class 12 mathematics 10 complete unsolved papers for final exams practice

1. Unsolved practice paper – 1
Section A
1. Fine the value (s) of x for which the matrix A = [
𝒙 𝟏 𝟐
𝟏 𝟎 𝟑
𝟓 −𝟏 𝟒
] is singular .
2. Given ∫ 𝒆 𝒙
( tan x + 1 ) sec x dx = 𝒆 𝒙
 (x) + c. Fine  (x).
3. Let A = { 1,2,3,} . The relation R on set A is defined as R = { ( 1,2 ), (1,3 ), (1,3) } . Check whether R is
reflexive or not.
4. Evaluate : cos [𝟐 𝐬𝐢𝐧−𝟏 𝟏
𝟐
] .
5. If a vector has direction angles 90o , 60o and 30o , then fine its direction cosines.
6. Evaluate the integral : ∫
𝟏
𝟏+ 𝒙 𝟐
√𝟑
𝟏
dx.
Section B
7. Evaluate :- ∫
𝒙 𝟐
+ 𝟏
(𝒙−𝟏) 𝟐 ( 𝒙+𝟑)
dx.
8. Differentiate the following w.r.t. x: 𝐜𝐨𝐭−𝟏
[
√𝟏+ 𝒙 𝒎
+ √𝟏− 𝒙 𝒎
√𝟏+ 𝒙 𝒎 − √𝟏− 𝒙 𝒎 ] .
9. A bag contains 2 white, 3 red and 4 blue balls . Two balls are drawn from the bag. Fine the
probability distributions of the number of white balls.
OR
A random variable X has the following probability distribution:
X 0 1 2 3 4 5 6 7
P(X) 0 K 2k 2k 3k k2
2k2
7k2
+ k
Determine (i) k. (ii) P (X < 3 ) . (iii) P (X > 6 ) (iv) P (0 < X < 3 ) .
10. Fine inverse of the matrix [
𝟎 −𝟔
−𝟑 𝟓
] by using Elementary Row Transformation.
11. If x = tan (
𝟏
𝒂
𝒍𝒐𝒈 𝒚 ) . Then prove that (1+ x2)
𝒅 𝟐
𝒚
𝒅𝒙 𝟐 + ( 𝟐𝒙 − 𝒂)
𝒅𝒚
𝒅𝒙
= 𝟎 .
OR
If y = 500e7x + 600 e-7x , then prove that
𝒅 𝟐
𝒚
𝒅𝒙 𝟐 = 49 y.
12. Proved that |
𝒂 + 𝒃 + 𝒄 −𝒄 −𝒃
−𝒄 𝒂 + 𝒃 + 𝒄 −𝒂
−𝒃 −𝒂 𝒂 + 𝒃 + 𝒄
| = 𝟐( 𝒂 + 𝒃)( 𝒃+ 𝒄)(𝒄 + 𝒂) .
13. Let A be the set of all 50 student of class XII in a central school. Let  : A → N be a function , defined
by  (x) = Roll number of student x. Show that  is one – one but not onto . All the students of this

2. class are participating in ‘SAVE ENVIRONMENT’ camping . Would you like to be a part of campaign ?
Why ?
14. Using differentials, find the approximate value of the following :
𝟏
(𝟑𝟑)
𝟏
𝟓
.
15. Evaluate : ∫
𝟏
𝒔𝒊𝒏 ( 𝒙−𝒂) 𝒔𝒊𝒏 (𝒙−𝒃)
dx.
16. If A- 1 =
𝟏
𝟓
[
𝟑 −𝟏
𝟏 𝟐
] 𝒂𝒏𝒅 𝑩 = [
𝟎 𝟑
𝟏 𝟒
] , then find (AB)-1 .
17. Solve the differential equation 2x2 𝒅𝒚
𝒅𝒙
- 2xy + y2 = 0.
OR
Solve the differential equation tan y
𝒅𝒚
𝒅𝒙
= cos (x + y) + cos (x – y).
18. If 𝒂⃗⃗ , 𝒃⃗⃗ , 𝒄⃗ are three vectors such that 𝒂⃗⃗  𝒃⃗⃗ = 𝒄⃗ , 𝒃⃗⃗  𝒄⃗ = 𝒂⃗⃗ , prove that 𝒂⃗⃗ , 𝒃⃗⃗ , 𝒄⃗ are mutually
perpendicular to each other with 𝒂⃗⃗ = 1 and | 𝒂⃗⃗ | = | 𝒄⃗ | .
19. Solve: tan-1 (2x) + tan-1 (3x) = n +
𝟑𝝅
𝟒
, where n  Z.
OR
Solve : sin-1 ( 6√ 𝟑 x ) + sin-1 (6x) =
𝝅
𝟐
.
Section - C
20. Fine the equation of the plane passing through the point ( 1, 1, 0, ) , ( 1, 2, 1 ) and (-2, 2, -1 ) .
21. Evaluate : ∫ |𝒙 𝟑
− 𝒙|
𝟐
−𝟏
dx.
22. Find the area enclosed between the parabola y2 = 4ax and the line y = mx.
OR
Find the area bounded by the curves y = x and y = x3 .
23. If product of distances of the point (1, 1, 1 ) from origin and plane 𝒓⃗ . (𝒊̂ − 𝒋̂ + 𝒌̂ ) = -p be 8 , then
find the value (s) of p.
24. A manufacturer of patent medicines is preparing a production plan for medicines M1 and M2 . There
is sufficient raw material available to fill 20000 bottles of M1 and 40000 bottles of M2 . But there
are only 45000 bottles in to which either of the medicines can be put . Further, It takes 3 hours to
prepare enough material to fill 1000 bottles of M1 and 1 hour to prepare enough material to fill
1000 bottles of M2 . There are 66 house available for this operation . The profit is Rs. 8 per bottle
for M1 and Rs.7 per bottle for M2 . How should the manufacturer schedule his production in order
to maximise his profit ? Formulate the above L.P.P. mathematically and then solve it graphical.
25. Suppose the reliability of an HIV test is specified as follows: Among people having HIV , 90% of the
tests dated the disease but 10% fail to do so. Among people not having HIV , 99%of the tests show

3. HIV – ive but 1% are diagnosed as HIV + ive . From a large population of which only 0.1% have HIV ,
one person is selected at random at random for an HIV test, and the pathologist reports him/her as
HIV + ive . what is the probability that the person actually has HIV ?
OR
For three persons A, B and C, the chances of being selected as a manager of a firm are in the ration
4 : 1 : 2 respectively . the respective probability for them tointroduce a radical change in marketing
strategy are 0.3, 0.8, and 0.5. if the change takes place , find the probability that it is due to the
appointment of B.
26. Show that the volume of the greatest cylinderthat can be inscribed in a given cone of height h and
semi – vertical angle  is
𝟒
𝟐𝟕
 h3 tan2 .
ANSWERS
1. -3 2. sec 3. no 4. ½ 5. 0, ½,
√𝟑
𝟐
6.
𝝅
𝟏𝟐
7.
𝟑
𝟖
log | 𝒙 − 𝟏|-
𝟏
𝟐(𝒙−𝟏)
+
𝟓
𝟖
𝒍𝒐𝒈 | 𝒙 + 𝟑| + 𝑪 8.
𝒎𝒙 𝒎−𝟏
𝟐 √ 𝟏− 𝒙 𝟐𝒎
9.
X 0 1 2
P(X) 𝟐𝟏
𝟑𝟔
𝟏𝟒
𝟑𝟔
𝟏
𝟑𝟔
OR
(i)
𝟏
𝟏𝟎
(ii)
𝟑
𝟏𝟎
(iii)
𝟕
𝟏𝟎𝟎
(iv)
𝟑
𝟏𝟎
10. [
−
𝟓
𝟏𝟖
−
𝟏
𝟑
−
𝟏
𝟔
𝟎
]
13. Yes , I would like to be a part of this camping because healthy environment will improve the
quality of our lives.
14. 0.4969 15. cosec (b – a ) log |
𝒔𝒊𝒏 (𝒙−𝒃)
𝒔𝒊𝒏 (𝒙−𝒂)
| + C
16. −
𝟏
𝟏𝟓
[
𝟗 −𝟏𝟎
−𝟑 𝟏
] 17 . −
𝟐𝒙
𝒚
+ 𝒍𝒐𝒈 | 𝒙| + 𝑪 OR sec y = 2 sin x +c
19. −
𝟏
𝟔
OR
𝟏
𝟏𝟐
20. -2x – 3y + 3z + 5 = 0 21.
𝟏𝟏
𝟒
22.
𝟖 𝒂 𝟐
𝟑𝒎 𝟐 sq. units OR ½ sq. units 23. 7 and – 9
24. Number of bottle of medicine M1 = 10500, number of bottle of medicine M2 = 34500 and
maximum profit = Rs. 325500 25.
𝟏𝟎
𝟏𝟐𝟏
𝑶𝑹
𝟒
𝟏𝟓
.

4. Unsolved Practice Paper – 2
Section A
1. Find the position vector of a point R which divided the line joining two points P and Q, whose
position vectors are 𝒊̂ + 𝟐𝒋̂ − 𝒌̂ and −𝒊̂ + 𝒋̂ + 𝒌̂ respectively , in the ration 2: 1 internally.
2. Evaluate :- ∫
( 𝒙+𝟏)(𝒙+𝒍𝒐𝒈 𝒙) 𝟐
𝒙
dx.
3. Evaluate : cos-1 (- ½ ) + sin-1 (- ½ ) .
4. Evaluate the integral : ∫ 𝐱 (𝟏 − 𝐱) 𝐧
dx.
5. Let A = { 1, 2, 3} . the relation R on set A is defined as R = { ( 1, 1 ), (1,2 ), (2,1 ),(2,2) } . Check whether
R is transitive or not .
6. If A is square matrix such that AT A = I , write the value of | 𝐴|.
Section B
7. Find the absolute maximum and absolute minimum values of  (x) = x + sin 2x , x  [0,2].
8. Show that the relation R on R , defined as R = { ( a,b ) : a < b2 } , is neither reflexive nor symmetric
nor transitive .
OR
Find go and  og , when  : R  R are g: R R are defined by  (x) = | 𝒙| and g (x) = | 𝟓𝒙 − 𝟐|.
9. In the first five months , the performance of a student in x months is governed by the relation
( x) = 2x3 - 9x2 + 12x + 1 . Find the months in which the performance of the student is increasing or
decreasing . What life skills should the student develop to improve his performance ?
10. If y = A cos (log x) + B sin (log x) , then prove that x2 y2 + xy1 + y = 0.
11. How many time must a man toss a fair coin so that the probability of having at least one head is
more than 80% ? OR
Suppose X has binomial distribution B ( 6, ½ ) . show that X = 3 is the most likely Outcome.
12. Evaluate : ∫( 𝐬𝐢𝐧−𝟏
𝒙) 𝟐
dx .
13. Show that the area of a parallelogram with diagonals 3𝒊̂ + 𝒋̂ − 𝟐𝒌̂ 𝒂𝒏𝒅 𝒊̂ − 𝟑𝒋̂ + 𝟒𝒌̂ is 5√ 𝟑 sq
units.
14. Write in the simplest from : tan-1 (
𝟑𝒂 𝟐
𝒙− 𝒙 𝟑
𝒂 𝟑− 𝟑𝒂𝒙 𝟐) ; 𝒂 > 0 ; −
𝒂
√𝟑
≤ 𝒙 ≤
𝒂
√𝟑
.
OR
Prove that tan-1 (
𝟑𝒙− 𝒙 𝟑
𝟏− 𝟑𝒙 𝟐 ) = tan-1 x + tan-1 (
𝟐𝒙
𝟏− 𝒙 𝟐 ) ; | 𝒙| <
𝟏
√𝟑
.
15. Discuss the continuity of the function  (x) = sin x – cos x.
16. Differentiate the following w.r.t. x: cot-1 √
𝒂+𝒙
𝒂−𝒙
.

