4𝑥−𝑥2 ⋅ [CBSE Term-2 2022] 16. Find: ∫ sin 2𝑥 √9−cos4 𝑥 𝑑𝑥. [CBSE Term-2 SQP 2022] 17. Find: ∫ 𝑑𝑥 √3−2𝑥−𝑥2 [CBSE SQP 2022] 18. Integrate ∫ 2𝑥 (𝑥2+1)(𝑥2+2) with respect of 𝑥. [CBSE Term-2 2022] 19. Find: ∫ 𝑥+1 (𝑥2+1)𝑥 𝑑𝑥 [CBSE Term-2 SQP 2022] 20. Find: ∫ 𝑑𝑥 √9−4𝑥2 [CBSE 2020] 21. Find: ∫ 𝑥4 log 𝑥𝑑𝑥 [CBSE 2020] 22. Find: ∫ sec3 𝑥𝑑𝑥. [CBSE 2020] 23. Evaluate: ∫ 𝑥+1 (𝑥+2)(𝑥+3) 𝑑𝑥 [CBSE 2020] 24. Integrate: ∫ 2cos 𝑥 (1−sin 𝑥)(1+sin2 𝑥) 𝑑𝑥 [CBSE 2020] CLICK HERE FOR SOLUTIONS
  • 77.
    25. Find: ∫√1 − sin 2𝑥𝑑𝑥, 𝜋 4 < 𝑥 < 𝜋 2 [CBSE 2019] 26. Evaluate: ∫ 𝑑𝑥 √5−4𝑥−2𝑥2 [CBSE 2019] 27. Integrate the function cos (𝑥+𝑎) sin (𝑥+𝑏) with respect to 𝑥. [CBSE 2019] 28. Find: ∫ sec2 𝑥 √tan2 𝑥+4 𝑑𝑥 [CBSE 2019] 29. Find: ∫ log 𝑥 (1+log 𝑥)2 𝑑𝑥. [CBSE Term-2 SQP 2022] 30. Integrate: ∫ 1 𝑒𝑥+1 𝑑𝑥 [CBSE Term-2 2022] Longs Qs (4 - 5 marks) 31. Find: ∫ sin−1 (2𝑥)𝑑𝑥. [CBSE 2019] 32. Find: ∫ 𝑥tan−1 𝑥𝑑𝑥 [CBSE 2019] 33. Find: ∫ 2cos 𝑥 (1−sin 𝑥)(2−cos2 𝑥) 𝑑𝑥. [CBSE 2019] 34. Find: ∫ 2𝑥 (𝑥2+1)(𝑥4+4) 𝑑𝑥. [CBSE 2017] 35. Find: ∫ cos 𝜃 (4+sin2 𝜃)(5−4cos2 𝜃) 𝑑𝜃 𝜃 [CBSE 2017] 36. Find: ∫ 𝑑𝑥 5−8𝑥−𝑥2. [CBSE 2017] 37. Find: ∫ (3sin 𝜃−2)cos 𝜃 5−cos2 𝜃−4sin 𝜃 𝑑𝜃. [CBSE 2016] CLICK HERE FOR SOLUTIONS
  • 78.
    38. Evaluate: ∫ √𝑥 √𝑎3−𝑥3 𝑑𝑥 [CBSE2016] 39. Integrate the following rational function: 𝑥 (𝑥−1)2(𝑥+2) with respect to 𝑥 [CBSE 2015] 40. Find ∫ 𝑥2+𝑥+1𝑑𝑥 (𝑥+2)(𝑥2+1) [CBSE 2015] 41. Find: ∫ sin6 𝑥 cos8 𝑥 𝑑𝑥 [CBSE 2014] 42. Find ∫ 𝑥3 (𝑥4+3𝑥2+2) 𝑑𝑥 [CBSE 2014] 43. Evaluate: ∫ (√cot 𝑥 + √tan 𝑥)𝑑𝑥. [CBSE 2014] 44. Integrate the function √𝑥2 + 1[log (𝑥2 + 1) − 2log 𝑥] 𝑥4 [CBSE 2014] 45. Find: ∫ 𝑥2 (𝑥2+1)(𝑥2+4) 𝑑𝑥 [CBSE 2014] 46. Evaluate: ∫ 1 cos4 𝑥+sin4 𝑥 𝑑𝑥 [CBSE 2014] 47. Evaluate: ∫ 1 sin4 𝑥+sin2 𝑥cos2 𝑥+cos4 𝑥 𝑑𝑥. [CBSE 2014] 3. DEFINITE INTEGRALS Objective Qs (1 mark) 48. For any integer 𝑛, the value of � 𝜋 −𝜋 𝑒cos2 𝑥 sin3 (2𝑛 + 1)𝑥𝑑𝑥 is: CLICK HERE FOR SOLUTIONS
  • 79.
    (a) -1 (b) 0 (c)1 (d) 2 [CBSE SQP 2023] 49. ∫−𝜋/4 𝜋/4 (sec2 𝑥)𝑑𝑥 is equal to: (a) -1 (b) 0 (c) 1 (d) 2 [CBSE 2020] Very Short & Short Qs (1 – 3 marks) 50. Evaluate: ∫−1 1 log𝑒 � 2−𝑥 2+𝑥 � 𝑑𝑥. [CBSE SQP 2023] 51. Evaluate: ∫0 𝜋 4 log (1 + tan 𝑥)𝑑𝑥. [CBSE SQP 2023] 52. Evaluate: ∫−1 1 |𝑥4 − 𝑥|𝑑𝑥. [CBSE 2023] 53. Evaluate: ∫log √2 log √3 1 (𝑒𝑥+𝑒−𝑥)(𝑒𝑥−𝑒−𝑥) 𝑑𝑥. [CBSE 2023] 54. Evaluate: ∫2 3 𝑥 𝑥2+1 𝑑𝑥 [CBSE SQP 2022] 55. Evaluate: ∫0 4 |𝑥 − 1|𝑑𝑥 [CBSE SQP 2022] 56. Evaluate: ∫𝜋/6 𝜋/3 𝑑𝑥 1+√tan 𝑥 [CBSE SQP 2022] 57. Evaluate: ∫1 4 {|𝑥| + |3 − 𝑥|}𝑑𝑥. [CBSE Term-2 2022] 58. Evaluate: ∫1 3 √𝑥 √𝑥+√4−𝑥 𝑑𝑥. [CBSE Term-2 2022] CLICK HERE FOR SOLUTIONS
  • 80.
    59. Evaluate: ∫−1 2 |𝑥3 −3𝑥2 + 2𝑥|𝑑𝑥 [CBSE Term-2 SQP 2022] 60. Evaluate: ∫1 2 � 1 𝑥 − 1 2𝑥2� 𝑒2𝑥 𝑑𝑥. [CBSE 2020] 61. Evaluate: ∫−2 2 |𝑥|𝑑𝑥. [CBSE 2020] 62. Evaluate: ∫1 3 |2𝑥 − 1|𝑑𝑥. [CBSE 2020] 63. Evaluate: ∫−𝜋 𝜋 (1 − 𝑥2)sin 𝑥 ⋅ cos2 𝑥𝑑𝑥. [CBSE 2019] 64. Evaluate: ∫−1 2 |𝑥| 𝑥 𝑑𝑥 [CBSE 2019] 65. Prove that: ∫0 𝑎 𝑓(𝑥)𝑑𝑥 = ∫0 𝑎 𝑓(𝑎 − 𝑥)𝑑𝑥, hence evaluate ∫0 𝜋 𝑥sin 𝑥 1+cos2 𝑥 𝑑𝑥 [CBSE 2019] 66. Prove that: ∫0 𝑎 𝑓(𝑥)𝑑𝑥 = ∫0 𝑎 𝑓(𝑎 − 𝑥)𝑑𝑥 and hence evaluate ∫0 1 𝑥2 (1 − 𝑥)𝑛 𝑑𝑥, [CBSE 2019] 67. Evaluate: ∫0 𝜋/4 sin 𝑥+cos 𝑥 16+9sin 2𝑥 𝑑𝑥. [CBSE 2018] 68. Evaluate: ∫2 3 3𝑥 𝑑𝑥 [CBSE 2017] 69. Evaluate: ∫0 𝜋 𝑥tan 𝑥 sec 𝑥+tan 𝑥 𝑑𝑥 [CBSE 2017] 70. Evaluate: ∫0 𝜋 𝑒2𝑥 ⋅ sin � 𝜋 4 + 𝑥� 𝑑𝑥. [CBSE 2016] 71. Evaluate: ∫−1 2 |𝑥3 − 𝑥|𝑑𝑥 [CBSE 2016] 72. Evaluate: ∫−𝜋 𝜋 (cos 𝑎𝑥 − sin 𝑏𝑥)2 𝑑𝑥. [CBSE 2015] 73. Evaluate: ∫0 𝜋/2 𝑑𝑥 1+√tan 𝑥 CLICK HERE FOR SOLUTIONS
  • 81.
    [CBSE 2015] 74. Evaluate:∫0 𝜋/4 𝑑𝑥 cos3 𝑥√2sin 2𝑥 𝑑𝑥. [CBSE 2015] 75. Evaluate: ∫0 1 𝑥𝑒𝑥2 𝑑𝑥 [CBSE 2014] 76. Evaluate: ∫2 4 𝑥 𝑥2+1 𝑑𝑥. [CBSE 2014] 77. If 𝑓(𝑥) = ∫0 𝑥 𝑡sin 𝑡𝑑𝑡, then write the value of 𝑓′ (𝑥). [CBSE 2014] 78. Evaluate: ∫0 𝜋/2 𝑥2 sin 𝑥𝑑𝑥. [CBSE 2014] 79. Evaluate: ∫𝜋/6 𝜋/3 sin 𝑥+cos 𝑥 √sin 2𝑥 𝑑𝑥. [CBSE 2014] CLICK HERE FOR SOLUTIONS
  • 82.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 83.
