केन्द्रीय विद्यालय संगठन, गरु

ु ग्राम संभाग
KENDRIYA VIDYALAYA SANGATHAN GURUGRAM REGION
कक्षा- XII Class-XII
2022-23

PRACTICE PAPER-V

Class: XII Max Marks: 80

Subject: MATHEMATICS (041) Time Allowed: 3 Hours

General Instructions:
1. This question paper contains-five sections A, B, C, D and E. Each section is
compulsory. However, there are internal choices in some questions.
2. Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark
each.
3. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.
4. Section C has 6 Short Answer (SA)-type questions of 3 marks each.
5. Section D has 4 Long Answer(LA)-type questions of 5 marks each.
6. Section E has 3 source based/case based/passage based/integrated units of
assessment (4 marks each) with sub parts.

SECTION-A
( Multiple Choice Questions)
(Each question carries 1 mark)
Q1 If A is square matrix such that A2= I , then (A-I)3+(A+I)2-7A is equal to
(a) -A (b) I-A (c) I+A (d) 3A
Q2
[5 x]
If A= y 0 and A=A’ ,then

(a) x = 0,y = 5 (b) x+y = 5 (c) x = y (d) none of these

 2
Q3 The value of the expression |⃗a × ⃗b| +( ⃗a . b⃗ ) is
2

2
(a) a⃗ . ⃗b ( b )|⃗a|.|⃗b|( c )|⃗a|2|b⃗| ( d ) ¿

{
Q4 x −9
2
, x≠3
If the function f defined by f(x) = x−3 is continuous at x=3, then the value of K is
k , x=3

(a) 6 (b) 3 (c) -6 (d) 3

3

Q5 If f ' ( x )=x 2 e x ,then f (x) is

1 x3 1 x 4
1 x3 1 x2
(a) 3 e +C (b) 3 e +C (c) 2 e +C (d) 2 e +C

Q6
{( ) }= 0,is
3
d dy
The sum of the order and degree of the following differential equation dx dx

(a) 5 (b) 4 (c) 3 (d) 2

Q7 Corner points of the feasible region for an LPP are (0, 3), (1,1) and (3,0).Let Z = px + qy,
where p,q>0, be the objective function. The condition on p and q so that the minimum of
Z occurs at (3,0) and (1,1) is
q
p=
(a) p = q (b) 2 (c) p = 3q (d) p=q
Q8 ^ μ ^j+ k^ and b⃗ = i+2
The value of μ such that the vectors a⃗ =2 i+ ^ ^j +3 k^ are orthogonal is

3 5
(a) 0 (b) 1 (c) 2
(d) - 2
√3
Q9 dx
The value of ∫ is
1 1+ x 2
π 2π π π
(a) 3 (b) 3
(c) 6
(d) 12

Q1 If A is a square matrix of order n ,then |adj( A)| =

0
(a) | A| (b) | A|
n−1
(c) | A|n (d) | A|n−2

Q1 The corner points of the shaded bounded feasible region of an LPP are (0,0),(30,0),(20,30)
1 and (0,50) as shown in the figure .

The maximum value of the objective function Z = 4x+y is

(a) 120 (b) 130 (c) 140 (d) 150
Q1
| x 2| | 6 2|
If 18 x = 18 6 ,then x is equal to
2
(a) 6 (b) ±6 (c) -6 (d) 0
Q1 If A is a square matrix of order 3, such that A(adjA) = 10 I ,then |adj A| is equal to
3 (a) 1 (b) 10 (c) 100 (d) 101
Q1 Let A and B be two events . If P(A)=0.2,P(B)=0.4 ,P(AUB)=0.6 then P B is equal to (A)
4
(a) 1 (b) 0 (c) 0.2 (d) 0.4
Q1 dy tan−1 x
The integrated factor of the differential equation: (1+x2) dx + y =e is
5
1 −1 −1 −1