5. 17. Evaluate : ∫
𝟏
𝒙
𝟏
𝟐+ 𝒙
𝟏
𝟑
dx .
18. Evaluate : ∫
𝒄𝒐𝒔 𝟓
𝒙
𝒄𝒐𝒔 𝟓 𝒙+ 𝒔𝒊𝒏 𝟓 𝒙
𝝅
𝟐
𝟎
dx.
OR
Evaluate:- ∫ | 𝒙 + 𝟐|
𝟓
−𝟓
dx.
19. Find the general solution of the differential equation ( 1 + x2 )
𝒅𝒚
𝒅𝒙
+ y = tan-1 x.
Section C
20. There are three coins. One is a two – headed coin (having head on both faces ), another is a biased
coin that come tails up 25% of the times and the third is an unbiased coin. One of the three coins is
chosen at random and tossed, it shows head , what is the probability that it was the two – headed
coin ?
OR
A bag contains 4 balls. Two balls are drawn at random and are found to be white. What is the
probability that all balls are white?
21. Find the equation of the plane through the line of intersection of 𝒓⃗ .( 𝒊̂ + 𝒋̂ + 𝒌̂) = 𝟏 and
𝒓⃗ .( 𝟐𝒊̂ + 𝟑𝒋̂ + 𝟒𝒌̂)− 𝟓 = 𝟎 and parallel to the line
𝒙−𝟏
𝟏
=
𝒚−𝟓
−𝟏
=
𝒛+𝟏
𝟏
.
22. Show that A = [
𝟓 𝟑
−𝟏 −𝟐
] satisfies A2 - 3A – 7I = O and hence find A-1 .
23. A factory owner purchases two types of machines, M1 and M2 for his factory. The requirements and
limitations for the machines are as follows:
Area Occupiedby Each
Machine
Labour Force for
Each Machine
Daily Output
Machine M1 1000 sq m 12 men 60 units
Machine M2 1200 sq m 8 men 40 units
He has an area of 9000 sq.m. and 72 skilled men who can operate the machines. How many
machines of each type should he buy to maximize the daily output? Formulate the above L.P.P.
mathematically and then solve it graphically.
24. Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y.
OR
Find the area bounded by the curves (x – 1)2 + y2 = 1 and x2 + y2 = 1.

6. 25. Find the distance of the point (2, 3, 4) from the plane 3x + 2y + 2z + 5 = 0 measured parallel to the
line
𝒙+𝟑
𝟑
=
𝒚−𝟐
𝟔
=
𝒛
𝟐
.
26. Prove that |
−𝒃𝒄 𝒃 𝟐
+ 𝒃𝒄 𝒄 𝟐
+ 𝒃𝒄
𝒂 𝟐
+ 𝒂𝒄 −𝒂𝒄 𝒄 𝟐
+ 𝒂𝒄
𝒂 𝟐
+ 𝒂𝒃 𝒃 𝟐
+ 𝒂𝒃 −𝒂𝒃
| = (ab + bc + ca) 3 .
ANSWERS
1.
−𝒊̂+ 𝟒𝒋̂+ 𝒌̂
𝟑
2.
(𝒙+𝒍𝒐𝒈 𝒙) 𝟑
𝟑
+ C 3.
𝝅
𝟐
4.
𝟏
𝒏+𝟏
−
𝟏
𝒏+𝟐
5. Yes 6.  1
7. Absolute maximum value of f(x) is 2 which occurs at x = 2 and absolute minimum value of f(x) is 0
which occurs at x = 0.
8. g o f : R → R such that (g o f) (x) = | 𝟓| 𝒙 |−𝟐| and f o g : R → R such that (f o g) (x) = | 𝟓𝒙 − 𝟐| .
10. The performance of the student is increasing during the first, third, fourth and fifth months,
whereas it is decreasing during the second month. The life skills the student must develop to
improve his performance are hard work, grif, determination, commitment, regularity and sincerity.
11. 3 12. (sin-1 x)2 + 2(sin-1 x) √𝟏 − 𝒙 𝟐 - 2x + C 14. 3 tan-1 (
𝒙
𝒂
)
15. f is continuous at all points of its domain. 16.
−𝟏
𝟐 √ 𝒂 𝟐− 𝒙 𝟐
17. 𝟐𝒙
𝟏
𝟐 − 𝟑𝒙
𝟏
𝟑 + 𝟔𝒙
𝟏
𝟔 − 𝟔 𝒍𝒐𝒈 |𝒙
𝟏
𝟔 + 𝟏 | + 𝑪 18.
𝝅
𝟒
OR 29
19. 𝒚𝒆𝐭𝐚𝐧−𝟏
𝒙
= 𝒆𝐭𝐚𝐧−𝟏
𝒙 ( 𝐭𝐚𝐧−𝟏
𝒙 − 𝟏)+ 𝑪 20.
𝟒
𝟗
𝑶𝑹
𝟑
𝟓
21. x – z + 2 = 0
22.
𝟏
𝟕
[
𝟐 𝟑
−𝟏 −𝟓
]
23. Number of machines M1 = 6, number of Machines M2 = 0 and maximum daily output = 320 units.
24. [
√𝟐
𝟔
+
𝟗
𝟒
𝐬𝐢𝐧−𝟏
(
𝟐√𝟐
𝟑
)] 𝒔𝒒. 𝒖𝒏𝒊𝒕𝒔 𝑶𝑹 (
𝟐𝝅
𝟑
−
√𝟑
𝟐
) sq. units 25. 7 units.

7. Unsolved Practice Paper – 3
Section – A
1. Find the angle between vectors 𝒂⃗⃗ 𝒂𝒏𝒅 𝒃⃗⃗ with magnitudes √ 𝟑 and 2 respectively having 𝒂⃗⃗ . 𝒃⃗⃗ = √ 𝟔.
2. Evaluate :- ∫ 𝒔𝒊𝒏 ( 𝒄𝒐𝒔 𝒙) 𝒔𝒊𝒏 𝒙 𝒅𝒙 .
3. Find the direction cosines of a line which makes equal angles with the coordinate axes.
4. Write the value of x – y + z from the following equations : [
𝒙 + 𝒚 + 𝒛
𝒙 + 𝒛
𝒚 + 𝒛
] = [
𝟗
𝟓
𝟕
] .
5. If 𝒂⃗⃗ 𝒂𝒏𝒅 𝒃⃗⃗ are two vectors such that |𝒂⃗⃗ × 𝒃⃗⃗ | = 𝒂⃗⃗ . 𝒃⃗⃗ , then what is the angle between 𝒂⃗⃗ 𝒂𝒏𝒅 𝒃⃗⃗ ?
6. Evaluate :- ∫
𝟏
𝒙
𝟑
𝟐
dx.
Section - B
7. Evaluate :- ∫
𝒍𝒐𝒈 𝒙
(𝟏+𝒍𝒐𝒈 𝒙) 𝟐 dx.
8. Find the angle between the line
𝒙+𝟏
𝟐
=
𝟑𝒚+𝟓
𝟗
=
𝟑−𝒛
−𝟔
and the plane 10x + 2y – 11z = 3.
9. If 𝒂⃗⃗ 𝒂𝒏𝒅 𝒃⃗⃗ are two vectors such that | 𝒂⃗⃗ | = 3, |𝒃⃗⃗ | = 4 and |𝒂⃗⃗ + 𝒃⃗⃗ |= √ 𝟓 , then find 𝒂⃗⃗ . 𝒃⃗⃗ .
10. Prove that :- 𝐬𝐢𝐧−𝟏 𝟑
𝟓
− 𝐬𝐢𝐧−𝟏 𝟖
𝟏𝟕
= 𝐜𝐨𝐬−𝟏 𝟖𝟒
𝟖𝟓
.
OR
Prove that :- 2 𝐭𝐚𝐧−𝟏 𝟏
𝟐
+ 𝐭𝐚𝐧−𝟏 𝟏
𝟕
= 𝒕𝒂𝒏−𝟏 𝟑𝟏
𝟏𝟕
11. Verify (if applicable) Lagrange’s mean value theorem for the function f(x) = sin4 x + cos4 x on [𝟎,
𝝅
𝟐
] .
OR
Prove that y =
𝟒 𝒔𝒊𝒏 
(𝟐+𝒄𝒐𝒔  )
-  is an increasing function of  on [𝟎,
𝝅
𝟐
] .
12. If F(x) = [
𝒄𝒐𝒔 𝒙 − 𝒔𝒊𝒏 𝒙 𝟎
𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝒙 𝟎
𝟎 𝟎 𝟏
] , then show that F(x) F(y) = F(x + y).
13. In a hostel, 60% of the students read Hindi newspaper , 40% read English newspaper and 20% read
both Hindi and English newspapers. A student is selected at random.
a. If she reads Hindi newspaper , find the probability that she reads English newspaper.
b. If she reads English newspaper, find the probability that the reads Hindi newspaper.
OR
An instructor has a question bank consisting of 300 easy True/ False questions, 200 difficult True/
False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a
question is selected at random from the question bank, what is the probability that it will be an easy
question given that it is multiple choice question?

8. 14. Find the equation of tangent to the curve y =
𝒙−𝟕
( 𝒙−𝟐)(𝒙−𝟑)
at the point where it cuts the x – axis.
15. If y = 3e2x + 2e3x , then prove that y2 – 5y1 + 6y = 0.
OR
If ey (1 + x) = 1, then prove that y2 = (y1 )2 .
16. Find the adjoint of the given matrix [
𝟐 −𝟏
𝟒 𝟑
] .
17. Using properties of determinants, prove that |
𝒔𝒊𝒏 𝜶 𝒄𝒐𝒔 𝜷 𝒄𝒐𝒔 (𝜶 + 𝜷)
𝒔𝒊𝒏 𝜷 𝒄𝒐𝒔 𝜷 𝒄𝒐𝒔 (𝜷 + 𝜶)
𝒔𝒊𝒏 𝜸 𝒄𝒐𝒔 𝜸 𝒄𝒐𝒔 (𝜸+ 𝜹)
| = 0.
18. Evaluate :-- ∫
𝒔𝒊𝒏 𝒙
𝒔𝒊𝒏 𝟒𝒙
dx.
19. A population grows at the rate of 8% per year. How long does it take for the population todouble?
The government runs various programmes to educate people about the disadvantages of large
families. Would you like to volunteer for these programmes?
Section - C
20. An open box with a square base is to be made out of a given quantity of cardboard of area c2 square
units. Show that the maximum volume of the box is
𝒄 𝟑
𝟔√𝟑
cubic units.
21. Bag I contains 3 red and 4 black balls and Bag II contains 5 red and 6 black balls. One ball is drawn at
random from one of the bags and is found to be red. Find the probability that it was drawn from Bag
II.
OR
Three bags A, B , C contain 6 red, 4 black; 4 red, 6 black and 5 red, 5 black balls respectively. One of
The bags is selected at random and a ball is drawn from it. If the ball drawn is red, find the
probability that it is drawn from the bag A.
22. If f(x) =
𝟒𝒙+𝟑
𝟔𝒙−𝟒
, 𝒙 ≠
𝟐
𝟑
, then show that (f o f) (x) = x, for all x ≠
𝟐
𝟑
. what is the inverse of f?
23. A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and
3 hours of craftsman’s time in making, while a cricket bat takes 3 hours of machine time and 1 hour
of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine
time and 24 hours of craftsman’s time. What number of rackets and bats must be made if the factory
is to work at full capacity? If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find
the maximum profit of the factory when it works at full capacity. Express it as an L.P.P. and then
solve it.

