This document contains a mathematics sample paper with 29 questions across 4 sections - A, B, C and D. The questions cover a wide range of mathematics topics including calculus, algebra, trigonometry, matrices, probability, linear programming, and geometry. The paper is designed to test students' mathematical skills and knowledge over 3 hours with the maximum marks being 100.

1. MATHEMATICS (Sample Paper)
Time allowed: 3 hours M.M.- 100
SECTION - A
1. Evaluate : 𝑙𝑜𝑔
2+𝑥
2−𝑥
1
−1
dx
2. If ‘*’ be a binary operation on ‘R’ defined by a*b =
𝑎𝑏
5
then find identity element for the binary operation ‘*’ if
exists.
3. If ∆=
2 −3 5
6 0 1
1 5 −7
𝑡𝑕𝑒𝑛 𝑓𝑖𝑛𝑑 ∶ 𝑎12. 𝐴32 + 𝑎11. 𝐴21
4. If 𝑎 + 𝑏 = 𝑎 − 𝑏 then what is the angle between 𝑎 and 𝑏
SECTION B
5. Show : tan−1 1+𝑥+ 1−𝑥
1+𝑥− 1−𝑥
=
𝜋
4
+
1
2
cos−1
𝑥 , −1 < 𝑥 < 1
6. If y = 𝑙𝑜𝑔 (𝑥 + 𝑥2 + 1) 2
then show : (1 + x2
)
𝑑2 𝑦
𝑑𝑥2 + 𝑥
𝑑𝑦
𝑑𝑥
− 2 = 0
7. Simplify: tan-1 a cos x−b sin x
b cos x+a sin x
8. Find 2 x 2 matrix B : B
1 −2
1 4
=
6 0
0 6
.
9. The radius of a spherical diamond is measured as 7 cm with an error of 0.04cm . find the approximate error in
calculating its volume.
10. The mean and variance of a binomial distribution are 12 and 3 respectively. Find the probability distribution.
11. Find the angle between the line
𝑥+1
2
=
3𝑦+5
9
=
3−𝑧
−6
and the plane 10x + 2y – 11z = 3.
12. Evaluate:
𝑑𝑥
7−6𝑥−𝑥2
SECTION -C
13. By using the properties of determinants show :
𝑎 𝑏 𝑐
𝑎 − 𝑏 𝑏 − 𝑐 𝑐 − 𝑎
𝑏 + 𝑐 𝑐 + 𝑎 𝑎 + 𝑏
= 𝑎3
+ 𝑏3
+ 𝑐3
− 3𝑎𝑏𝑐
14. Find the intervals in which f(x) is strictly increasing or strictly decreasing where f(x) = -2x3
-9x2
-12x+1
OR
Find the equation of tangent to the curve y = (3𝑥 − 2) which is parallel to line 4x – 2y + 5 = 0
15. If x =a Sin2t(1+Cos2t), y = bcos2t(1-Cos2t) find
𝑑𝑦
𝑑𝑥
16. Roger Federer wins thrice as often as he loses. If he plays two matches then find the probability distribution of
number of wins.
17. Evaluate :
𝑥2+𝑥+1
(𝑥+2)(𝑥2+1)
dx
18. Find the equation of plane which contains the point (1, -1, 2) and perpendicular to each of the planes 2x+3y-2z =
5 and x+2y-3z = 11

2. 19. f(x) =
𝑠𝑖𝑛 3𝑥
𝑡𝑎𝑛 2𝑥
, 𝑥 < 0
𝑘, 𝑥 = 0
𝑒3𝑥2
−1
1−𝑐𝑜𝑠 2𝑥
, 𝑥 > 0
, find k if f(x) is continuous at x = 0
20. Prove that volume of largest cone that can be inscribed in a sphere of radius ‘R’ is
8
27
of the volume of sphere.
Also find the height of cone.
OR
Given the sum of perimeter of a square and circle. Show that sum of their areas is least when the side of the
square is equal to the diameter of circle.
21. Three bags contain balls as shown in the table below :
Bag Number of white balls Number of green balls Number of red balls
I 1 2 3
II 2 1 1
III 4 3 2
A bag chosen at random and two balls are drawn from it. These balls happen to be white and green.
What is the probability that these balls came from bag III?
22. If 𝑎 , 𝑏, 𝑐 are mutually perpendicular vectors of equal magnitudes then show that the vector
𝑎 + 𝑏 + 𝑐 is equally inclined to 𝑎 , 𝑏 𝑎𝑛𝑑 𝑐.
OR
If𝑎 = 𝑖 + 𝑗 + 𝑘 and 𝑏 = 𝑗 − 𝑘 then find a vector 𝑐 such that 𝑎 × 𝑐 = 𝑏 𝑎𝑛𝑑 𝑎. 𝑐 = 3
23. Solve the following differential equation : x dy – y dx = 𝑥2 + 𝑦2 dx
SECTION D
24. Find the area bounded by the circles x2
+ y2
= 4 and x2
+ (y-2)2
= 4.
25. Show that relation ‘R’ defined on the set N x N given by : (a.b) R (c.d) ⇒a+d = b+c is a equivalence relation.
26. Evaluate :
𝑥 𝑡𝑎𝑛 𝑥
𝑠𝑒𝑐 𝑥+𝑡𝑎𝑛 𝑥
𝑑𝑥
𝜋
0
OR
𝑎−𝑥
𝑎+𝑥
𝑎
−𝑎
dx
27. Find the foot of perpendicular drawn from the point (0, 2, 3) to the line
𝑥+3
5
=
𝑦−1
2
=
𝑧+4
3
. Also find the image of
the point w.r.t. the line.
28. If A =
3 2 1
4 −1 2
7 3 −3
find A-1
and by using A-1
solve :
3𝑥 + 4𝑦 + 7𝑧 = 14
2𝑥 − 𝑦 + 3𝑧 = 4
𝑥 + 2𝑦 − 3𝑧 = 0
29. Prerna Singhal has Rs. 1500 for purchase the bags of chocolates and sweets. A bag of chocolates and a bag of
sweets cost Rs. 180 and Rs. 120 respectively. She has the storage capacity of 10 bags only. She earns a profit of
Rs. 11 and Rs. 9 respectively per bag of chocolates and sweets. Formulate it as a LPP and solve it graphically for
the maximum profit she can get for her birthday party.