    Application of Integrals 1.APPLICATION OF DEFINITE INTEGRALS Very Short & Short Qs (1 -3 marks) 1. Using integration, find the area of the region {(𝑥, 𝑦): 𝑦2 ≤ 𝑥 ≤ 𝑦}. [CBSE Term-2 2022] 2. Find the area bounded by the curves 𝑦 = |𝑥 − 1| and 𝑦 = 1, using integration. [CBSE Term-2 2022] 3. Using integration, find the smaller area enclosed by the circle 𝑥2 + 𝑦2 = 4 and the line 𝑥 + 𝑦 = 2. [CBSE 2020] 4. Find the area enclosed between the parabola 4𝑦 = 3𝑥2 and the straight line 3𝑥 − 2𝑦 + 12 = 0. [CBSE 2015] 5. Using integration, find the area of the region bounded by the line 𝑥 − 𝑦 + 2 = 0, the curve 𝑥 = �𝑦 and 𝑦-axis. [CBSE 2015] 6. Find the area of the smaller region bounded by the ellipse 𝑥2 9 + 𝑦2 4 = 1 and the line 𝑥 3 + 𝑦 2 = 1 [CBSE 2014] Long Qs (4 – 5 marks) 7. Make a rough sketch of the region {(𝑥, 𝑦): 0 ≤ 𝑦 ≤ 𝑥2 + 1,0 ≤ 𝑦 ≤ 𝑥 + 1,0 ≤ 𝑥 ≤ 2} and find the area of the region, using the method of integration. [CBSE SQP 2023] 8. Using Integration, find the area of triangle whose vertices are (−1,1), (0,5) and (3,2). [CBSE 2023] 9. Make a rough sketch of the region {(𝑥, 𝑦) : 0 ≤ 𝑦 < 𝑥2 , 0 ≤ 𝑦 < 𝑥, 0 ≤ 𝑥 ≤ 2} and find the area of the region using integration. [CBSE SQP 2022] 10. Using integration, find the area of the region bounded by the curves 𝑥2 + 𝑦2 = 4, 𝑥 = √3𝑦 and 𝑥- axis lying in the first quadrant. [CBSE Term-2 2022] CH-8 CLICK HERE FOR SOLUTIONS
  • 84.
    11. Using integration,find the area of the region in the first quadrant enclosed by the 𝑥-axis, the line 𝑦 = 𝑥 and the circle 𝑥2 + 𝑦2 = 32. [CBSE 2018, 2015, 2014] 12. Find the area bounded by the circle 𝑥2 + 𝑦2 = 16 and the line √3𝑦 = 𝑥 in the first quadrant, using integration. [CBSE 2017] 13. Using integration, find the area bounded by the curves: 𝑦 = |𝑥 + 1| + 1, 𝑥 = −3, 𝑥 = 3 and 𝑦 = 0. [CBSE 2014] 14. Using integration, find the area bounded by the curve 𝑥2 = 4𝑦 and the line 𝑥 = 4𝑦 − 2. [CBSE 2014] 15. Using integration, find the area of the region in the first quadrant enclosed by the 𝑥-axis, the line 𝑦 = 𝑥 and the circle 𝑥2 + 𝑦2 = 18. [CBSE 2014] 16. Using integration, find the area of the region bounded by the triangle whose vertices are (−1,2), (1,5) and (3,4). [CBSE 2014] CLICK HERE FOR SOLUTIONS
  • 85.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 86.
    Differential Equations 1. ORDERAND DEGREE OF DIFFERENTIAL EQUATION Objective Qs (1 mark) 1. The degree of the differential equation �1 + � 𝑑𝑦 𝑑𝑥 � 2 � 3 = � 𝑑2𝑦 𝑑𝑥2� 2 is: (a) 4 (b) 3 2 (c) 2 (d) not defined [CBSE SQP 2023] 2. Degree of the differential equation sin 𝑥 + cos � 𝑑𝑦 𝑑𝑥 � = 𝑦2 is: (a) 2 (b) 1 (c) not defined (d) 0 [CBSE 2023] 3. If 𝑚 and 𝑛 respectively, are the order and the degree of the differential equation 𝑑 𝑑𝑥 �� 𝑑𝑦 𝑑𝑥 �� 4 = 0, then 𝑚 + 𝑛 = (a) 1 (b) 2 (c) 3 (d) 4 [CBSE SQP 2022] Very Short & Short Qs (1 - 3 marks) 4. Write the sum of the order and the degree of the following differential equation: 𝑑 𝑑𝑥 � 𝑑𝑦 𝑑𝑥 � = 5. [CBSE SQP Term-2 2022] 5. The degree of the differential equation 1 + � 𝑑𝑦 𝑑𝑥 � 2 = 𝑥 is [CBSE 2020] 6. Find the order and the degree of the differential equation 𝑥2 𝑑2𝑦 𝑑𝑥2 = �1 + � 𝑑𝑦 𝑑𝑥 � −2 � 4 . [CBSE 2019] CH-9 CLICK HERE FOR SOLUTIONS
  • 87.
    7. Write thedegree of the differential equation 𝑥3 � 𝑑2𝑦 𝑑𝑥2� 2 + � 𝑑𝑦 𝑑𝑥 � 4 = 0 [CBSE 2019] 8. Find the order and degree (if defined) of the differential equation 𝑑2 𝑦 𝑑𝑥2 + 𝑥 � 𝑑𝑦 𝑑𝑥 � 2 = 2𝑥3 log � 𝑑2 𝑦 𝑑𝑥2 � [CBSE 2019] 9. Write the sum of the order and degree of the differential equation 𝑑 𝑑𝑥 �� 𝑑𝑦 𝑑𝑥 � 3 � = 0. [CBSE 2015] 10. Write the sum of the order and degree of the differential equation � 𝑑2𝑦 𝑑𝑥2� 2 + � 𝑑𝑦 𝑑𝑥 � 3 + 𝑥4 = 0. [CBSE 2015] 11. Write the sum of order and degree of differential equation, 1 + � 𝑑𝑦 𝑑𝑥 � 4 = 7 � 𝑑2 𝑦 𝑑𝑥2 � 3 [CBSE 2015] 2. SOLUTIONS OF FIRST ORDER, FIRST DEGREE DIFFERENTIAL EQUATIONS Objective Qs (1 mark) 12. The general solution of the differential equation 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 0; (Given 𝑥, 𝑦 > 0 ), is of the form: (a) 𝑥𝑦 = 𝑐 (b) 𝑥 = 𝑐𝑦2 (c) 𝑦 = 𝑐𝑥 (d) 𝑦 = 𝑐𝑥2 (Where ' 𝑐 ' is an arbitrary positive constant of integration) [CBSE SQP 2023] 13. The integrating factor of the differential equation (1 − 𝑦2) 𝑑𝑥 𝑑𝑦 + 𝑥𝑦 = 𝑎𝑦, (−1 < 𝑦 < 1) is: (a) 1 𝑦2−1 (b) 1 �𝑦2−1 CLICK HERE FOR SOLUTIONS
  • 88.
    (c) 1 1−𝑦2 (d) 1 �1−𝑦2 [CBSE 2023] 14. Theintegrating factor of the differential equation (𝑥 + 3𝑦2) 𝑑𝑦 𝑑𝑥 = 𝑦 is: (a) 𝑦 (b) −𝑦 (c) 1 𝑦 (d) − 1 𝑦 [CBSE 2020] 15. The number of arbitrary constants in the particular solution of a differential equation of second order is (are): (a) 0 (b) 1 (c) 2 (d) 3 [CBSE 2020] Very Short & Short Qs (1 - 3 marks) 16. Solve the differential equation: 𝑦𝑒 𝑥 𝑦𝑑𝑥 = �𝑥𝑒 𝑥 𝑦 + 𝑦2 � 𝑑𝑦, (𝑦 ≠ 0) [CBSE SQP 2023] 17. Solve the differential equation: (cos2 𝑥) 𝑑𝑦 𝑑𝑥 + 𝑦 = tan 𝑥; �0 ≤ 𝑥 < 𝜋 2 � [CBSE SQP 2023] 18. If (𝑎 + 𝑏𝑥)𝑒 𝑦 𝑥 = 𝑥, then prove that 𝑥 𝑑2 𝑦 𝑑𝑥2 = � 𝑎 𝑎 + 𝑏𝑥 � 2 [CBSE SQP 2023] 19. Find the general solution of the differential equation (𝑥𝑦 − 𝑥2)𝑑𝑦 = 𝑦2 𝑑𝑥 [CBSE 2023] CLICK HERE FOR SOLUTIONS
  • 89.
    20. Find thegeneral solution of the differential equation: (𝑥2 + 1) 𝑑𝑦 𝑑𝑥 + 2𝑥𝑦 = �𝑥2 + 4 [CBSE 2023] 21. Solve the differential equation: 𝑦𝑑𝑥 + (𝑥 − 𝑦2)𝑑𝑦 = 0 [CBSE SQP 2022] 22. Find the general solution of the following differential equation: 𝑥 𝑑𝑦 𝑑𝑥 = 𝑦 − 𝑥sin � 𝑦 𝑥 � [CBSE Term-2 SQP 2022] 23. Find the particular solution of the following differential equation, given that 𝑦 = 0 when 𝑥 = 𝜋 4 𝑑𝑦 𝑑𝑥 + 𝑦cot 𝑥 = 2 1 + sin 𝑥 [CBSE Term-2 SQP 2022] 24. Solve the differential equation: 𝑥𝑑𝑦 − 𝑦𝑑𝑥 = �𝑥2 + 𝑦2𝑑𝑥. [CBSE SQP 2022] 25. Find the general solution of the differential equation: 𝑑𝑦 𝑑𝑥 = 𝑒𝑥−𝑦 + 𝑥2 𝑒−𝑦 [CBSE Term-2 2022] 26. Find the general solution of the differential equation: sec2 𝑥tan 𝑦𝑑𝑥 + sec2 𝑦tan 𝑥𝑑𝑦 = 0 [CBSE Term-2 2022] 27. For what value of ' 𝑛 ' is the following a homogeneous differential equation? 𝑑𝑦 𝑑𝑥 = 𝑥3 − 𝑦𝑛 𝑥2𝑦 + 𝑥𝑦2 [CBSE SQP 2020] CLICK HERE FOR SOLUTIONS
  • 90.