9. 24. Find the vector and Cartesian equations of the line passing through the point (1, 2, -4) and
perpendicular to the two lines
𝒙−𝟖
𝟑
=
𝒚+𝟏𝟗
−𝟏𝟔
=
𝒛−𝟏𝟎
𝟕
𝒂𝒏𝒅
𝒙−𝟏𝟓
𝟑
=
𝒚−𝟐𝟗
𝟖
=
𝒛−𝟓
−𝟓
.
25. Find the area of the region lying in the second quadrant and bounded by y = 4x2 , x = 0, y = 1 and y=4.
OR
Find the area bounded by the curve y = cos x between x = 0 and x - 2 .
26. Evaluate :- ∫ 𝐜𝐨𝐭−𝟏
(𝟏 − 𝒙 + 𝒙 𝟐
)
𝟏
𝟎
dx
ANSWERS
1. 45o 2. Cos (cos x) + C 3.
𝟏
√𝟑
,
𝟏
√𝟑
,
𝟏
√𝟑
𝑶𝑹 −
𝟏
√𝟑
, −
𝟏
√𝟑
, −
𝟏
√𝟑
4. 1 5. 45o 6. log
𝟑
𝟐
7.
𝒙
𝟏+𝒍𝒐𝒈 𝒙
+ C
8. sin-1 (
𝟖
𝟐𝟏
) 9. -10 13. (i)
𝟏
𝟑
, (ii) ½ OR
𝟓
𝟗
14. x – 20 y = 7 16. [
𝟑 𝟏
−𝟒 𝟐
] 18.
𝟏
𝟖
𝒍𝒐𝒈 |
𝒔𝒊𝒏 𝒙−𝟏
𝒔𝒊𝒏 𝒙+𝟏
| −
𝟏
𝟒√𝟐
𝒍𝒐𝒈 |
√𝟐 𝒔𝒊𝒏 𝒙−𝟏
√𝟐 𝒔𝒊𝒏 𝒙+𝟏
| + 𝑪
19.
𝟐𝟓
𝟐
log 2 years. Yes, I would to like to volunteer for these programmes because we can stop
population explosion only by educating people about the disadvantages of large families.
21.
𝟑𝟓
𝟔𝟖
𝑶𝑹
𝟐
𝟓
22. f-1 (x) =
𝟒𝒙+𝟑
𝟔𝒙−𝟒
, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 ≠
𝟐
𝟑
23. Number of tennis rackets = 4, number of cricket bats = 12 and maximum profit = Rs. 200.
24. Vector form : 𝒓⃗ = (𝒊̂ + 𝟐𝒋̂ − 𝟒𝒌̂ ) + (𝟐𝒊̂ + 𝟑𝒋̂ + 𝟔𝒌̂ ) 𝒂𝒏𝒅 𝑪𝒂𝒓𝒕𝒆𝒔𝒊𝒂𝒏 𝒇𝒐𝒓𝒎 ∶
𝒙−𝟏
𝟐
=
𝒚−𝟐
𝟑
=
𝒛+𝟒
𝟔
.
25.
𝟕
𝟑
sq. units OR 4 sq. units 26.
𝝅
𝟐
- log 2

10. Unsolved Practice Paper – 4
Section A
1. The side of a square sheet is increasing at the rate of 4 cm/min. at what rate is the area increasing,
when the side is 5cm long?
2. Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (-5, 7).
3. If A = [
−𝟏 𝟒
𝟏 𝟑
] and BT = [
𝟎 𝟑
𝟏 𝟐
] , then find (7A + 5B)T .
4. Evaluate :- ∫
𝒍𝒐𝒈 𝒙 𝟐
𝒙
dx.
5. From the differential equation corresponding to y2 = (x – a)3 , by eliminating a.
6. Evaluate :- ∫ 𝒔𝒊𝒏 𝟓
𝒙 𝒄𝒐𝒔 𝟓
𝒙 𝒅𝒙
𝟏
−𝟏
.
Section - B
7. If x = a(cos  +  sin  ) , y – a (sin  -  cos  ) , then prove that
𝒅 𝟐
𝒚
𝒅𝒙 𝟐 =
𝒔𝒆𝒄 𝟑

𝒂
.
OR
If y = tan x + sec x, then prove that
𝒅 𝟐
𝒚
𝒅𝒙 𝟐 =
𝒄𝒐𝒔 𝒙
( 𝟏−𝒔𝒊𝒏 𝒙) 𝟐 .
8. A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into a shape of a
square and the other into a shape of an equilateral triangle. What should be the lengths of the two
pieces so that the sum of the areas of the square and the triangle is minimum?
9. Evaluate :- ∫
𝒔𝒊𝒏 𝟐𝒙 𝒄𝒐𝒔 𝟐𝒙
√ 𝟗− 𝒄𝒐𝒔 𝟒 𝟐𝒙
dx
10. If the function f(x) = {
𝟑𝒂𝒙 + 𝒃 𝒊𝒇 𝒙 > 1
𝟏𝟏 𝒊𝒇 𝒙 = 𝟏
𝟓𝒂𝒙 − 𝟐𝒃 𝒊𝒇 𝒙 < 1
is continuous at x = 1, find the values of a and b.
11. A driver starts a car from a point P at time t = 0 seconds and stops at point Q. the distance x (in
meters) covered by it in t seconds is given by x = t2 ( 2 -
𝒕
𝟑
) . find the time taken by it to reach Q and
also find the distance between P and Q. the driver has stopped the car at the time Q on the roadside
to take the call on his mobile phone. Has he done right in doing so?
12. Differentiate cos xx w.r.t. x.
13. Evaluate :- ∫
𝟏
𝟑+𝟐 𝒄𝒐𝒔 𝒙
dx
14. If the vertices A, B and C of a ABC are (1, 2, 3), (-1, 0, 0) , (0, 1, 2) respectively, then find ABC using
vectors.
15. Solve :- 4 sin-1 x =  - cos-1 x. OR Solve :- sin-1 x =
𝝅
𝟔
+ cos-1 x
16. Evaluate :- ∫ ( 𝟐𝒍𝒐𝒈 𝒔𝒊𝒏𝒙 − 𝒍𝒐𝒈 𝒔𝒊𝒏 𝟐𝒙) 𝒅𝒙
𝝅
𝟐
𝟎
. OR Evaluate :- ∫ 𝒍𝒐𝒈 (𝟏 + 𝒕𝒂𝒏 𝒙) 𝒅𝒙
𝝅
𝟒
𝟎

11. 17. Find the area of the region in the first quadrant enclosed by the x – axis, line x = √ 𝟑 y and the circle
x2 + y2 = 4.
18. Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls.
Find the probability that
a. Both are red
b. First is black and second is red
c. One of them is black and other is red.
OR
A bag contains 3 white, 5 black and 2 red balls. Three balls are drawn from it. Find the probability
that
a. One is white, one is black and one is red.
b. Balls drawn are white, black and red respectively.
c. One is red and two are white
19. Find the particular solution of the differential equation (x3 + x2 + x + 1)
𝒅𝒚
𝒅𝒙
= 2x2 + x given that y = 1
when x = 0.
Section - C
20. A brick manufacturer has two depots, P and Q with stocks of 30000 and 20000 bricks respectively. He
receives orders from three builders A, B and C for 15000, 20000 and 15000 bricks respectively. The
cost in Rs. For transporting 1000 bricks to the builders from the depots is given in the following
tables.
To
From
A B C
Q
40 20 30
P
20 60 40
How should the manufacturer fulfil the orders so as to keep the cost of transportation minimum?
Formulate the above L.P.P. mathematically and then solve it graphically.
21. Find the equation of the plane that contains the lines 𝒓⃗ = ( 𝒊̂ + 𝒋̂) +  (𝒊̂ − 𝒋̂ + 𝟐𝒌̂) and 𝒓⃗ =
( 𝒊̂ + 𝒋̂) +  (𝒊̂ + 𝟐𝒋̂ − 𝒌̂) .
22. If A = [
𝟐 −𝟑 𝟓
𝟑 𝟐 −𝟒
𝟏 𝟏 −𝟐
] , then find A-1 . using A-1 , solve the system of equations:

12. 2x - 3y + 5z = 11, 3x + 2y - 4z = -5, x + y – 2z = -3.
23. Let L be the set of all lines in the XY plane and R be the relation on L defined as R = {(L1 , L2 ) : L1 is
parallel to L2 }. Show that R is an equivalence relation. Find the set of all the lines related to the line y
= 2x + 4 (i.e., find the equivalence class of the line y = 2x + 4).
OR
Show that the given relation R is defined on the set A = {x  Z : 0  x  12}, given by
R = {(a, b) : | 𝒂 − 𝒃| is multiple of 4}, is an equivalence relation. Write the set of all elements related
to 1.
24. Find the equation of the plane passing through the point (3, 0, -1) and parallel to the lines
𝒙−𝟑
𝟏
=
𝒚−𝟏
𝟐
=
𝒛
𝟑
and 𝒓⃗ = (−𝒊̂ + 𝟒𝒋̂ − 𝟐𝒌̂) +  (𝟐𝒊̂ − 𝟑𝒋̂ + 𝟒𝒌̂).
25. Suppose 5 men out of 100 and 25 women out of 1000 are orators. An orators is chosen at random.
Find the probability of a male person being selected, assuming that there are equal number of men
and women.
OR
Suppose 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random.
What is the probability of this person being male? Assume that there are equal number of males and
females.
26. Without expanding, prove that |
𝒂 + 𝒃𝒙 𝒄 + 𝒅𝒙 𝒑 + 𝒒𝒙
𝒂𝒙 + 𝒃 𝒄𝒙 + 𝒅 𝒑𝒙 + 𝒒
𝒖 𝒗 𝒘
| = ( 𝟏 − 𝒙 𝟐)|
𝒂 𝒄 𝒑
𝒃 𝒅 𝒒
𝒖 𝒗 𝒘
| .
ANSWER :
1. 40 cm2 / min
2. Scalar components of 𝑨𝑩⃗⃗⃗⃗⃗⃗ are -7 ,6 and vector components of 𝑨𝑩⃗⃗⃗⃗⃗⃗ are -7𝒊̂ ,6𝒋̂ .
3. [
−𝟕 𝟐𝟐
𝟑𝟑 𝟑𝟏
] 4.
𝟏
𝟒
(logx2 )2 + C
5. 2
𝒅𝒚
𝒅𝒙
= 3𝒚
𝟏
𝟑 6. 0
8.
𝟖𝟎√𝟑
𝟗+𝟒√𝟑
m and
𝟏𝟖𝟎
𝟗+𝟒√𝟑
m 9.
𝟏
𝟒
sin-1 (
𝒄𝒐𝒔 𝟐
𝟐𝒙
𝟑
) + C
10. a = 3, b= 2
11. 4s,
𝟑𝟐
𝟑
m . Yes , he has done right . One should not the mobile phone while driving

13. 12. – sin xx xx ( 1 + log x ) 13.
𝟐
√𝟓
𝐭𝐚𝐧−𝟏
[
𝒕𝒂𝒏
𝒙
𝟐
√𝟓
] + C
15.
√𝟑
𝟐
OR ½ 16.
𝝅
𝟒
log ¼ OR
𝝅
𝟖
log 2
17.
𝝅
𝟑
sq. units
18. (i)
𝟏𝟔
𝟖𝟏
, (ii)
𝟐𝟎
𝟖𝟏
(iii)
𝟒𝟎
𝟖𝟏
OR (i) ¼ (ii)
𝟏
𝟐𝟒
(iii)
𝟏
𝟐𝟎
19. y =
𝟏
𝟐
log | 𝒙 + 𝟏|+ ¾ log |𝒙 𝟐
+ 𝟏|- ½ tan-1 x + 1
20. 15000, 0 and 15000 bricks should be transported from P to A,B and C respectively ; 0, 20000 and 0
bricks should be transported from Q to A, B and C respectively and minimum cost of transportation
= Rs. 1300.
21. –x + y + z = 0 22. A-1 = [
𝟎 𝟏 −𝟐
−𝟐 𝟗 −𝟐𝟑
−𝟏 𝟓 −𝟏𝟑
] ; x = 1 , y = 2, z = 3
23. { y = 2x + c : c  R } OR { 1,5,9} 24. 17x +2y – 7z – 58 = 0
25.
𝟐
𝟑
𝑶𝑹
𝟐𝟎
𝟐𝟏

14. Unsolved Practice Paper – 5
Section A
1. If A = B are square matrices of the same order such that | 𝑨| = 6 and AB = I, then write the value of | 𝑩|.
2. Evaluate the integral: ∫ 𝒙 𝒔𝒊𝒏 𝒙
𝝅
𝟐
𝟎
dx.
3. Let A = {1,2,3,) . The reaction R on set A is defined as R = { ( 1,1) , (1,2)} , check whether R is transitive or
not.
4. Evaluate : sin [
𝝅
𝟑
− 𝐬𝐢𝐧−𝟏
(−
𝟏
𝟐
)] .
5. Find the position vector of a point R which divides the line joining two point P and Q , whose position
vector are 𝒊̂ + 2𝒋̂ – 𝒌̂ and - 𝒊̂ + 𝒋̂ + 𝒌̂ respectively ,in the ratio 2: 1 externally.
6. Evaluate :
𝟏
𝒙− √ 𝒙
dx
Section B
7. Test whether the relation R on z define by R = { ( a ,b), : | 𝒂 − 𝒃| < 5 } is reflexive , symmetric and
transitive .
OR
Let  : R  R be the signum function defined as  (x) = ∫
𝟏 𝒊𝒇 𝒙 > 0
𝟎 𝒊𝒇 𝒙 = 𝟎
−𝟏 𝒊𝒇 𝒙 < 0
and g : R  R be the
greatest integer function given by, g (x) = x . Do  o g and g o  coincide in ( 0, 1 ] ?
8. Find the area of parallelogram whose adjacent sides are determined by the vector 𝒂⃗⃗ = 𝒊̂ - 𝒋̂ + 3𝒌̂ and
𝒃⃗⃗ = 2𝒊̂ -7𝒋̂ + 𝒌̂
9. Find the absolute maximum and absolute minimum values of  (x) = 12𝒙
𝟒
𝟑 – 6𝒙
𝟏
𝟑, x  [ -1 ,1].
OR
Find the point of local maxima and local minima, if any , of  (x) = x √ 𝟏 − 𝒙 , 0,< x < 1 using first
derivative test .Also , find the local maximum and local minimum values, as the case may be.
10. Three groups of children contain 3 girls and 1 boy ; 2 girls and boys ; 1 girl and 3 boys respectively . One
child is selected at random from each group . Find the chance that the three selected comprise one girl
and two boys. These three selected will participate in a debate competition on ‘CLEANLINESS”. what
are your views on cleanliness?
11. If y = sin (sin x) , then proved that
𝒅 𝟐
𝒚
𝒅𝒙 𝟐 + tan x
𝒅𝒚
𝒅𝒙
+ cos2 xy = 0.
12. Find the intervals on which  (x) = tan-1 ( sin x + cos x ) x  ( 0,
𝝅
𝟒
) is
(i) strictly increasing or strictly decreasing (ii) increasing or decreasing

15. OR
Find the point on the curve y = x3 – 11x + 5 at which the equation of tangent is y = x - 11.
13. Prove that cos [𝟐 𝐭𝐚𝐧−𝟏 𝟏
𝟕
] = sin [𝟒 𝐭𝐚𝐧−𝟏 𝟏
𝟑
] .
14. Differentiate the following w.r.t. x: cos-1 (
𝒙+ √ 𝟏− 𝒙 𝟐
√𝟐
) .
15. Evaluate the integral :  x sin-1 x dx
16. Show that the function  (x) = | 𝒙 − 𝟏|is not differentiable at x = 1.
17. Find the particular solution of the differential equation ( x – y ) ( dx – dy ) , given that y = 1 when x = 0 .
18. Evaluate : ∫ (𝒔𝒊𝒏 | 𝒙| + 𝒄𝒐𝒔 | 𝒙|
𝝅
𝟐
−
𝝅
𝟐
dx
OR
Evaluate: ∫
𝒔𝒊𝒏𝒙+𝒄𝒐𝒔 𝒙
√𝒔𝒊𝒏 𝟐𝒙
𝝅
𝟑
𝝅
𝟔
dx
19. Evaluate : ∫
𝟏
√ 𝒔𝒊𝒏 𝟑 𝒙 𝒔𝒊𝒏(𝒙+ 𝜶)
dx
Section C
20. Find the equation of the plane which passes through the line of intersection of the planes x + y + z =
and 2x + 3y + 4z = 5 and parallel to x – axis .
21. Prove that |
𝒂 𝒂 + 𝒄 𝒂 − 𝒃
𝒃 − 𝒄 𝒃 𝒃 + 𝒂
𝒄 + 𝒃 𝒄 − 𝒂 𝒄
| = ( a+ b + c ) ( a2 + b2 +c2 ).
22. A fruit grower can use two types of fertilisers in his garden, brand P and Q. The amounts (in kg) of
nitrogen , phosphoricacid , potash and chlorine in a bag of each brand are given in the table . Tests
indicate that the garden need at least 240 kg of phosphoricacid , at least 270 kg of potash and at most
310 kg of chlorine . If the grower wants to minimize wants to minimise the amount of nitrogen added
to the garden , how many bags of each brand should be used ? What is the minimum amount of
nitrogen added in the garden ? Formulate the above L.P.P. mathematically and solve it graphically
Brans P Brand Q
Nitrogen 3 3.5
Phosphoricacid 1 2
Potash 3 1.5
Chlorine 1.5 2
23. Sketch the graph of the curve y = | 𝑥 + 3| and evaluate | 𝑥 − 3|dx .

16. OR
Find the area bounded by the curve y = x  x  , x –axis and the ordinates x = 1 and x = -1.
24. Find the distance of the point ( -2, 3, -4 ) from the line
𝒙+𝟐
𝟑
=
𝟐𝒚+𝟑
𝟒
=
𝟑𝒛+𝟒
𝟓
measured parallel to the
plan 4x + 12 y – 3z + 1 = 0.
25. Suppose a girl throws a die . If she gets a 5 or 6, she tosses a coin three times and notes the number of
heads . If she gets a 1, 2, 3, or 4,: she tosses a coin once and notes whether a head or tail is obtained . If
she obtains exactly one head , what is the probability that she threw 1,2,3 or 4 with the die ?
OR
An insurance company insures 3000 scooters ,4000 cars and 5000 trucks. The probability of an accident
involving a scooters , a car and a truck is 0.02 .0.03, and 0.04 respectively . one of the insured meet
vehicles with an accident . Find the probability that it is a truck .
26. Find the inverse of [
𝟎 𝟐 −𝟏
𝟎 𝟑 𝟏
𝟑 𝟐 𝟏
] , using Elementary Row Transformation method .
ANSWER :-
1.
𝟏
𝟔
2. 1 3. Yes 4. 1
5. -3𝒊̂ + 0𝒋̂ + 3𝒌̂ 6. 2 log |√ 𝒙 − 𝟏|+ C
7. R is reflexive, symmetric but not transitive. OR NO
8. 15√ 𝟐 sq. units
9. Absolute maximum value of  (x) is 18 which occurs at x = -1 and absolute minimum value of  (x) is
−
𝟗
𝟏𝟒
which occurs at x =
𝟏
𝟖
. OR
f(x) has local maximum at x =
𝟐
𝟑
and local maximum values is f(
𝟐
𝟑
) =
𝟐√𝟑
𝟗
.
10.
𝟏𝟑
𝟑𝟐
. Everyone must acquire a habit of cleanliness so keep the surrounding clean.
12. (i) (x) is strictly increasing on ( 0,
𝝅
𝟒
) , (ii) f(x) is increasing on ( 0,
𝝅
𝟒
) .
OR ( 2,- 9 ) and ( -4 ,-15 )
14.
𝟏
√ 𝟏− 𝒙 𝟐
15. -
𝐬𝐢𝐧−𝟏
𝒙 (𝟏−𝟐𝒙 𝟐
)
𝟒
+
𝒙 √ 𝟏− 𝒙 𝟐
𝟒
+ C
17. log x- y  x +y + 1 18. 4. OR sin-1 (
√𝟑
𝟐
−
𝟏
𝟐
) - sin-1 (
𝟏
𝟐
−
√𝟑
𝟐
)
19. -
𝟐
𝒔𝒊𝒏 𝜶
√ 𝒄𝒐𝒔 𝜶+ 𝒔𝒊𝒏 𝜶 𝒄𝒐𝒕 𝜶 + C 20. –y -2z + 3 = 0
22. Number of bags of brand P fertilizer = 40 , number of page of brand Q fertilizes = 100
and minimum amount of nitrogen added in the garden = 470 kg.

17. 23. 9 OR
𝟐
𝟑
sq. units 24.
𝟏𝟕
𝟐
units 25.
𝟖
𝟏𝟏
OR
𝟏𝟎
𝟏𝟗
26.
[
𝟐
𝟑
−
𝟐
𝟑
𝟏
𝟑
𝟏
𝟓
𝟏
𝟓
𝟎
−
𝟑
𝟓
𝟐
𝟓
𝟎]
.