    28. The integratingfactor of the differential equation 𝑥 𝑑𝑦 𝑑𝑥 + 2𝑦 = 𝑥2 is [CBSE 2020] 29. Solve the following differential equation: 𝑑𝑦 𝑑𝑥 = 𝑥3 cosec 𝑦, given that 𝑦(0) = 0. [CBSE SQP 2020] 30. Find the particular solution of the differential equation 𝑥 𝑑𝑦 𝑑𝑥 = 𝑦 − 𝑥tan � 𝑦 𝑥 �, given that 𝑦 = 𝜋 4 at 𝑥 = 1. [CBSE 2020] 31. Solve the differential equation 𝑥𝑑𝑦 𝑑𝑥 + 𝑦 = 𝑥cos 𝑥 + sin 𝑥, given that 𝑦 = 1 when 𝑥 = 𝜋 2 . [CBSE 2019] 32. Solve the following differential equation: 𝑑𝑦 𝑑𝑥 + 𝑦 = cos 𝑥 − sin 𝑥 [CBSE 2019] 33. Solve the differential equation cos � 𝑑𝑦 𝑑𝑥 � = 𝑎, 𝑎 ∈ R [CBSE 2018] 34. Solve the differential equation log � 𝑑𝑦 𝑑𝑥 � = 3𝑥 + 4𝑦 [CBSE 2017] 35. Write the integrating factor of the following differential equation: (1 + 𝑦2) + (2𝑥𝑦 − cot 𝑦) 𝑑𝑦 𝑑𝑥 = 0. [CBSE 2015] 36. Write the solution of the differential equation: 𝑑𝑦 𝑑𝑥 = 2−𝑦 [CBSE 2015] 37. Find the integrating factor of the differential equation � 𝑒−2√𝑥 √𝑥 − 𝑦 √𝑥 � 𝑑𝑥 𝑑𝑦 = 1. [CBSE 2015] 38. Find the solution of the differential equation 𝑑𝑦 𝑑𝑥 = 𝑥3 𝑒−2𝑦 . [CBSE 2015] CLICK HERE FOR SOLUTIONS
  • 91.
    39. If 𝑦(𝑥)is a solution of � 2+sin 𝑥 1+𝑦 � 𝑑𝑦 𝑑𝑥 = −cos 𝑥 and 𝑦(0) = 1, then find the value of 𝑦 � 𝜋 2 �. [CBSE 2014] 40. Find the particular solution of the differential equation 𝑒𝑥 �1 − 𝑦2𝑑𝑥 + 𝑦 𝑥 𝑑𝑦 = 0, given that 𝑦 = 1, when 𝑥 = 0. [CBSE 2014] Long Qs (4 – 5 marks) 41. Solve: (1 + 𝑦2)𝑑𝑥 = (tan−1 𝑦 − 𝑥)𝑑𝑦. [CBSE 2015] 42. Solve the following differential equation �𝑦 − 𝑥cos � 𝑦 𝑥 �� 𝑑𝑦 + �𝑦cos � 𝑦 𝑥 � −2𝑥sin � 𝑦 𝑥 �� 𝑑𝑥 = 0 [CBSE 2015] 43. Show that the following differential equation is homogeneous and solve it. �𝑥sin2 � 𝑦 𝑥 � − 𝑦� 𝑑𝑥 + 𝑥𝑑𝑦 = 0 𝑦 = 𝜋 4 when 𝑥 = 1 [CBSE 2015, 14] 44. Find the general solution of the following differential equations: 𝑥log 𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 = 2 𝑥 log 𝑥 [CBSE 2014] CLICK HERE FOR SOLUTIONS
  • 92.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 93.
    Vector Algebra 1. BASICCONCEPTS OF VECTORS Objective Qs (1 mark) 1. 𝐴𝐵𝐶𝐷 is a rhombus whose diagonals intersect at 𝐸. Then 𝐸𝐴 �����⃗ + 𝐸𝐵 �����⃗ + 𝐸𝐶 �����⃗ + 𝐸𝐷 �����⃗ equals to: (a) 0 �⃗ (b) 𝐴𝐷 �����⃗ (c) 2BD �����⃗ (d) 2𝐴𝐷 �����⃗ [CBSE SQP 2023] 2. If 𝑎 ⃗ = 4𝚤 ˆ + 6𝚥 ˆ and 𝑏 �⃗ = 3𝚥 ˆ + 4𝑘 ˆ, then the vector form of the component of 𝑎 ⃗ along 𝑏 �⃗ is: (a) 18 5 (3𝚤 ˆ + 4𝑘 ˆ) (b) 18 25 (3𝚥 ˆ + 4𝑘 ˆ) (c) 18 5 (3𝚤 ˆ + 4𝑘 ˆ) (d) 18 25 (4𝚤 ˆ + 6𝚥 ˆ) [CBSE SQP 2023] 3. If in △ ABC, BA �����⃗ = 2𝑎 ⃗ and BC �����⃗ = 3𝑏 �⃗, then AC �����⃗ is: (a) 2𝑎 ⃗ + 3𝑏 �⃗ (b) 2𝑎 ⃗ − 3𝑏 �⃗ (c) 3𝑏 �⃗ − 2𝑎 ⃗ (d) −2𝑎 ⃗ − 3𝑏 �⃗ [CBSE 2023] 4. If two vectors 𝑎 ⃗ and 𝑏 �⃗ are such that |𝑎 ⃗| = 2, |𝑏 �⃗| = 3 and 𝑎 ⃗ ⋅ 𝑏 �⃗ = 4, then |𝑎 ⃗ − 2𝑏 �⃗| is equal to: (a) √2 (b) 2√6 (c) 24 (d) 2√2 [CBSE SQP 2022] 5. The value of 𝑝 for which 𝑝(𝚤 ˆ + 𝚥 ˆ + 𝑘 ˆ) is a unit vector is: (a) 0 (b) 1 √3 (c) 1 (d) √3 [CBSE 2020] CH-10 CLICK HERE FOR SOLUTIONS
  • 94.
    Very Short &Short Qs (1-3 marks) 6. If 𝑎 ⃗ = 𝚤 ˆ − 𝚥 ˆ + 7𝑘 ˆ and 𝑏 �⃗ = 5𝚤 ˆ − 𝚥 ˆ + 𝜆𝑘 ˆ, then find the value of 𝜆 so that vectors 𝑎 ⃗ + 𝑏 �⃗ and 𝑎 ⃗ − 𝑏 �⃗ are orthogonal. [CBSE SQP 2022] 7. The position vectors of two points 𝐴 and B are 𝑂𝐴 �����⃗ = 2𝚤 ˆ − 𝚥 ˆ − 𝑘 ˆ and 𝑂𝐵 �����⃗ = 2𝚤 ˆ − 𝚥 ˆ + 2𝑘 ˆ respectively. The position vector of a point 𝑃 which divides the line segment joining 𝐴 and 𝐵 in the ratio 2: 1 is [CBSE 2020] 8. 𝑋 and 𝑌 are two points with position vectors 3𝑎 ⃗ + 𝑏 �⃗ and 𝑎 ⃗ − 3𝑏 �⃗ respectively. Write the position vector of a point 𝑍 which divides the line segment 𝑋𝑌 in the ratio 2:1 externally. [CBSE 2019] 9. Find the value of 𝑝 for which the vectors 3𝚤 ˆ + 2𝚥 ˆ + 9𝑘 ˆ and 𝚤 ˆ − 2𝑝𝚥 ˆ + 3𝑘 ˆ are parallel. [CBSE 2017] 10. Write the position vector of the point which divides the join of the points with position vectors 3𝑎 ⃗ − 2𝑏 �⃗ and 2𝑎 ⃗ + 3𝑏 �⃗ in the ratio 2: 1. [CBSE 2016] 11. Find the unit vector in the direction of the sum of the vectors 2𝚤 ˆ + 3𝚥 ˆ − 𝑘 ˆ and 4𝚤 ˆ − 3𝚥 ˆ + 2𝑘 ˆ. [CBSE 2016] 12. Find the position vector of a point which divides the join of points with position vectors 𝑎 ⃗ − 2𝑏 �⃗ and 2𝑎 ⃗ + 𝑏 �⃗ externally in the ratio 2 : 1 . [CBSE 2016] 13. Find a vector in the direction of vector 𝑎 ⃗ = 𝚤 ˆ − 2𝚥 ˆ that has magnitude 7 units. [CBSE 2015] 14. Write the direction ratios of the vector 3𝑎 ⃗ + 2𝑏 �⃗ where 𝑎 ⃗ = 𝚤 ˆ + 𝚥 ˆ − 2𝑘 ˆ and 𝑏 �⃗ = 2𝚤 ˆ − 4𝚥 ˆ + 5𝑘 ˆ. [CBSE 2015] 15. Write the value of cosine of the angle which the vector 𝑎 ⃗ = 𝚤 ˆ + 𝚥 ˆ + 𝑘 ˆ makes with the 𝑦-axis. [CBSE 2014] 16. Find a vector 𝑎 ⃗ of magnitude 5√2, making an angle of 𝜋 4 with 𝑥-axis, 𝜋 2 with 𝑦-axis and an acute angle 𝜃 with 𝑧-axis. [CBSE 2014] CLICK HERE FOR SOLUTIONS
  • 95.