18. Unsolved practice Paper - 6
Section A
1. Let 𝒂⃗⃗ = 𝒊̂ + 𝟐𝒋̂ 𝒂𝒏𝒅 𝒃⃗⃗ = 𝟐𝒊̂ + 𝒋̂ . 𝒊𝒔 | 𝒂⃗⃗ | = |𝒃⃗⃗ | ? Are the vectors 𝒂⃗⃗ and 𝒃⃗⃗ equal ?
2. Evaluate : ∫
𝟐
𝟏+𝒄𝒐𝒔 𝟐𝒙
dx .
3. It is given that at x = 1 , the function  (x) = x4 – 62 x2 + ax + 9 attains its maximum value on the
interval [ 0,2 ] . find the value of a.
4. Evaluate : ∫( 𝟏 − 𝒙)√ 𝒙 dx .
5. Determine the order and degree ( if defined ) of the following differential equation :
y = xp + √𝒙 𝟐 𝒑 𝟐 + 𝟒 ; p =
𝒅𝒚
𝒅𝒙
.
6. If A is an invertible matrix of order 3  3 and | 𝑨| = 7 , then find adj ( adj A ) .
Section B
7. A man 2 m tall is curious to see his shadow increasing . He walks at a uniform speed of 5km/h
away from a lamppost 6 metres high . Find the rate at which the length of his shadow increases
when he is 1 metre away from the pole . what value is highlighted in this question ?
8. If x = a (  - sin  ) and y = a ( 1 + cos  ) , then prove that
𝒅 𝟐
𝒚
𝒅𝒙 𝟐 =
𝟏
𝒂 (𝟏−𝒄𝒐𝒔  ) 𝟐
OR
If x = a cos  and y = b sin  , then prove that
𝒅 𝟐
𝒚
𝒅𝒙 𝟐 = -
𝒃 𝟒
𝒂 𝟐 𝒚 𝟑 .
9. Show that of all the rectangles inscribed in a given circle , the square has maximum perimeter .
10. Evaluate : ∫
𝟏
√−𝟐𝒙 𝟐 + 𝟑𝒙+𝟏
dx.
11. For what value of is the function defined by f(x) = {
 (𝒙 𝟐
− 𝟐𝒙) 𝒊𝒇 𝒙 ≤ 𝟎
𝟒𝒙 + 𝟏 𝒊𝒇 𝒙 > 0
Continuous at x = 0?
What about continuity at x = 1 ?
12. Evaluate : ∫
𝟏
𝟑−𝟓 𝒄𝒐𝒔 𝟏𝟐 𝒙
dx.
13. Differentiate 𝒙 𝒙 𝒙
w.r.t. x.
14. Find the area between the curves y = x and y = x2 .
15. Find x and if ( 2𝒊̂ + 6𝒋̂ + 27𝒌̂ )  (𝒊̂ +  𝒋̂ + 𝒌̂ ) = 0 .
16. Solve : 2 tan-1 x = sin-1 (
𝟐𝒂
𝟏+ 𝒂 𝟐)+ sin-1 (
𝟐𝒃
𝟏+ 𝒃 𝟐) .
OR
Solve : tan-1 (2 + x) + tan-1 (2 – x) = tan2-1 𝟐
𝟑

19. 17. Find the particular solution of the deferential equation cos (
𝒅𝒚
𝒅𝒙
) = a , given that y = 1 when x = 0.
18. Evaluate the integral : ∫ 𝒍𝒐𝒈 [
𝟏
𝒙
− 𝟏]
𝟏
𝟎
dx
OR
Evaluate : ∫ 𝒍𝒐𝒈 [
𝟒+𝟑 𝒔𝒊𝒏 𝒙
𝟒+𝟑 𝒄𝒐𝒔 𝒙
]
𝝅
𝟐
𝟎
dx
19. Two cards are drawn from a pack of 52 cards . What is the probability of getting
(i) first card red and second card king ? (ii) a red card and a king card ?
OR
Two cards are drawn from a pack of 52 cards . Find the probability that
(i) First is heart card and second is red card .
(ii) One is hear card and other is red card.
Section C
20. Prove that the lines
𝒙+𝟏
−𝟏
=
𝒚−𝟐
𝟐
=
𝒛−𝟓
𝟓
𝒂𝒏𝒅
𝒙+𝟑
−𝟑
=
𝒚−𝟏
𝟏
=
𝒛−𝟓
𝟓
are coplanar . Also find the plane
containing these two lines.
21. An oil company has two deport , P and Q with capacities of 7000 liters and 4000 liters respectively .
The company is to supply oil to three petrol pumps D, E and F whose requirements are 4500 litres,
3000 litres and 3500 litres respectively . The distance ( in km ) between the depots and petrol pumps
is given in the following table:
Distance( in km)
To From P Q
D 7 3
E 6 4
F 3 2
Accounting that the transportation cost per km is Rs. 2 per litre, how should the delivery be
scheduled in order that the transportation cost is minimum? Formulate the above L.P.P.,
mathematically and then solve it graphically.
22. Let S be the set of all points in a plane and R be a relation on S, defined by
R = {(P, Q): Distance between P and Q is less than 4.5 units}.
OR
Show that the relation R defined on the set A = {1, 2, 3, 4, 5}, given by R = {(a, b): | 𝒂 − 𝒃| is even}, is
an equivalence relation. Show that all the elements of {1, 3, 5} are related so each other all the
elements of {2, 4} are related to each other, but no element of {1, 3, 5} is related to element of {2, 4}.
23. Solve given system of equation by using matrix method:

20. 𝟐
𝒙
+
𝟑
𝒚
+
𝟏𝟎
𝒛
= 𝟒 ,
𝟒
𝒙
−
𝟔
𝒚
+
𝟓
𝒛
= 𝟏,
𝟔
𝒙
+
𝟗
𝒚
−
𝟐𝟎
𝒛
= 𝟐 ; x, y, z  0.
24. In a class 5% of the boys and 10% of the girls have an IQ of more than 150. In this class, 60% of the
students are boys. If a student is selected at random and found to have an IQ of more than 150. Find
the probability that the student is a boy.
OR
In a certain college, 4% of boys and 1% of girls are taller than 1.75 meters. Furthermore, 60% of the
students in the college are girls. A student is selected at random and is found to be taller than 1.75
meters. Find the probability that the selected student is girl.
25. Prove , using properties if determinants: |
𝒂 𝒂 + 𝒃 𝒂 + 𝒃 + 𝒄
𝟐𝒂 𝟑𝒂 + 𝟐𝒃 𝟒𝒂 + 𝟑𝒃 + 𝟐𝒄
𝟑𝒂 𝟔𝒂 + 𝟑𝒃 𝟏𝟎𝒂 + 𝟔𝒃+ 𝟑𝒄
| = a3 .
26. Show that the plane whose vectorequation is 𝒓⃗ .(𝒊̂ + 𝟐𝒋̂− 𝒌̂ ) = 6 contains the line whose vectors
equation is 𝒓⃗ .( 𝟒𝒊̂ + 𝟒𝒋̂) + (𝟐𝒊 + 𝒋̂ + 𝟒𝒌̂ ).
ANSWER :-
1. | 𝒂⃗⃗ | = |𝒃⃗⃗ | 𝒃𝒖𝒕 𝒂⃗⃗ ≠ 𝒃⃗⃗ . 2. tan x + C 3. 120
4.
𝟐𝒙
𝟑
𝟐
𝟑
-
𝟐𝒙
𝟓
𝟐
𝟓
+ C 5. Order is 1 and degree is 1. 6. 7A
7. 2.5 km/h. Curiosity of the man is highlighted in this question. 10.
𝟏
√𝟐
𝐬𝐢𝐧−𝟏
[
𝟒𝒙−𝟑
√𝟏𝟕
] + 𝑪
11. f is not continuous at x = 0, for any real value of  and f is continuous at x = 1, for every real value of
 .
12.
𝟏
𝟐√𝟔
𝒍𝒐𝒈 |
√𝟑 𝒕𝒂𝒏 𝒙− √𝟐
√𝟑 𝒕𝒂𝒏 𝒙+ √𝟐
| + C 13. xx 𝒙 𝒙 𝒙
[
𝟏
𝒙
+ 𝒍𝒐𝒈 𝒙 (𝟏 + 𝒍𝒐𝒈 𝒙)] 14.
𝟏
𝟔
sq. units
15.  = 3 and  =
𝟐𝟕
𝟐
16.
𝒂−𝒃
𝟏−𝒂𝒃
𝑶𝑹 ± 𝟑 17. Y = (cos-1 a) x + 1
18. 0 OR 0 19. (i)
𝟏
𝟐𝟔
, ( 𝒊𝒊)
𝟏
𝟏𝟑
OR (i)
𝟐𝟓
𝟐𝟎𝟒
, (ii)
𝟐𝟓
𝟏𝟎𝟐
20. x – 2y + z = 0
21. 500, 3000 and 3500 litres of oil should be transported from P to D, E and F respectively; 4000, 0 and 0
Litres of oil should be transported from Q to D, E and F respectively and minimum cost of
transportation = Rs. 88000.
23. x = 2, y = 3, z = 5 24.
𝟑
𝟏𝟏
OR
𝟑
𝟕
.

21. Unsolved practice paper – 7
Section A
1. Evaluate: ∫ 𝒆 𝒔𝒊𝒏 𝟐
𝒙
sin 2x dx.
2. If the operation * is defined on Q as a * b = 2a + b – ab ; for all a, b  Q , find the value of 3 * 4 .
3. Show that the point (1,0) , (6,0) , (0,0) are collinear .
4. Fine the direction cosines of the vector joining the point A (1,2 – 3 ) and B (-1,-2 , 1 ) , directed from A
to B.
5. Let A = {1,2.3} . the relation R on set A is defined as R = { (1,1)} . Check whether R is reflexive or not.
6. Evaluate : ∫
𝒔𝒆𝒄 𝟐
𝒙
𝒄𝒐𝒔𝒆𝒄 𝟐 𝒙
dx .
Section B
7. If A and B commuted , than prove that ABn = Bn A , for all n  N.
8. Ten eggs are drawn successively with replacement from a lot containing 10% rotten eggs. What is the
probability that there is at least one rotten egg?
OR
Suppose that 90% of people are right –handed . what is the probability that at most 8 of a random
sample that 90% people are right – handed ?
9. If y = 𝒆𝐭𝐚𝐧−𝟏
𝒙
, then prove that ( 1+ x2 ) y2 + (2x- 1) y1 = 0 .
10. Differentiate the following w.r.t. x. cos-1 (x√ 𝟏 − 𝒙 + √ 𝒙 √𝟏 − 𝒙 𝟐 )
OR
Differentiate the following w.r.t.x: cos-1 [
𝟑𝒙+𝟒√ 𝟏− 𝒙 𝟐
𝟓
]
11. Prove that the relation R on Z , defined by (a,b)  R  a-b is divisible by 5, is an equivalence relation
on Z.
OR
Show that the signum function  : R  R , given by  (x) = {
𝟏 𝒊𝒇 𝒙 > 0
𝟎 𝒊𝒇 𝒙 = 𝟎
−𝟏 𝒊𝒇 𝒙 < 0
is neither one – one or
not.
12. Using differentials ,find the approximate value of ( 0.037)1/2 .
13. Evaluate : ∫
𝒙 𝟐
+ 𝟏
𝒙 𝟒 + 𝟏
dx
14. Prove that |
𝟏 𝟏 𝟏
𝑨 𝑩 𝑪
𝒂 𝟑
𝒃 𝟑
𝒄 𝟑
| = (a – b ) ( b- c ) (c – a) ( a + b+ c ).

22. 15. Find the particular solution of the differential equation :- x
𝒅𝒚
𝒅𝒙
–y + x sin (
𝒚
𝒙
) = 0 given that when x = 2 ,
y = .
16. If 𝒂⃗⃗ ≠ 𝟎⃗⃗ , 𝒂⃗⃗ . 𝒃⃗⃗ = 𝒂⃗⃗ . 𝒄⃗ 𝒂𝒏𝒅 𝒂⃗⃗ × 𝒃⃗⃗ = 𝒂⃗⃗ × 𝒄⃗ , then show that 𝒃⃗⃗ = 𝒄⃗ .
17. Solve : sin-1 (
𝒙
𝟐
)+ cos-1 x =
𝝅
𝟔
.
OR
Solve : cos -1 (
𝒙 𝟐
+ 𝟏
𝒙 𝟒+ 𝟏
) +
𝟏
𝟐
𝐭𝐚𝐧−𝟏
(
𝟐𝒙
𝟏− 𝒙 𝟐) =
𝟐𝝅
𝟑
.
18. A book store has 20 mathematics books, 15 physics book and 12 chemistry books. Their selling
prices are Rs. 300 , Rs, 320 and Rs.340 each respectively . Find the total amount the store will
receive from selling all the items . Do you thing that we must books our best friend ? Why ?
19. Evaluate : ∫
𝟏
𝒔𝒊𝒏 ( 𝒙−𝒂) 𝒄𝒐𝒔 (𝒙−𝒃)
dx
Section C
20. If the lengths of three sides of a trapezium other than base are equal to 10 cm, then find the area of
trapezium when it is maximum.
21. Find the coordinates of the point where the line though (5,1,6) and (3, 4, 1 ) crosser the x z – plane .
22. Evaluate : ∫
𝒄𝒐𝒔 𝟐
𝒙
𝒄𝒐𝒔 𝟐 𝒙+𝟒 𝒔𝒊𝒏 𝟐 𝒙
𝝅
𝟐
𝟎
dx.
23. A doctor is to visit a patient. From past experience , It is known that the probabilities that he will
come by train , bus scooter or by other means of transport are
𝟑
𝟏𝟎
,
𝟏
𝟓
,
𝟏
𝟏𝟎
𝒂𝒏𝒅
𝟐
𝟓
respectively . The
probabilities that he will be late are
𝟏
𝟒
,
𝟏
𝟑
,
𝟏
𝟏𝟐
if he comes by train , bus and scooter respectively . But
if he comes by other means of transport, then he will not be late . When he arrives , he is late. What
is the probability that he comes by train?
OR
In an examination , an examinee either guesses or copies or knows the answer of MCQs with four
choices . The probability that he makes a guess is
𝟏
𝟑
, and the probability that he copies answer is
𝟏
𝟔
The probability that his answer is correct , given that copied it , is
𝟏
𝟖
, . Find the probability that he
copies the answer to question , given that he correctly answered it.
24. Show that the point (0,-1,0) , (1,1,1 ) , (3,3,0) and (0,1,3) are coplanar . Also , find the plane
containing them.
25. Let O,A and O B be the intercept of the ellipse 9x2 +y2 = 36 in the first quadrant such that OA = 2 and
O B = 6 , find the area between the arc AB and the chord A B.
OR

23. Find the area bounded by the curves y = 6x – x2 and y = x2 – 2x2 .
26. A toy company manufactures two types of gift items , A and B . Market tests and the available
resources have indicated that the combined production level should not exceed 1200 gift item per
week and the demand for gift item of type B is at most half of for gift items type A. Further , the
production level of gift items A can exceed three times the production of gift items of other type by
at most 600 units the . If the company makes profit of Rs. 12 and 16 per item respectively on gift
items A and B , how many of each should be produced weekly in order to maximise the profit ?
Formulate The above L.P.P mathematically and then solve it graphically .
ANSWER :-
1. 𝒆 𝒔𝒊𝒏 𝟐
𝒙
+ 𝑪 2. -2 4. -
𝟏
𝟑
,
𝟐
𝟑
,
𝟐
𝟑
5. No
6. tan x – x + C 8. 1 - (
𝟗
𝟏𝟎
)
𝟏𝟎
OR 1 -
𝟏𝟗
𝟏𝟎
(
𝟗
𝟏𝟎
)
𝟗
10. −
𝟏
√ 𝟏− 𝒙 𝟐
−
𝟏
𝟐√ 𝒙 √𝟏−𝒙
OR −
𝟏
√ 𝟏− 𝒙 𝟐
12. 0.1925
13.
𝟏
√𝟐
𝐭𝐚𝐧−𝟏
(
𝒙 𝟐
− 𝟏
√𝟐𝒙
) + C 15. cosec (
𝒚
𝒙
) − 𝒄𝒐𝒕(
𝒚
𝒙
) =
𝟐
𝒙
17. 1 OR No solution
18. Rs. 14880. Yes, we must make books our best friends because books helps us increase our
knowledge.
19. sec (b – a) log | 𝒔𝒊𝒏 ( 𝒙 − 𝒂) 𝒔𝒆𝒄 (𝒙 − 𝒃)| + C 20. 75 √ 𝟑 cm2 21. (
𝟏𝟕
𝟑
, 𝟎,
𝟐𝟑
𝟑
)
22.
𝝅
𝟔
23.
𝟏
𝟐
OR
𝟏
𝟐𝟗
24. 4x – 3y + 2z = 3
25. (3 - 6) sq. units OR
𝟔𝟒
𝟑
sq. units
26. Number of gifts items of type A = 800, number of gift items of type B = 400 and maximium profit =
Rs. 16000.

24. Unsolved Practice Paper – 8
Section – A
1. Evaluate : ∫
𝒆 𝟐𝒙
− 𝒆−𝟐𝒙
𝒆 𝟐𝒙+ 𝒆−𝟐𝒙 dx .
2. If A is a square matrix of order 3  3 such that | 𝐴|= 3 , then find | 𝑨 (𝒂𝒅𝒋 𝑨)|.
3. Evaluate : ∫
𝟐
𝟏−𝒄𝒐𝒔 𝟐𝒙
dx
4. Let A = { 1,2,3 } . The relation R on set A is as R = {(1,1), (2,3)}. Check whether R is transitive or not.
5. Find the principal value of cos-1 (
√𝟑
𝟐
).
6. Find are vector 𝒓⃗ , prove that 𝒓⃗ = ( 𝒓⃗ . 𝒊̂) 𝒊̂ + ( 𝒓⃗ . 𝒋̂) 𝒋̂ + (𝒓⃗ . 𝒌̂)𝒌̂ .
Section B
7. For what value of a the vectors 2 𝒊̂ – 3𝒋̂ + 4𝒌̂ and a𝒊̂ + 6𝒋̂ – 8𝒌̂ are collinear ?
8. Let R be a relation on set A of ordered pairs of positive integers defined by
(a,b) R (c,d )  ad = bc, for all (a,b ) , (c,d)  A  A. Show that R is an equivalence relation on A  A.
OR
Let  : R  R be defined as  (x) = 10x + 7, find the function g: RR such that g o  = o g = IR.
9. If the value of derivative of tan-1 (a + bx ) at x = 0 is 1, then prove that 1 + a2 = b.
10. A and b appeared for an interview for two vacancies. The probability of A’s selection is
𝟏
𝟓
and that of
B’s selection is 1/3. Find the probability that (i) only one of them will be selected (ii) at least one will
by selected .
Name two qualities that a person should possess while appearing for an interview .
OR
A speaks trust in 80% cases and 90% cases . Find the probability that
(i) they contradict each other in stating the same fact.
(ii) they favour each other in stating the same fact .
Who is more trustworthy : A or B?
11. Find the absolute maximum and absolute minimum values of  (x) = (3x2 – x2 )
𝟏
𝟑
,x  [-1,1 ].
12. Solved : tan-1 (
𝟏−𝒙
𝟏+𝒙
) =
𝟏
𝟐
𝐭𝐚𝐧−𝟏
𝒙; x > 0.
OR
Prove that : tan[
𝝅
𝟒
+
𝟏
𝟐
𝐜𝐨𝐬−𝟏
(
𝒂
𝒃
)] + 𝒕𝒂𝒏[
𝝅
𝟒
−
𝟏
𝟐
𝐜𝐨𝐬−𝟏
(
𝒂
𝒃
)] =
𝟐𝒂
𝒃
13. Find the intervals on which  (x) =
𝟑
𝟏𝟎
𝒙 𝟒
−
𝟒
𝟓
𝒙 𝟑
− 𝟑𝒙 𝟐
+
𝟑𝟔
𝟓
𝒙 + 𝟏𝟏 is
a. strictly increasing or strictly decreasing ; b. increasing or decreasing .

25. 14. Evaluate:- ∫ √
𝟏− √ 𝒙
𝟏+ √ 𝒙
dx
15. Evaluate the integral : ∫ 𝐬𝐢𝐧−𝟏
(
𝟐𝒙
𝟏+ 𝒙 𝟐 )
𝟏
𝟎
dx
OR
Evaluate the integral : |𝐬𝐢𝐧 𝐱 – 𝐜𝐨𝐬 𝐱| dx
16. Show that the function  defined by  (x) = {
𝟑𝒙 − 𝟐 𝒊𝒇 𝟎 < 𝑥 ≤ 1
𝟐𝒙 𝟐
− 𝒙 𝒊𝒇 𝟏 < 𝑥 ≤ 2
𝟓𝒙 − 𝟒 𝒊𝒇 𝒙 > 2
is not differentiable at x=2.
17. Evaluate :  x2 tan-1 x dx .
18. Find the general solution of the differential equation sin-1 (
𝒅𝒚
𝒅𝒙
) = x + y .
19. Differentiate w.r.t. x: sin [𝟐 𝐭𝐚𝐧−𝟏
√
𝟏−𝒙
𝟏+𝒙
] .
Section C
20. Prove , using properties of determinants : |
( 𝒃 + 𝒄) 𝟐
𝒃𝒂 𝒄𝒂
𝒂𝒃 ( 𝒄 + 𝒂) 𝟐
𝒄𝒃
𝒂𝒄 𝒃𝒄 ( 𝒂 + 𝒃) 𝟐
| = 2abc(a + b + c)3 .
21. A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of
food P contains 12unit of calcium ,4 units of iron, 6 units of cholesterol and 6 units of vitamin A.
Each packet of the same quantity of food Q contains 3 units of calcium ,20 units of iron,4 units of
cholesterol and 3 unit of vitamin A. The diet require at least 240 unit of calcium, at least 460 units of
iron and at most 300 units of cholesterol . How many packets of each food should be used to
minimize the amount of vitamin A in the diet? What is the minimum amount of vitamin A?
Formulate the above L.P.P mathematically and then solve it graphically.
22. There are 3 urns having the following composition of white and black balls: Urn I contains 7 white
and 3 black balls; Urn II contains 4 white and 6 black balls; Urn III contains 2 white and 8 black balls .
One of these urns is chosen with probabilities 0.2 ,0.6 , and 0.2 respectively . from the chosen urn ,
two balls are drawn at random without replacement . Both the balls happened to be white calculate
the probability that the balls drawn were from Urn III.
OR
There are 3 urns having the following composition of white and black balls; Urn I contains 7 white
and 3 black balls; Urn II contains 4 white and 6 black ; urn III contains 2 white and 8 black balls. One
of these urns is chosen with probabilities 0.2,0.6, and 0.2 respectively . from the chosen urn, two
balls are drawn at random with replacement.
Both the balls happened to be white calculate the probability that the balls drawn were from Urn III.