    Long Qs (4– 5 marks) 17. The two vectors 𝚥 ˆ + 𝑘 ˆ and 3𝚤 ˆ − 𝚥 ˆ + 4𝑘 ˆ represent the two sides 𝐴𝐵 and 𝐴𝐶, respectively of a △ 𝐴𝐵𝐶. Find the length of the median through 𝐴. [CBSE 2016, 15] 2. PRODUCT OF TWO VECTORS Objective Qs (1 mark) 18. The value of 𝜆 for which two vectors 2𝚤 ˆ − 𝚥 ˆ + 2𝑘 ˆ and 3𝚤 ˆ + 𝜆𝚥 ˆ + 𝑘 ˆ are perpendicular is: (a) 2 (b) 4 (c) 6 (d) 8 [CBSE SQP 2023] 19. Unit vector along 𝑃𝑄 �����⃗, where coordinates of 𝑃 and 𝑄 respectively are (2,1, −1) and (4,4, −7), is: (a) 2𝚤 ˆ + 3𝚥 ˆ − 6𝑘 ˆ (b) −2𝚤 ˆ − 3𝚥 ˆ + 6𝑘 ˆ (c) − 2 7 𝚤 ˆ − 3 7 𝚥 ˆ + 6 7 𝑘 ˆ (d) 2 7 𝚤 ˆ + 3 7 𝚥 ˆ − 6 7 𝑘 ˆ [CBSE 2023] 20. If |𝑎 ⃗ × 𝑏 �⃗| = √3 and 𝑎 ⃗ ⋅ 𝑏 �⃗ = −3, then angle between 𝑎 ⃗ and 𝑏 �⃗ is: (a) 2𝜋 3 (b) 𝜋 6 (c) 𝜋 3 (d) 5𝜋 6 [CBSE 2023] 21. The scalar projection of the vector 3𝚤 ˆ − 𝚥 ˆ − 2𝑘 ˆ on the vector 𝚤 ˆ + 2𝚥 ˆ − 3𝑘 ˆ is: (a) 7 √14 (b) 7 14 (c) 6 13 (d) 7 2 [CBSE SQP 2022] 22. If the projection of 𝑎 ⃗ = 𝚤 ˆ − 2𝚥 ˆ + 3𝑘 ˆ on 𝑏 �⃗ = 2𝚤 ˆ + 𝜆𝑘 ˆ is zero, then the value of 𝜆 is: (a) 0 (b) 1 CLICK HERE FOR SOLUTIONS
  • 96.
    (c) − 2 3 (d) − 3 2 [CBSE2020] 23. If 𝚤 ˆ, 𝚥 ˆ, 𝑘 ˆ are unit vectors along three mutually perpendicular directions, then: (a) 𝚤 ˆ. 𝚥 ˆ = 1 (b) 𝚤 ˆ × 𝚥 ˆ = 1 (c) 𝚤 ˆ ⋅ 𝑘 ˆ = 0 (d) 𝚤 ˆ × 𝑘 ˆ = 0 [CBSE 2020] Case Based Qs (4 – 5 marks) 24. Teams 𝐴, 𝐵, 𝐶 went for playing a tug of war game. Teams A, B, C have attached a rope to a metal ring and is trying to pull the ring into their own area. Team A pulls with force 𝐹1 = 6𝚤 ˆ + 0𝚥 ˆ𝑁. Team 𝐵 pulls with force 𝐹2 = −4𝚤 ˆ + 4𝚥 ˆ𝑁. Team C pulls with force 𝐹3 = −3𝚤 ˆ − 3𝚥 ˆ𝑁. Based on the above information, answer the following questions: (A) What is the magnitude of the force of Team A? (B) Which team will win the game? (C) Find the magnitude of the resultant force exerted by the teams. OR In what direction is the ring getting pulled? Very Short & Short Qs (1 − 3 marks) 25. If 𝑎 ⃗, 𝑏 �⃗, 𝑐 ⃗ are three non-zero unequal vectors such that 𝑎 ⃗ ⋅ 𝑏 �⃗ = 𝑎 ⃗ ⋅ 𝑐 ⃗, then find the angle between 𝑎 ⃗ and 𝑏 �⃗ − 𝑐 ⃗. CLICK HERE FOR SOLUTIONS
  • 97.
    [CBSE 2023] 26. If𝑎 ⃗ and 𝑏 �⃗ are two vectors of equal magnitude and 𝛼 is the angle between them, then prove that � 𝑎 �⃗+𝑏 �⃗ 𝑎 �⃗−𝑏 �⃗ � = cot � 𝛼 2 �. [CBSE Term-2 2022] 27. If 𝑎 ⃗ and 𝑏 �⃗ are two vectors such that |𝑎 ⃗ + 𝑏 �⃗| = |𝑏 �⃗|, then prove that (𝑎 ⃗ + 2𝑏 �⃗) is perpendicular to 𝑎 ⃗. [CBSE Term-2 2022] 28. If 𝑎 ˆ and 𝑏 ˆ are unit vectors, then prove that |𝑎 ˆ + 𝑏 ˆ| = 2cos 𝜃 2 , where 𝜃 is the angle between them. [CBSE Term-2 SQP 2022] 29. Find |𝑥 ⃗| if (𝑥 ⃗ − 𝑎 ⃗) ⋅ (𝑥 ⃗ + 𝑎 ⃗) = 12, where 𝑎 ⃗ is a unit vector. [CBSE SQP 2022] 30. The area of the parallelogram whose diagonals are 2𝚤 ˆ and −3𝑘 ˆ is ............. square units. [CBSE 2020] 31. The value of 𝜆 for which the vectors 2𝚤 ˆ − 𝜆𝚥 ˆ + 𝑘 ˆ and 𝚤 ˆ + 2𝚥 ˆ − 𝑘 ˆ are orthogonal is [CBSE 2020] 32. Show that the vectors 2𝚤 ˆ − 𝚥 ˆ + 𝑘 ˆ, 3𝚤 ˆ + 7𝚥 ˆ + 𝑘 ˆ and 5𝚤 ˆ + 6𝚥 ˆ + 2𝑘 ˆ form the sides of a right-angled triangle. [CBSE 2020] 33. If the sum of two unit vectors 𝑎 ⃗ and 𝑏 �⃗ is a unit vector, show that the magnitude of their difference is √3. [CBSE 2019] 34. If 𝑎 ⃗ + 𝑏 �⃗ + 𝑐 ⃗ = 0 and |𝑎 ⃗| = 5, |𝑏 �⃗| = 6 and |𝑐 ⃗| = 9, then find the angle between 𝑎 ⃗ and 𝑏 �⃗. [CBSE 2018] 35. If 𝜃 is the angle between two vectors 𝚤 ˆ − 2𝚥 ˆ + 3𝑘 ˆ and 3𝚤 ˆ − 2𝚥 ˆ + 𝑘 ˆ, find sin 𝜃. [CBSE 2018] 36. Find the magnitude of each of the two vectors 𝑎 ⃗ and 𝑏 �⃗, having the same magnitude, such that the angle between them is 60∘ and their scalar product is 9 2 . [CBSE 2018] 37. Write the vectors of unit length perpendicular to both the vectors 𝑎 ⃗ = 2𝚤 ˆ + 𝚥 ˆ + 2𝑘 ˆ and 𝑏 �⃗ = 𝚥 ˆ + 𝑘 ˆ. [CBSE 2016] CLICK HERE FOR SOLUTIONS
  • 98.
    38. Write thenumber of vectors of unit length perpendicular to both the vector 𝑎 ⃗ = 2𝚤 ˆ + 𝚥 ˆ + 2𝑘 ˆ and vector 𝑏 �⃗ = 𝚥 ˆ + 𝑘 ˆ. [CBSE 2016] 39. If 𝑎 ⃗ = 7𝚤 ˆ + 𝚥 ˆ − 4𝑘 ˆ and 𝑏 �⃗ = 2𝚤 ˆ + 6𝚥 ˆ + 3𝑘 ˆ, then find the projection of 𝑎 ⃗ and 𝑏 �⃗. [CBSE 2015] 40. If vector 𝑎 ⃗ and vector 𝑏 �⃗ are two unit vectors such that vector (𝑎 ⃗ + 𝑏 �⃗) is also a unit vector, then find the angle between vector 𝑎 ⃗ and vector 𝑏 �⃗. [CBSE 2014] 41. Vectors 𝑎 ⃗ and 𝑏 �⃗ are such that |𝑎 ⃗| = √3, |𝑏 �⃗| = 2 3 and (𝑎 ⃗ × 𝑏 �⃗) is a unit vector. Write the angle between 𝑎 ⃗ and 𝑏 �⃗. [CBSE 2014] Long Qs 4 - 5 marks 42. If vector 𝑎 ⃗ = 𝚤 ˆ + 2𝚥 ˆ + 3𝑘 ˆ and vector 𝑏 �⃗ = 2𝚤 ˆ + 4𝚥 ˆ − 5𝑘 ˆ repesent two adjacent sides of a parallelogram, find unit vectors parallel to the diagonals of the parallelogram. [CBSE 2020] 43. If 𝚤 ˆ + 𝚥 ˆ + 𝑘 ˆ, 2𝚤 ˆ + 5𝚥 ˆ, 3𝚤 ˆ + 2𝚥 ˆ − 3𝑘 ˆ and 𝚤 ˆ − 6𝚥 ˆ − 𝑘 ˆ are the position vectors of points 𝐴, 𝐵, 𝐶 and 𝐷 respectively, then find the angle between 𝐴𝐵 �����⃗ and 𝐶𝐷 �����⃗. Deduce that 𝐴𝐵 �����⃗ and 𝐶𝐷 �����⃗ are collinear. [CBSE 2019] 44. Show that the points 𝐴, 𝐵, 𝐶 with position vectorts 2𝚤 ˆ − 𝚥 ˆ + 𝑘 ˆ, 𝚤 ˆ − 3𝚥 ˆ − 5𝑘 ˆ, 3𝚤 ˆ − 4𝚥 ˆ − 4𝑘 ˆ, respectively, are the vertices of a right angled triangle. Hence find the area of the triangle. [CBSE 2018] 45. Let vector 𝑎 ⃗ = 4𝚤 ˆ + 5𝚥 ˆ + 𝑘 ˆ, vector 𝑏 �⃗ = 𝚤 ˆ − 4𝚥 ˆ + 5𝑘 ˆ and vector 𝑐 ⃗ = 3𝚤 ˆ + 𝚥 ˆ + −𝑘 ˆ Find 𝑎 ⃗ vector 𝑑 ⃗ which is perpendicular to both vector 𝑐 ⃗ and vector 𝑏 �⃗ and 𝑑 ⃗ ⋅ 𝑎 ⃗ = 21 [CBSE 2018] 46. If 𝑎 ⃗, 𝑏 �⃗ and 𝑐 ⃗ are three mutually perpendicular vectors of the same magnitude, then prove that 𝑎 ⃗ + 𝑏 �⃗ + 𝑐 ⃗ is equally inclined with the vectors 𝑎 ⃗, 𝑏 �⃗ and 𝑐 ⃗. [CBSE 2017] 47. The two adjacent sides of a parallelogram are 2𝚤 ˆ − 4𝚥 ˆ − 5𝑘 ˆ and 2𝚤 ˆ + 2𝚥 ˆ + 3𝚥 ˆ. Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram. [CBSE 2016] CLICK HERE FOR SOLUTIONS
  • 99.