26. 23. Find the equation of plane passing through the point (1,1,-1) and perpendicular to the planes
x+ 2y+3z-7 = 0 and 2x – 3y 4x = 0 .
24. Find inverse of [
𝟎 𝟏 𝟐
𝟏 𝟐 𝟑
𝟑 𝟏 𝟏
] by using Elementary Row transformation method .
25. Using the method of integration , find the area bounded by the curve | 𝒙| + | 𝒚| = 1.
OR
Find the area bounded by the curve y = x3 , the x – axis and the ordinates x = 1.
26. Show that the lines 𝒓⃗ = (−𝒊̂ + 𝟐𝒋̂+ 𝟓𝒌̂) +  (−𝒊̂ + 𝟐𝒋̂ + 𝟓𝒌̂) and 𝒓⃗ = (−𝟑𝒊̂ + 𝒋̂ + 𝟓𝒌̂) +
 (−𝟑𝒊̂ + 𝒋̂ + 𝟓𝒌̂) are coplanar. Also, find the equation of the plane containing these two lines.
ANSWERS:-
1.
𝟏
𝟐
𝒍𝒐𝒈 |𝒆 𝟐𝒙
+ 𝒆−𝟐𝒙
| + C 2. 27 3. –cot x + C
4. Yes 5.
𝟓𝝅
𝟔
7. -4
8. g : R  R defined by g(y) =
𝒚−𝟕
𝟏𝟎
10. (i)
𝟐
𝟓
, (ii)
𝟕
𝟏𝟓
. A person appearing for an interview should be intelligent and honest. OR
(i)
𝟏𝟑
𝟓𝟎
, (ii)
𝟑𝟕
𝟓𝟎
. B is more trustworthy because B peaks more truth than A.
11. Absolute maximum value of f(x) is 𝟒
𝟏
𝟑 which occurs at x = -1 and absolute minimum value of f(x) is 0
which occurs at x = 0.
12.
𝟏
√𝟑
13. (i) f(x) is strictly increasing on (-2, 1)  (3, ) and strictly decreasing on (- , -2]  [1, 3] .
(ii) f(x) is increasing on [-2, 1]  [3, ) and decreasing on (- , -1]  [1, 3].
14. -2 √ 𝟏 − 𝒙 – sin-1
√ 𝒙 + √ 𝒙 √ 𝟏 − 𝒙 + C 15.
𝝅
𝟐
- log 2 OR 2√ 𝟐 - 2
17.
𝒙 𝟑
𝟑
tan-1 x -
𝒙 𝟐
𝟔
+
𝟏
𝟑
log |√𝟏 + 𝒙 𝟐| + C 18. −
𝟐
𝟏+𝒕𝒂𝒏 (
𝒙+𝒚
𝟐
)
= x + C 19. −
𝒙
√ 𝟏+ 𝒙 𝟐
21. Quantity of food P = 15 packets, quantity of food Q = 20 packets and minimum amount of vitamin A
= 150 units.
22.
𝟏
𝟒𝟎
𝑶𝑹
𝟒
𝟏𝟎𝟏
23. 17x + 2y – 7z – 26 = 0
24. [
𝟏
𝟐
−
𝟏
𝟐
𝟏
𝟐
−𝟒 𝟑 −𝟏
𝟓
𝟐
−
𝟑
𝟐
𝟏
𝟐
] 25. 2 sq. units OR
𝟏
𝟐
sq. units 26. X – 2y + z = 0.

27. Unsolved practice Paper -9
Section A
1. Differentiate w.r.t. x: cos √ 𝒙 .
2. If A [
𝒂 𝟎 𝟎
𝟎 𝒂 𝟎
𝟎 𝟎 𝒂
] , then find An .
3. Find the magnitude and rectangular(or scalar ) components of the position vector of the point
( 3, -1 , 2 ).
4. Differentiate the following w.r.t. x: sin-1 (
𝟐𝒙
𝟏+ 𝒙 𝟐).
5. The total cost C (x) in rupees , associated with the plantation of x trees is given by
C (x) = 0.007x3 – 0.003x2 + 15x + 4000. Find the marginal cost when 17 trees are planted.
6. Evaluate :  sec x ( secx + tan x ) dx.
Section B
7. An Apache helicopter of enemy is fling along the curve given by y = x2 + 7. A soldier , placed at (3,7) ,
wants to shoot down the helicopter when it is nearest to him . Fine the distance .
8. Evaluate : ∫
𝒙+𝒔𝒊𝒏 𝒙
𝟏+𝒄𝒐𝒔 𝒙
dx .
9. Find the value of K so that  (x) = {
𝒌𝒙 𝟐
𝒊𝒇 𝒙 ≤ 𝟐
𝟑 𝒊𝒇 𝒙 > 2
is continuous at x = 2.
10. Evaluate : ∫
𝟏
𝟓+𝟑 𝒔𝒊𝒏 𝟐 𝒙
dx .
11. If log (x2 + y2 ) = 2 tan-1 (
𝒚
𝒙
) , than prove that
𝒅𝒚
𝒅𝒙
=
𝒙+𝒚
𝒙−𝒚
.
OR
If x = 2 cos  - cos 2 and y = 2 sin  - sin 2 , then prove that
𝒅𝒚
𝒅𝒙
= tan (
𝟑
𝟐
) .
12. Evaluate : ∫
𝟏
𝟏+𝒔𝒊𝒏 𝒙
𝝅
𝟎
dx OR Evaluate : ∫ √ 𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝟓
𝒙
𝝅
𝟐
𝟎
dx.
13. If a, b, and c are three mutually perpendicular unit vector , that prove that |𝒂⃗⃗ + 𝒃⃗⃗ + 𝒄⃗ | = √ 𝟑 .
14. Solve : 𝐬𝐢𝐧−𝟏 𝟖
𝟏𝟕
= 𝐬𝐢𝐧−𝟏
𝒙 − 𝐬𝐢𝐧−𝟏 𝟑
𝟓
OR
Solve : 𝐭𝐚𝐧−𝟏 𝟑𝟐
𝟒𝟑
= 𝐭𝐚𝐧−𝟏 𝟏
𝒙
− 𝐭𝐚𝐧−𝟏 
𝟒
15. A stone is dropper into a quiet lake and waves move in a circle at a speed of 5cm/s. At the instant
when the radius of the circular wave is 8cm, how fast is the enclosed area increasing ? Like the store
, people throw garbage in the lake . Do you favour this type of act?
16. Find the area of the region bounded by the following curves : y = 1+| 𝒙 + 𝟏|; x = -2 ; x = 3 ; y = 0.

28. 17. One card is draw at random form a well – shuffled deck of 52 cards . Let E : The cards drawn is a king
or queen, F : The card drawn is a queen or jack . Are the events E and independent?
OR
Three cards are drawn with replacement from a well – shuffled pack of card . Find the probability
that
(i) the cards drawn are king , queen and jack respectively .
(ii) The cards drawn are king , queen and jack.
18. Find the particular solution of the differential equation
𝒅𝒚
𝒅𝒙
= 𝟏 + x2 + y2 +x2 y2 given that y = 1 when
x = 0.
19. If ( cos x )y = (cos y )x that find
𝒅𝒚
𝒅𝒙
.
Section C
20. Show that the lines
𝒙−𝒂+𝒅
𝜶− 𝜹
+
𝒚−𝒂
𝜶
+
𝒛−𝒂−𝒅
𝜶+ 𝜹
and
𝒙−𝒃+𝒄
 − 𝜸
+
𝒚−𝒃
𝜷
+
𝒛−𝒃−𝒄
𝜷+ 𝜸
are coplanar .
21. There are two types of fertilisers , F1 and F2 , F1 consists of 10% nitrogen and 6% phosphoric acid
and F2 consists of 5% nitrogen and 10% phosphoricacid . After testing the soil conditions, a farmer
finds that he needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for his crop . If F1 costs Rs.
6 per kg and F2 costs Rs. 5 per kg. determine how much of each type of fertiliser should be used so
that nutrient requirement are met at a minimum cost ? What is the minimum cost? Formulate the
above L.P.P . mathematically and then solve it graphically .
22. The given relation R is defined on the set of real number as a R b  | 𝒂|  b. Fine whether the given
relation is reflexive ,symmetric and transitive.
OR
Show that the function  : N  N , given by  (x) = x (-1)x , is a bijection .
23. If A = [
𝟐 −𝟑 𝟓
𝟑 𝟐 −𝟒
𝟏 𝟏 −𝟐
] , then find A-1 . Using A2-1 ,solve the system of equations:
2x + 3y + z = 11 , -3x +2y + z = -5, 5x -4y – 2z = 13.
24. Find the vector equation of the plane which is at a distance of
𝟔
√𝟐𝟗
from the origin and its normal
from the origin is 2 𝒊̂ - 3𝒋̂ + 4 𝒌̂ : also , find its Cartesian from.
25. Prove that |
𝟏 𝒂 𝟐
+ 𝒃𝒄 𝒂 𝟑
𝟏 𝒃 𝟐
+ 𝒄𝒂 𝒃 𝟑
𝟏 𝒄 𝟐
+ 𝒂𝒃 𝒄 𝟑
| = -( a – b ) (b – c ) ( c – a ) ( a2 + b2 +c2 ).
26. In a factory machine a produced 30% of the total output , machine B produces 25% and machine C
prodxuces the remaining output . The defective items produces by machinery A, B and C are 1%,