    48. The scalarproduct of the vector 𝑎 ⃗ = 𝚤 ˆ + 2𝚥 ˆ + 3𝑘 ˆ with a unit vector along the sum of vectors 𝑏 �⃗ = 2𝚤 ˆ + 4𝚥 ˆ − 5𝑘 ˆ and 𝑐 ⃗ = 𝜆𝚤 ˆ + 2𝚥 ˆ + 3𝑘 ˆ is equal to one. Find the value of 𝜆 and hence find the unit vector along 𝑏 �⃗ + 𝑐 ⃗. [CBSE 2014] CLICK HERE FOR SOLUTIONS
  • 100.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 101.
    Three Dimensional Geometry 1.BASIC CONCEPTS Objective Qs (1 mark) 1. The lines 𝑟 ⃗ = 𝚤 ˆ + 𝚥 ˆ − 𝑘 ˆ + 𝜆(2𝚤 ˆ + 3𝚥 ˆ − 6𝑘 ˆ) and 𝑟 ⃗ = 2𝚤 ˆ − 𝚥 ˆ − 𝑘 ˆ + 𝜇(6𝚤 ˆ + 9𝚥 ˆ − 18𝑘 ˆ); (where 𝜆 and 𝜇 are scalars) are: (a) coincident (b) skew (c) intersecting (d) parallel [CBSE SQP 2023] 2. If the direction cosines of a line are � 1 𝑐 , 1 𝑐 , 1 𝑐 �, then: (a) 0 < 𝑐 < 1 (b) 𝑐 > 2 (c) 𝑐 = ±√2 (d) 𝑐 = ±√3 [CBSE SQP 2023] 3. Equation of line passing through origin and making 30∘ , 60∘ and 90∘ with 𝑥, 𝑦, 𝑧-axes respectively, is: (a) 2𝑥 √3 = 𝑦 2 = 𝑧 0 (b) 2𝑥 √3 = 2𝑦 1 = 𝑧 0 (c) 2𝑥 = 2𝑦 √3 = 𝑧 1 (d) 2𝑥 √3 = 2𝑦 1 = 𝑧 1 [CBSE 2023] 4. The length of the perpendicular drawn from the point (4, −7,3) on the 𝑦-axis is: (a) 3 units (b) 4 units (c) 5 units (d) 7 units [CBSE 2020] 5. Assertion (A): If a line makes angles 𝛼, 𝛽, 𝛾 with positive direction of the coordinate axes, then sin2 𝛼 + sin2 𝛽 + sin2 𝛾 = 2. Reason (R): The sum of squares of the direction cosines of a line is 1. [CBSE 2023] (a) Both (A) and (R) are true and (R) is the correct explanation of (𝐴). CH-11 CLICK HERE FOR SOLUTIONS
  • 102.
    (b) Both (A)and (R) are true, but (R) is not the correct explanation of (A). (c) (A) is true, but (R) is false. (d) (A) is false, but (R) is true. Very Short & Short Qs (1 − 3 marks) 6. If the equation of a line is 𝑥 = 𝑎𝑦 + 𝑏, 𝑧 = 𝑐𝑦 + 𝑑 then find the direction ratios of the line and a point on the line. [CBSE 2023] 7. The cartesian equation of a line 𝐴𝐵 is: 2𝑥 − 1 12 = 𝑦 + 2 2 = 𝑧 − 3 3 Find the direction cosines of a line parallel to line AB. [CBSE Term 2 2022] 8. Find all the possible vector of magnitude 5√3 which are equally inclined to the coordinate axes. [CBSE Term-2 2022] 9. If a line makes 60∘ and 45∘ angles with the positive directions of 𝑥-axis and 𝑧-axis respectively, then find the angle that it makes with the positive direction of 𝑦-axis. Hence, write the direction cosines of the line. [CBSE Term-2 2022] 10. Find the direction cosines of the following line: 3 − 𝑥 −1 = 2𝑦 − 1 2 = 𝑧 4 [CBSE Term-2 SQP 2022] 11. Find the value of 𝑘 so that the lines 𝑥 = −𝑦 = 𝑘𝑧 and 𝑥 − 2 = 2𝑦 − 1 = −𝑧 + 1 are perpendicular to each other. [CBSE 2020] 12. The line of shortest distance between two skew lines is to both the lines. [CBSE 2020] 13. The vector equation of the line through the points (3,4, −7) and (1, −1,6) is ...... . [CBSE 2020] 14. Find the vector equation of a line passing through the point (2,3,2) and parallel to the line 𝑟 ⃗ = (−2𝚤 ˆ + 3𝚥 ˆ) + 𝜆(2𝚤 ˆ − 3𝚥 ˆ + 6𝑘 ˆ). Also, find the distance between these two lines. [CBSE 2019] CLICK HERE FOR SOLUTIONS
  • 103.
    15. Find thevector equation of a line which passes through the point (3,4,5) and is parallel to the vector (2𝚤 ˆ + 2𝚥 ˆ − 3𝑘 ˆ). [CBSE 2019] 16. If a line makes angles 90∘ , 135∘ and 45∘ with the 𝑥, 𝑦 and 𝑧 axes respectively, find its direction cosines. [CBSE 2019] 17. Find the shortest distance between the lines vector 𝑟 ⃗ = (4𝚤 ˆ − 𝚥 ˆ) + 𝜆(𝚤 ˆ + 2𝚥 ˆ − 3𝑘 ˆ) 𝑟 ⃗ = (𝚤 ˆ − 𝚥 ˆ + 2𝑘 ˆ) + 𝜇(2𝚤 ˆ + 4𝚥 ˆ − 5𝑘 ˆ) [CBSE 2018] 18. The 𝑥-coordinate of a point on the line joining the points 𝑃(2,2,1) and 𝑄(5,1, −2) is 4. Find its 𝑧- coordinate. [CBSE 2017] 19. Find the vector equation of the line passing through the point 𝐴(1,2, −1) and parallel to the line 5𝑥 − 25 = 14 − 7𝑦 = 35𝑧. [Delhi 2017] 20. If a line makes angles 90∘ , 60∘ and 𝜃 with 𝑥, 𝑦 and 𝑧-axes respectively, where 𝜃 is acute, then find 𝜃. [CBSE 2015] 21. The equations of a line is 5𝑥 − 3 = 15𝑦 + 7 = 3 - 10z. Write the direction cosines of the line. [CBSE 2015] 22. Prove that the line 𝑟 ⃗ = (𝚤 ˆ + 𝚥 ˆ − 𝑘 ˆ) + 𝜆(3𝚤 ˆ − 𝚥 ˆ) and 𝑟 ⃗ = (4𝚤 ˆ − 𝑘 ˆ) + 𝜇(2𝚤 ˆ + 3𝑘 ˆ) intersect and find their point of intersection. [CBSE 2014] 23. Find the vector and cartesian equations of the line passing through the point (2,1,3) and perpendicular to the lines 𝑥−1 1 = 𝑦−2 2 = 𝑧−3 3 and 𝑥 −3 = 𝑦 2 = 𝑧 5 . [CBSE 2014] 24. A line passes through (2, −1,3) and is perpendicular to the lines vector 𝑟 ⃗ = (𝚤 ˆ + 𝚥 ˆ − 𝑘 ˆ) + 𝜆(2𝚤 ˆ − 2𝚥 ˆ + 𝑘 ˆ) and vector 𝑟 ⃗ = (2𝚤 ˆ − 𝚥 ˆ − 3𝑘 ˆ) + 𝜇(𝚤 ˆ + 2𝚥 ˆ + 2𝑘 ˆ). Obtain its equation in vector and cartesian form. [CBSE 2014] 25. Find the direction cosines of the line 𝑥+2 2 = 2𝑦−7 6 = 5−𝑧 6 . Also find the vector equation of the line through the point 𝐴(−1,2,3) and parallel to the given line. [CBSE 2014] CLICK HERE FOR SOLUTIONS
  • 104.
    26. Find thevalue of 𝑝 so that the lines 1−𝑥 3 = 7𝑦−14 2𝑝 = 𝑧−3 2 and 7−7𝑥 3𝑝 = 𝑦−5 1 = 6−𝑧 5 are at right angles. [CBSE 2014] 27. If the cartesian equations of a line are 3−𝑥 5 = 𝑦+4 7 = 2𝑧−6 4 , write the vector equation for the line. [CBSE 2014] 28. Write the equation of the straight line through the point (𝛼, 𝛽, 𝛾) and parallel to z-axis. [CBSE 2014] Long Qs (4 - 5 marks) 29. Find the coordinates of the image of the point (1,6,3) with respect to the line 𝑟 ⃗ = (𝚥 ˆ + 2𝑘 ˆ) + 𝜆(𝚤 ˆ + 2𝚥 ˆ + 3𝑘 ˆ); where ' 𝜆 ' is a scalar. Also, find the distance of the image from the 𝑦-axis. [CBSE SQP 2023] 30. An aeroplane is flying along the line 𝑟 ⃗ = 𝜆(𝚤 ˆ − 𝚥 ˆ + 3𝑘 ˆ); where ' 𝜆 ' is a scalar and another aeroplane is flying along the 𝑟 ⃗ = 𝚤 ˆ − 𝚥 ˆ + 𝜇(−2𝚥 ˆ + 𝑘 ˆ); where ' 𝜇 ' is a scalar. At what points on the lines should they reach, so that the distance between them is the shortest? Find the shortest possible distance between them. [CBSE SQP 2023] 31. A line 𝑙 passes through point (−1,3, −2) and is perpendicular to both the lines 𝑥 1 = 𝑦 2 = 𝑧 3 and 𝑥+2 −3 = 𝑦−1 2 = 𝑧+1 5 . Find the vector equation of the line 𝑙. Hence, obtain its distance from origin. [CBSE 2023] 32. Find the equations of the diagonals of the parallelogram PQRS whose vertices are 𝑃(4,2, −6), 𝑄(5, −3,1), 𝑅(12,4,5) and 𝑆(11,9, −2). Use these equations to find the point of intersection of diagonals. [CBSE 2023] 33. Find the vector and Cartesian equations of a line passing through (1,2, −4) and perpendicular to the two lines 𝑥 − 8 3 = 𝑦 + 9 −16 = 𝑧 − 10 7 and 𝑥 − 15 3 = 𝑦 − 29 8 = 𝑧 − 5 −5 [CBSE 2017] CLICK HERE FOR SOLUTIONS
  • 105.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 106.
    1. BASIC CONCEPTSOF LINEAR PROGRAMMING Objective Qs (1 mark) 1. The corner points of the bounded feasible region determined by a system of linear constraints are (0,3), (1,1) and (3,0). Let 𝑍 = 𝑝𝑥 + 𝑞𝑦, where 𝑝, 𝑞 > 0. The condition on 𝑝 and 𝑞 so that the minimum of 𝑍 occurs at (3,0) and (1,1) is: (a) 𝑝 = 2𝑞 (b) 𝑝 = 𝑞 2 (c) 𝑝 = 3𝑞 (d) 𝑝 = 𝑞 [CBSE SQP 2023] 2. The feasible region corresponding to the linear constraints of a Linear Programming Problem is given below. Which of the following is not a constraint to the given Linear Programming Problem? (a) 𝑥 + 𝑦 ≥ 2 (b) 𝑥 + 2𝑦 ≤ 10 (c) 𝑥 − 𝑦 ≥ 1 (d) 𝑥 − 𝑦 ≤ 1 [CBSE SQP 2023] 3. The objective function 𝑍 = 𝑎𝑥 + 𝑏𝑦 of an LPP has maximum value 42 at (4,6) and minimum value 19 at (3,2). Which of the following is true? (a) 𝑎 = 9, 𝑏 = 1 (b) 𝑎 = 5, 𝑏 = 2 (c) 𝑎 = 3, 𝑏 = 5 (d) 𝑎 = 5, 𝑏 = 3 [CBSE 2023] 4. The corner points of the feasible region of a linear programming problem are (0,4), CH-12 CLICK HERE FOR SOLUTIONS
  • 107.
    (8,0) and � 20 3 , 4 3 �.If 𝑍 = 30𝑥 + 24𝑦 is the objective function, then (maximum value of 𝑍 - minimum value of 𝑍 ) is equal to: (a) 40 (b) 96 (c) 120 (d) 136 [CBSE 2023] 5. The corner points of the shaded unbounded feasible region of an LPP are (0,4), (0.6,1.6) and (3,0) as shown in the figure. The minimum value of the objective function 𝑍 = 4𝑥 + 6𝑦 occurs at: (a) (0.6,1.6) only (b) (3,0) only (c) (0.6,1.6) and (3,0) only (d) every point of the line-segment joining the points (0.6,1.6) and (3,0) [CBSE SQP 2022] 6. The solution set of the inequality 3𝑥 + 5𝑦 < 4 is: (a) an open half-plane not containing the origin. (b) an open half-plane containing the origin. (c) the whole 𝑋𝑌-plane not containing the line 3𝑥 + 5𝑦 = 4 (d) a closed half plane containing the origin. [CBSE SQP 2022] 7. In the given graph, the feasible region for a LPP is shaded. CLICK HERE FOR SOLUTIONS
  • 108.
    The objective function𝑍 = 2𝑥 − 3𝑦 will be minimum at: (a) (4,10) (b) (6,8) (c) (0,8) (d) (6,5) [CBSE Term-1 SQP 2021] 8. Based on the given shaded region as the feasible region in the graph, at which point(s) is the objective function Z = 3𝑥 + 9𝑦 maximum? (a) Point B (b) Point 𝐶 (c) Point D (d) Every point on the line segment 𝐶𝐷 [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
  • 109.
    9. In alinear programming problem, the constraints on the decision variables 𝑥 and 𝑦 are 𝑥 − 3𝑦 ≥ 0, 𝑦 ≥ 0,0 ≤ 𝑥 ≤ 3. The feasible region: (a) is not in the first quadrant. (b) is bounded in the first quadrant. (c) is unbounded in the first quadrant. (d) does not exist. [CBSE Term-1 SQP 2021] 10. For an objective function 𝑍 = 𝑎𝑥 + 𝑏𝑦, where 𝑎, 𝑏 > 0; the corner points of the feasible region determined by a set of constraints (linear inequalities) are (0,20), (10,10), (30,30) and (0,40). The condition on 𝑎 and 𝑏 such that the maximum 𝑍 occurs at both the points (30,30) and (0,40) is: (a) 𝑏 − 3𝑎 = 0 (b) 𝑎 = 3𝑏 (c) 𝑎 + 2𝑏 = 0 (d) 2𝑎 − 𝑏 = 0 [CBSE Term-1 SQP 2021] 11. The corner points of feasible region for a linear programming problem are P(0,5), 𝑄(1,5), 𝑅(4,2) and 𝑆(12,0). The minimum value of the objective function 𝑍 = 2𝑥 + 5𝑦 is at the point: (a) P (b) Q (c) R (d) 𝑆 [CBSE Term-1 SQP 2021] 12. A linear programming problem is as follows: Minimise 𝑍 = 30𝑥 + 50𝑦 subject to the constraints, 3𝑥 + 5𝑦 ≥ 15 2𝑥 + 3𝑦 ≤ 18 𝑥 ≥ 0, 𝑦 ≥ 0 In the feasible region, the minimum value of 𝑍 occurs at: (a) a unique point (b) no point (c) infinitely many poins (d) two points only [CBSE Term-1 SQP 2021] CLICK HERE FOR SOLUTIONS
  • 110.
    Very Short &Short Qs (1-3 marks) 13. Solve the following Linear Programming Problem graphically: Maximise: 𝑍 = −𝑥 + 2𝑦, subject to the constraints: 𝑥 ≥ 3, 𝑥 + 𝑦 ≥ 5, 𝑥 + 2𝑦 ≥ 6, 𝑦 ≥ 0. [CBSE SQP 2023] 14. Solve the following Linear Programming Problem graphically: Minimise: 𝑍 = 𝑥 + 2𝑦, subject to the constraints: 𝑥 + 2𝑦 ≥ 100, 2𝑥 − 𝑦 ≤ 0,2𝑥 + 𝑦 ≤ 200, 𝑥, 𝑦 ≥ 0. [CBSE SQP 2023] 15. Solve the following linear programming problem graphically: Maximise: Z = x + 2y Subject to constraints: 𝑥 + 2𝑦 ≥ 100, 2𝑥 − 𝑦 ≤ 0 2𝑥 + 𝑦 ≤ 200 𝑥 ≥ 0, 𝑦 ≥ 0 [CBSE 2023] 16. Solve the following Linear Programming Problem graphically: Maximise 𝑍 = 400𝑥 + 300𝑦 Subject to 𝑥 + 𝑦 ≤ 200, 𝑥 ≤ 40, 𝑥 ≥ 20, 𝑦 ≥ 0 [CBSE SQP 2022] 17. The corner points of the feasible region determined by the system of linear constraints are as shown below: CLICK HERE FOR SOLUTIONS
  • 111.