29. 1.2% and 2% respectively . Three machines working together produce 10000 items in a day . An item
is drawn at random from a day’s output and found to be defective . Fine the probability that it was
produced by machine B or C .
OR
A factory has three machine X,Y and Z producing 1000, 2000 and 3000 bolts per day respectively .
The machine X produces 1% defective bolts, Y produces 1.5% and Z produces 2% defective bolts . As
the end a day , a bolt is drawn at random and is found to be defective . What is the probability that
this defective bolt has been produced by machine X ?
ANSWER :-
1. −
𝒔𝒊𝒏 √ 𝒙
𝟐√ 𝒙
2. [
𝒂 𝒏
𝟎 𝟎
𝟎 𝒂 𝒏
𝟎
𝟎 𝟎 𝒂 𝒏
]
3. Magnitude is √ 𝟏𝟒 and rectangular (or scalar) components are 3, -1, 2. 4.
𝟐
𝟏+ 𝒙 𝟐
5. Rs. 20.967 6. Tan x + sec x + C 7. √ 𝟓 units
8. x tan
𝒙
𝟐
+ C 9.
𝟑
𝟒
10.
𝟏
√𝟒𝟎
𝐭𝐚𝐧−𝟏
(
√𝟖 𝒕𝒂𝒏 𝒙
√𝟓
) + C
12. 2 OR
𝟔𝟒
𝟐𝟑𝟏
14.
𝟕𝟕
𝟖𝟓
OR −
𝟒𝟔𝟏
𝟗
15. 80  cm2 / s. No, I do not favour this type of act because it pollutes water which is an essential
Components for the survival of the living beings.
16.
𝟐𝟕
𝟐
sq. units 17. No OR (i)
𝟏
𝟏𝟗𝟕
, (ii)
𝟔
𝟐𝟏𝟗𝟕
18. Tan-1 y = x +
𝒙 𝟑
𝟑
+
𝝅
𝟒
.
19.
𝒍𝒐𝒈 (𝒄𝒐𝒔 𝒚𝟎+𝒚 𝒕𝒂𝒏 𝒙
𝒍𝒐𝒈 ( 𝒄𝒐𝒔 𝒙)+ 𝒙 𝒕𝒂𝒏 𝒚
21. Quantity of fertilizer F1 = 100 kg, quantity of fertilizer F2 = 80 kg and minimum cost = Rs. 1000.
22. R is not reflexive, not symmetric but transitive.
23. A-1 = [
𝟎 𝟏 −𝟐
−𝟐 𝟗 −𝟐𝟑
−𝟏 𝟓 −𝟏𝟑
] , x = 13, y = -49, z = 132.
24. Vector form is 𝒓⃗ .( 𝟐𝒊̂ − 𝟑𝒋̂ + 𝟒𝒌̂ ) = 6 and Cartesian form is 2x – 3y + 4z = 6.
26.
𝟒
𝟔
𝑶𝑹
𝟗
𝟏𝟎
.

30. Unsolved Practice Paper – 6
Section A
1. Evaluate : ∫
𝒆 𝒎 𝐬𝐢𝐧−𝟏 𝒙
√ 𝟏− 𝒙 𝟐
dx
2. Find the angle between two vector a and b with magnitude 1 and 2 respectively and |𝒂⃗⃗ × 𝒃⃗⃗ |= √ 𝟑.
3. Evaluate : ∫ 𝒆 𝒙 ( 𝒙 + 𝟏) 𝒔𝒊𝒏 (𝒙𝒆 𝒙
) dx .
4. Find the Cartesian equation of the plane 𝒓⃗ .[( 𝒔 − 𝟐𝒕) 𝒊̂ + ( 𝟑 − 𝒕) 𝒋̂ + ( 𝟐𝒔 + 𝒕) 𝒌̂] = 15.
5. Simplify : cos 𝜷 [
𝒄𝒐𝒔 𝜷 𝒔𝒊𝒏 𝜷
− 𝒔𝒊𝒏 𝜷 𝒄𝒐𝒔 𝜷
] + 𝒔𝒊𝒏 𝜷 [
𝒔𝒊𝒏 𝜷 − 𝒄𝒐𝒔 𝜷
𝒄𝒐𝒔 𝜷 𝒔𝒊𝒏 𝜷
] .
6. Find a vector of magnitude 5 units and parallel to resultants of the vectors 𝒂⃗⃗ = 𝟐𝒊̂ + 𝟑𝒋̂ − 𝒌̂ and
𝒃⃗⃗ = 𝒊̂ − 𝟐𝒋̂+ 𝒌̂.
Section B
7. The government is running a campaign ‘MAKE INDIA POLIO FREE’ To spread awareness in a particular
society ,it is displaying an air balloon with the above tag line printed on it. The balloon is in the from
of a right circular surmounted by a hemisphere , having a diameter equal to the height of the cone .
It is being inflated by a pump . How fast is its volume changing with respect to its total height h,
when h is 3 cm ? Do you thing we should work seriously towards polioeradication ? Write any
four values that been highlighted here.
8. If u = sin ( m cos-1 x ) and v = cos ( m sin-1 x ) , then prove that
𝒅𝒖
𝒅𝒗
= √
𝟏− 𝒖 𝟐
𝟏− 𝒗 𝟐 .
OR
If y = √ 𝒙 +
𝟏
√ 𝒙
, then prove that 2x
𝒅𝒚
𝒅𝒙
= √ 𝒙 −
𝟏
√ 𝒙
9. Evaluate : ∫ [𝒍𝒐𝒈 𝒍𝒐𝒈 𝒙 +
𝟏
(𝒍𝒐𝒈 𝒙) 𝟐] dx
10. If 𝒂⃗⃗ , 𝒃⃗⃗ 𝒂𝒏𝒅 𝒄⃗ are three vectors such that 𝒂⃗⃗ + 𝒃⃗⃗ + 𝒄⃗ = 𝟎⃗⃗ and | 𝒂⃗⃗ | = 5, |𝒃⃗⃗ | = 12, | 𝒄⃗ | = 13, then find
𝒂⃗⃗ . 𝒃⃗⃗ + 𝒄⃗ . 𝒃⃗⃗ + 𝒄⃗ . 𝒂⃗⃗ .
11. Prove that : tan-1 𝟏
𝟒
+ tan-1 𝟐
𝟗
= ½ cos-1 𝟑
𝟓
.
OR
If cos-1 𝒙
𝒂
+ cos-1 𝒚
𝒃
=  ,
𝒙 𝟐
𝒂 𝟐 −
𝟐𝒙𝒚
𝒂𝒃
cos 𝜶 +
𝒚 𝟐
𝒃 𝟐 = sin2  .
12. Find the coordinates of the point where the line through the points ( 3, -4 , -5 ) and ( 2,-3, 1) crosses
the plane 3x + 2y + z + 14 = 0 .
13. Using determinants , find the equation of the joining the point (1,2) and (3,6).

31. 14. Mother, father and son line up at random for a family picture .Determine P (E F ) , where E /; son at
one end and F : father in the middle.
OR
An electronic assembly consists of two sub stems , say A and B . From previous testing procedures ,
the following probabilities are assumed to be known : P (A fails ) = 0.2 . ( B fail alone ) = 0.15 ,.
Evaluate the probabilities :
(i) P ( A fails  B has failed ) (ii) P ( A fails along )
15. Using probabilities of determinants , solve for x : |
𝒙 − 𝟐 𝟐𝒙 − 𝟑 𝟑𝒙 𝟒
𝒙 − 𝟒 𝟐𝒙 − 𝟗 𝟑𝒙 − 𝟏𝟔
𝒙 − 𝟖 𝟐𝒙 − 𝟐𝟕 𝟑𝒙 − 𝟔𝟒
| = 0.
16. Check the applicability of (A) Rolle’s theorem (B) Lagrange’s mean value theorem for  (x) = | 𝒙| on
[ -1 ,1 ].
OR
Check the applicability of (A) Roll’s theorem (B) Lagrange’s mean value the theorem for  (x) = | 𝒙| on
[ 5,9 ] , where [x] is the greatest integer less than or equal to x.
17. Write A-1 for A = [
𝟐 𝟓
𝟏 𝟑
] .
18. From the differential equation representing the family of ellipses having foci on x – axis and center
at the origin.
19. Evaluate : ∫ 𝒕𝒂𝒏 𝟒
𝒙 𝒅𝒙 dx
Section C
20. Consider  : R+  [ -5 ,  ) given by  (x) = 9x2 + 6x – 5 . Show that  is invertible with f-1 (y) =
√𝒚+𝟔 − 𝟏
𝟑
, where R+ is the set of all non – negative real numbers.
21. Three bags contain balls as shown in the table below :
Bag No. of white Balls No. of Black Balls No. of Red Balls
I 1 2 3
II 2 1 1
III 4 3 2
A bag is chosen at random and two balls are drawn from it , They happen to be white and red .
What is probability that they came from the Bag III ?
OR

32. Three bags contain balls as shown in the table below:
Bag No. of white Balls No. of Black Balls No. of Red Balls
I 1 2 3
II 2 1 1
III 4 3 2
A bag is chosen at random and two balls are drawn from it with replacement. They happen to be
white and red . What is probability that they came from the Bag III ?
22. Evaluate : ∫
𝒙
( 𝟏+𝒙)(𝟏+ 𝒙 𝟐 )
∞
𝟎
dx
23. Show that the surface area of a closed cuboids with the square base and given volume is minimum
when it is cube.
24. One king of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of
flour and 50 g of fat . Find the maximum number of cakes which can made from 5 kg of flour and 1 kg
of fat, assuming that there is no shortage of the other ingredients used in making the cakes.
Formulate the above L.P.P. mathematically and then solve it graphically .
25. Find the coordinates of the foot of the perpendicularand length of the perpendicular drawn from
the point P ( 5,4,2) to the line = - + 3j + k + ( 2 + 3j – k ) . Also, find the image of P in the line .
26. The area between x = y2 and x = 4 , which is divided into two equal parts by the line x= a . Find the
value of a .
OR
Using integration , find the area of the triangular region whose sides have the equations
y = 2x + 1 , y = 3x + 1 and x = 4.
ANSWER :-
1.
𝒆 𝒎 𝐬𝐢𝐧−𝟏 𝒙
𝒎
+ C 2. 600 3. –cos ( xex ) + C
4. ( s – 2t ) x + ( 3 – t ) y + (2s + t )z = 15 5. [
𝟏 𝟎
𝟎 𝟏
] OR I2
6.
𝟓
√𝟏𝟎
( 𝟑𝒊̂ + 𝒋̂)
7.
𝟒𝝅
𝟑
cm2 . Yes we should work seriously towards polio eradication. The value that have been
highlighted here are awareness , helping other , care and empathy for children.
9. x log log x –
𝒙
𝒍𝒐𝒈 𝒙
+ C 10. – 169 12. (5,-6 ,- 17)
13. 2x – y = 0 14. 1 OR (i) ½ (ii) 0.05 15. 4
16. (A) Roll’s theorem is not applicable , (B) Lagrange mean value theorem is not applicable
OR

33. (A) Rolle’s theorem is not applicable , (B) Lagrange’s mean value theorem is not applicable .
17. [
𝟑 −𝟓
−𝟏 𝟐
] 18. Xy
𝒅 𝟐
𝒚
𝒅𝒙 𝟐 + 𝒙 (
𝒅𝒚
𝒅𝒙
)
𝟐
− 𝒚
𝒅𝒚
𝒅𝒙
= 𝟎
19.
𝒕𝒂𝒏 𝟑
𝒙
𝟑
− 𝒕𝒂𝒏 𝒙+ 𝒙 + 𝑪 21.
𝟓
𝟏𝟕
OR
𝟔𝟒
𝟏𝟗𝟗
22.
𝝅
𝟒
24. Number of cakes of I kind = 20 , number of cakes of II kind = 10 and and maximum number of cakes
= 30.
25. (1, 6, 0) ; √ 𝟐𝟒 units; (-3, 8, -2) 26. (𝟒)
𝟐
𝟑 OR 8 sq. units