    (A) Let 𝑍= 3𝑥 − 4𝑦 be the objective function. Find the maximum and minimum value of Z and also the corresponding point at which the maximum and minimum value occurs. (B) Let X = 𝑝x + 𝑞𝑦, where 𝑝, 𝑞 > 0 be the objective function. Find the condition on 𝑝 and 𝑞 so that the maximum value of 𝑍 occurs at 𝐵(4,10) and 𝐶(6,8). Also mention the number of optimal solutions in this case. [CBSE SQP 2020] 18. Solve the following LPP graphically: Minimise, 𝑍 = 5𝑥 + 7𝑦 Subject to the constraints 2𝑥 + 𝑦 ≥ 8 𝑥 + 2𝑦 ≥ 10 𝑥, 𝑦 ≥ 0 [CBSE 2020] 19. A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large van is ₹ 400 and each small van is ₹ 200. Not more than ₹ 3000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimise cost. [CBSE 2017] 20. Two tailors, A and B, earn ₹ 300 and ₹ 400 per day, respectively. A can stitch 6 shirts and 4 pairs of trousers while 𝐵 can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP. [CBSE 2017] CLICK HERE FOR SOLUTIONS
  • 112.
    21. A smallfirm manufactures necklace and bracelets. The total number of necklace and bracelet that it can handle per day is at most 24. It takes 1 hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is ₹ 100 and that on a bracelet is ₹ 300, how many of each should be produced daily to maximise the profit? It is being given that at least one of each must be produced. [CBSE 2017] 22. Solve the following linear programming problem (LPP) graphically: Maximise and Minimise 𝑍 = 5𝑥 + 10𝑦 subject to the constraints 𝑥 + 2𝑦 ≤ 120, 𝑥 + 𝑦 ≥ 60, 𝑥 − 2𝑦 ≥ 0, 𝑥 ≥ 0, 𝑦 ≥ 0 [CBSE 2017] 23. If a 20 year old girl drives her car at 25 km/h, she has to spend ₹4/km on petrol. If she drives her car at 40 km/h, the petrol cost increases to ₹ 5/km. She has ₹ 200 to spend on petrol and wishes to find the maximum distance she can travel within one hour. Express the above problem as Linear Programming Problem. Write any one value reflected in the problem. [CBSE SQP 2016] Long Qs (4 - 5 marks) 24. A manufacturer has three machine I, II, and III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for atleast 5 hours a day. She produces only two items M and N each requiring the use of all the three machines. The number of hours required for producing 1 unit each of M and N on the three machines are given in the following table: Items Number of hours required on machines I II III M 1 2 1 N 2 1 1.25 She makes a profit of ₹ 600 and ₹ 400 on items 𝑀 and 𝑁 respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produce? What will be the maximum profit? [CBSE 2020] 25. A small manufacturer has employed 5 skilled men and 10 semi-skilled men and makes an article in two qualities deluxe model and an ordinary model. The making of a deluxe model requires 2hrs. worked by a skilled man and 2 hrs. worked by a semiskilled man. The ordinary model CLICK HERE FOR SOLUTIONS
  • 113.
    requires 1hr bya skilled man and 3hrs. by a semi-skilled man. By union rules no man may work more than 8hrs. per day. The manufacturers clear profit on deluxe model is ₹ 15 and on an ordinary model is ₹ 10. How many of each type should be made in order to maximise his total daily profit. [CBSE 2019] 26. Find graphically, the maximum value of 𝑧 = 2𝑥 + 5𝑦, subject to constraints given below: 2𝑥 + 4𝑦 ≤ 8,3𝑥 + 𝑦 ≤ 6, 𝑥 + 𝑦 ≤ 4; 𝑥 ≥ 0, 𝑦 ≥ 0 [CBSE 2015] 27. Maximise 𝑍 = 8𝑥 + 9𝑦 subject to the constraints given below: 2𝑥 + 3𝑦 ≤ 6,3𝑥 − 2𝑦 ≤ 6, 𝑦 ≤ 1; 𝑥, 𝑦 ≥ 0 [CBSE 2015] CLICK HERE FOR SOLUTIONS
  • 114.
    CLASS 12 MATHEMATICS CHAPTER WISE, TOPIC WISE SOLVED PAPERS (FROM 2014 TO 2023) PYQ
  • 115.
    Probability 1. BASIC CONCEPTSAND CONDITIONAL PROBABILITY Objective Qs (1 mark) 1. If 𝐴 and 𝐵 are two events such that 𝑃(𝐴/𝐵) = 2 × 𝑃(𝐵/𝐴) and 𝑃(𝐴) + 𝑃(𝐵) = 2 3 , then 𝑃(𝐵) is equal to: (a) 2 9 (b) 7 9 (c) 4 9 (d) 5 9 [CBSE 2023] 2. A problem in Mathematics is given to three students whose chances of solving it are 1 2 , 1 3 , 1 4 , respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is: (a) 1 4 (b) 1 3 (c) 1 2 (d) 3 4 [CBSE SQP 2023] 3. Given two independent events 𝐴 and 𝐵 such that 𝑃(𝐴) = 0.3, 𝑃(𝐵) = 0.6 and 𝑃(𝐴′ ∩ 𝐵′) is: (a) 0.9 (b) 0.18 (c) 0.28 (d) 0.1 [CBSE SQP 2022] 4. If 𝐴 and 𝐵 are two independent events with 𝑃(𝐴) = 1 3 and 𝑃(𝐵) = 1 4 , then 𝑃(𝐵′ /𝐴) is: (a) 1 4 (b) 1 8 (c) 3 4 (d) 1 [CBSE 2020] CH-13 CLICK HERE FOR SOLUTIONS
  • 116.
    5. A dieis thrown once. Let 𝐴 be the event that the number obtained is greater than 3 . Let 𝐵 be the event that the number obtained is less than 5 . Then 𝑃(𝐴 ∪ 𝐵) is: (a) 2 5 (b) 3 5 (c) 0 (d) 1 [CBSE 2020] 6. A card is picked at random from a pack of 𝟓𝟐 playing cards. Given that the picked card is a queen, the probability of this card to be a card of spade is: (a) 1 3 (b) 4 13 (c) 1 4 (d) 1 2 [CBSE 2020] Very Short & Short Qs (1 -3 marks) 7. 𝐴 and 𝐵 throw a die alternately till one of them gets 𝑎 ' 6 ' and wins the game. Find their respective probabilities of winning, if 𝐴 starts the game first. [CBSE 2023] 8. Three friends go for coffee. They decide who will pay the bill, by each tossing a coin and then letting the "odd person" pay. There is no odd person if all three tosses produce the same result. It there is no odd person in the first round, they make a second round of tosses and they continue to do so until there is an odd person. What is the probability that exactly three rounds of tosses are made? [CBSE SQP 2022] 9. Events 𝐴 and 𝐵 are such that 𝑃(𝐴) = 1 2 , 𝑃(𝐵) = 7 12 and 𝑃( not 𝐴 or not 𝐵) = 1 4 . State whether A and B are independent? [CBSE Term-2 2022] 10. A box 𝐵1 contains 1 white ball and 3 red balls. Another box 𝐵2 contains 2 white balls and 3 red balls. If one ball is drawn at random from each of the boxes 𝐵1 and 𝐵2, then find the probability that the two balls drawn are of the same colour. [CBSE Term-2 2022] 11. Two cards are drawn at random from a pack of 52 cards one-by-one without replacement. What is the probability of getting first card red and second card jack? [CBSE Term-2 SQP 2022] 12. Two cards are drawn at random and oneby-one without replacement from a well shuffled pack of 52 playing cards. Find the probability that one card is red and other is black. CLICK HERE FOR SOLUTIONS
  • 117.
    [CBSE 2020] 13. Abag contains 3 black, 4 red and 2 green balls. If three balls are drawn simultaneously at random, then the probability that the balls are of different colours is . [CBSE 2020] 14. The probability of finding a green signal on a busy crossing 𝑋 is 30%. What is the probability of finding a green signal on 𝑋 on two consecutive days out of three? [CBSE 2020] 15. A speaks truth in 80% cases and B speak truth in 90% cases. In what percentage of cases are they likely to agree with each other in stating the same fact? [CBSE SQP 2020] 16. If 𝐴 and 𝐵 are two events such that 𝑃(𝐴) = 0.4, 𝑃(𝐵) = 0.3 and 𝑃(𝐴 ∪ 𝐵) = 0.6, then find P(B′ ∩ 𝐴). [CBSE 2020] 17. Out of 8 outstanding students of a school, in which there are 3 boys and 5 girls, a team of 4 students is to be selected for a quiz competition. Find the probability that 2 boys and 2 girls are selected. [CBSE 2019] 18. Prove that if 𝐸 and 𝐹 are independent events, then the events 𝐸′ and 𝐹′ are also independent. [CBSE 2019] 19. 12 cards numbered 1 to 12 (one number on one card), are placed in a box and mixed up thoroughly. Then a card is drawn at random from the box. If it is known that the number on the drawn card is greater than 5, find the probability that the card bears an odd number. [CBSE 2019] 20. A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let 𝐴 be the event, 'the number is even', and 𝐵 be the event, 'the number is red'. Are A and B independent? [CBSE 2019] 21. A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. [CBSE 2018] 22. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Given that (A) the youngest is a girl. (B) at least one is a girl. [CBSE 2014] CLICK HERE FOR SOLUTIONS
  • 118.
    23. A bagcontains 3 red and 7 black balls. Two balls are drawn one by one at a time at random without replacement. If second drawn ball is red then what is the probability the first drawn ball is also red? [CBSE 2014] 24. A couple has two children. Find the probability that both are boys, it is known that: (A) one of the children is a boy (B) older child is a boy. [CBSE 2014] Long Qs (4 – 5 marks) 25. In a game of Archery, each ring of the Archery target is valued. The centre most ring is worth 10 points and rest of the rings are allotted points 9 to 1 in sequential order moving outwards. Archer 𝐴 is likely to earn 10 points with a probability of 0.8 and Archer B is likely to earn 10 points with a probability of 0.9. Based on the above information, answer the following questions: If both of them hit the Archery target, then find the probability that (A) Exactly one of them earns 10 points. (B) both of them earn 10 points. [CBSE Term-2 2022] 26. Probability of solving a specific problem independently by 𝐴 and 𝐵 are 1 2 and 1 3 respectively. If both try to solve the problem independently, find the probability that: (A) the problem is solved. CLICK HERE FOR SOLUTIONS
  • 119.
    (B) exactly oneof them solves the problem. [CBSE 2018] 27. A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and 𝐵 wins the game if he gets a total of 10. If 𝐴 starts the game, then find the probability that 𝐵 wins. [CBSE 2016] 28. If 𝐴 and 𝐵 are two independent events such that 𝑃(𝐴 ‾ ∩ 𝐵) = 2 15 and 𝑃(𝐴 ∩ 𝐵 ‾) = 1 6 , then find 𝑃(𝐴) and 𝑃(𝐵). [CBSE 2015] 29. Consider the experiment of tossing a coin. If the coin shows head, toss it again, but if it shows tail, then throw a die. Find the conditional probability of the event that 'the die shows a number greater than 4' given that 'there is atleast one tail'. [CBSE 2014] 2. BAYES' THEOREM, RANDOM VARIABLE AND ITS PROBABILITY DISTRIBUTION Case Based Qs 30. In an Office three employees Jayant, S onia and Oliver process incoming copies of a certain form. Jayant processes 50% of the forms, Sonia processes 20% and Oliver the remaining 30% of the forms. Jayant has an error rate of 0.06, Sonia has an error rate of 0.04 and Oliver has an error rate of 0.03. Based on the above information, answer the following questions: (A) Find the probability that Sonia processed the form and committed an error. (B) Find the total probability of committing an error in processing the form. (C) The manager of the Company wants to do a quality check. During inspection, he selects a form at random from the days output of processed form. If the form selected at random has an error, find the probability that the form is not processed by Jayant. OR CLICK HERE FOR SOLUTIONS
  • 120.
    Let E bethe event of committing an error in processing the form and let 𝐸1, 𝐸2 and 𝐸3, be the events that Jayant, Sonia and Oliver processed the form. Find the value of ∑𝑖=1 3 𝑃(𝐸𝑖 ∣ 𝐸). [CBSE SQP 2023] 31. An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company's statistics show that an accident-prone person will have an accident at sometime within a fixed one-year period with probability 0.6, whereas this probability is 0.2 for a person who is not accident prone. The company knows that 20 percent of the population is accident prone. (A) What is the probability that a new policyholder will have an accident within a year of purchasing a policy? (B) Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone? [CBSE Term-2 SQP 2022] 32. There are two antiaircraft guns, named as 𝐴 and 𝐵. The probabilities that the shell fired from them hits an airplane are 0.3 and 0.2 respectively. Both of them fired one shell at an airplane at the same time. (A) What is the probability that the shell fired from exactly one of them hit the plane? (B) If it is known that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from B? CLICK HERE FOR SOLUTIONS
  • 121.
    [CBSE SQP 2022] 33.Recent studies suggest that roughly 12% of the world population is left handed. Depending upon the parents, the chances of having a left handed child are as follows: 𝐴 : When both father and mother are left handed: Chances of left handed child is 24%. 𝐵 : When father is right handed and mother is left handed: Chances of left handed child is 22%. C : When father is left handed and mother is right handed: Chances of left handed child is 17%. 𝐷 : When both father and mother are right handed: Chances of left handed child is 9%. Assuming that 𝑃(𝐴) = 𝑃(𝐵) = 𝑃(𝐶) = 𝑃(𝐷) = 1 4 and 𝐿 denotes the event that child is left handed. (A) Find 𝑃(𝐿/𝐶) (B) Find 𝑃(𝐿 ‾/𝐴) (C) Find 𝑃(𝐴/𝐿) Find the probability that a randomly selected child is left handed given that exactly one of the parents is left handed. [CBSE 2023] Very Short & Short Qs (1 - 3 marks) 34. Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, find the mean of random variable. CLICK HERE FOR SOLUTIONS
  • 122.
    [CBSE 2023] 35. Therandom variable 𝑋 has a probability distribution 𝑃(𝑋) of the following form, where ' 𝑘 ' is some real number: 𝑃(𝑋) = � 𝑘, if 𝑥 = 0 2𝑘, if 𝑥 = 1 3𝑘, if 𝑥 = 2 0, otherwise (A) Determine the value of 𝑘. (B) Find 𝑃(𝑋 < 2). (C) Find 𝑃(𝑋 > 2). [CBSE 2023] 36. A bag contains 1 red and 3 white balls. Find the probability distribution of the number of red balls if 2 balls are drawn at random from the bag one-by-one without replacement. [CBSE Term-2 SQP 2022] 37. There are two bags. Bag I contain 1 red and 3 white balls, and Bag II contains 3 red and 5 white balls. A bag is selected at random and a ball is drawn from it. Find the probability that the ball so drawn red in colour. [CBSE Term-2 2022] 38. A coin is tossed twice. The following table shows the probability distribution of number of tails: 𝑋 0 1 2 𝑃(𝑋) 𝐾 6𝐾 9𝐾 (A) Find the value of 𝐾. (B) Is the coin tossed biased or unbiased? Justify your answer. [CBSE Term-2 2022] 39. Three rotten apples are mixed with seven fresh apples. Find the probability distribution of the number of rotten apples, if three apples are drawn one by one with replacement. Find the mean of the number of rotten apples. [CBSE 2020] 40. In a shop 𝑋, 30 tins of ghee of type 𝐴 and 40 tins of ghee of type B which look alike, are kept for sale. While in shop 𝑌, similar 50 tins of ghee of type 𝐴 and 60 tins of ghee of type 𝐵 are there. One tin of ghee is purchased from one of the randomly selected shop and is found to be of type B. Find the probability that it is purchased from shop Y. [CBSE 2020] 41. Find the probability distribution of 𝑋, the number of heads in a simultaneous toss of two coins. [CBSE 2019] CLICK HERE FOR SOLUTIONS
  • 123.
    42. Of thestudents in a school, it is know that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance? Is regularity required only school? Justify your answer. [CBSE 2017] 43. Three persons 𝐴, 𝐵 and 𝐶 apply for a job of manager in a private company. Chances of their selection (A, B, C) are in the ratio 1: 2: 4. The probabilities that 𝐴, 𝐵 and 𝐶 can introduce changes to improve profits of the company are 0.8,0.5 and 0.3 respectively. If the changes does not take place, find the probability that it is due to the appointment of C. [CBSE 2016] 44. Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spades. Hence, find the mean of the distribution. [CBSE 2015] 45. A bag A contains 4 black and 6 red balls and bag 𝐵 contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag 𝐴 is choosen, otherwise bag B. If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black. [CBSE 2015] Long Qs (4 – 5 marks) 46. There are two boxes, namely box-I and box-II. Box-I contains 3 red and 6 black balls. Box-II contains 5 red and 5 black balls. One of the two boxes is selected at random and a ball is drawn at random. The ball drawn is found to be red. Find the probability that this red ball comes out from box-II. [CBSE Term-2 2022] 47. An insurance company insured 3000 cyclists, 6000 scooter drivers, and 9000 car drivers. The probability of an accident involving a cyclist, a scooter driver, and a car driver are 0.3,0.05 and 0.02 respectively. One of the insured persons meets with an accident. What is the probability that he is a cyclist? [CBSE 2019] 48. A manufacturer has three machine operators A, B and C. The first operator A produces 1% of defective items, whereas the other two operators 𝐵 and 𝐶 produces 5% and 7% defective items respectively. 𝐴 is on the job for 50% of the time, B is on the job 30% of time and 𝐶 on the job for 20% of the time. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by 𝐴 ? [CBSE 2019] 49. Often it is taken that a truthful person commands more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six. Find the probability that it is actually a six. Do you also agree that the value of truthfulness leads to more respect in the society? CLICK HERE FOR SOLUTIONS
  • 124.
    [CBSE 2017] 50. Abag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white? [CBSE 2016] 51. A class has 15 students whose ages are 14,17,15,14,21,17,19,20,16,18,20,17, 16,19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age 𝑋 of the selected student is recorded. What is the probability distribution of the random variable X? Find the mean of X. [CBSE 2014] 52. In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins loses. [CBSE 2014] 53. Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable X and hence find the mean of the distribution. [CBSE 2014] CLICK HERE FOR SOLUTIONS
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    SKILL MODULES BEINGOFFERED IN MIDDLE SCHOOL Artificial Intelligence Beauty & Wellness Design Thinking & Innovation Financial Literacy Handicrafts Information Technology Marketing/Commercial Application Mass Media - Being Media Literate Travel & Tourism Coding Data Science (Class VIII only) Augmented Reality / Virtual Reality Digital Citizenship Life Cycle of Medicine & Vaccine Things you should know about keeping Medicines at home What to do when Doctor is not around Humanity & Covid-19 Blue Pottery Pottery Block Printing
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    SKILL SUBJECTS ATSR. SEC. LEVEL (CLASSES XI – XII) Retail InformationTechnology Web Application Automotive Financial Markets Management Tourism Beauty & Wellness Agriculture Food Production Front Office Operations Banking Marketing Health Care Insurance Horticulture Typography & Comp. Application Geospatial Technology Electrical Technology Electronic Technology Multi-Media
  • 140.
    Taxation Cost AccountingOffice Procedures & Practices Shorthand (English) Shorthand (Hindi) Air-Conditioning & Refrigeration Medical Diagnostics Textile Design Design Salesmanship Business Administration Food Nutrition & Dietetics Mass Media Studies Library & Information Science Fashion Studies Applied Mathematics Yoga Early Childhood Care & Education Artificial Intelligence Data Science Physical Activity Trainer(new) Land Transportation Associate (NEW) Electronics & Hardware (NEW) Design Thinking & Innovation (NEW)
  • 141.