(a) −1
tan x (b) 2 e tan x
(c) 3e tan x
(d) e
tan x
e

Q1 d2 y
If y = 5e7x + 6e-7x,show that is equal to
d x2
6
(a) 7y (b) 6 y (c) 49y (d) 36y
Q1 The projection of the vector 2i+3j+2k on the vector i+2j+k is
7 (a) √ ( b ) ( c ) (d) √
5 6 5 6 6
3 6 5 19
Q1 If the direction cosines of a line are k,k,k then
8 1 1
(a) k > 0 (b) 0 < k< 1 (c) k = 1 (d) k= or k = -
√3 √3
 ASSERTION- REASON BASED QUESTIONS

In the following questions, a statement of assertion (A) is followed by a statement of

Reason (R).Choose the correct 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 and R is not the correct explanation of A
(c ) A is true but R is false.
(d ) A is true but R is false.

Q1
9
Assertion(A): The value of cos 2 +sin ( π −1
( −12 ))= 12
Reason(R) : sin−1 (−θ )=−sin−1 θ
Q2 x+2 y−1 z−2 x−3 y z +1
Assertion(A): −2 = 3 = 1 and −3 = −2 = 2 are coplaner.
0
Reason (R) : Let line l1 passes through the point ( x 1 , y 1 , z 1) and parallel to the vector whose
direction ratios are a 1, b1 , ∧c 1, : Let line l2 passes through the point ( x 2 , y 2 , z 2) and parallel to
the vector whose direction ratios are a 2, b2 ,∧c 2 ,. Then both lines l1∧l2 are coplaner if and

| |
x 2−x 1 y 2− y 1 z 2 −z1
only if a1 b1 c1 =0
a2 b2 c2

SECTION -B
This section comprises of very short answer type-question (VSA) of 2 marks each

Q2 33 π
Find the value of sin-1[cos( 5 )]
1
OR

( x−2 )
Let y = R-{ 3 } and B = R- { 1 } . Consider the function f:A→B defined by f(x)= x−3 . Is f is one-
 one and onto ? Justify your Answer.

Q2 An edge of a variable cube is increasing at the rate of 5cm per second. How fast is the
2 volume increasing when the side is 15 cm.
Q2 Find the vector of magnitude 6,which is perpendicular to both the vectors
3 ^ ^j+2 k^ ∧4 i−
2 i− ^ ^j+3 k^

OR
Find the equation of a line in vector and cartesian form which passes through the point
^ ^j−2 k^
(1,2,3) and is parallel to the vector3i+2
2
Q2 If x sin ( a+ y ) +sina cos ( a+ y )=0 , then prove that dy =
sin (a+ y )
dx sina
4

Q2 If a⃗ +b⃗ +c⃗ =0⃗ and |⃗a| =3, |b⃗| =5 and |c⃗| =7 then what is the angle between a⃗ and b⃗ .
5
SECTION C
(This section comprises of short type questions (SA) of 3 marks each)
Q2 dx
Find:∫
√ 5−4 x−2 x2
6

Q2 1 1
Probabilities of solving specific problem independently by A and B are 2 and 3 respectively
7
. If both try to solve the problem independently .Find the probability that (i) the problem
is solved (ii) exactly one of them solves the problem.
OR
From a lot of 30 bulbs which include 6 defectives ,a sample of 4 bulbs is drawn at random
with replacement. Find the probability distribution of the number of defective bulbs.

π
Q2 x sinx
Evaluate :∫ 2
dx
0 1+cos x
8
 OR
5

Evaluate:∫|x +2| dx
−5

Q2 Solve the differential equation:

9 ( 1-y2) (1+logx) dx + 2xydy = 0

OR
Solve the differential equation x dy - y dx =√ x 2+ y 2 dx

Q3 Solve the following Linear Programming Problem graphically:

0 Maximize: Z =100 x+120 y
Subject to : 5 x +8 y ≤ 200,5 x+ 4 y ≤120 , x , y ≥ 0
Q3 x2
Evaluate:∫ 2
dx ¿
( x ¿¿ 2+ 4)(x + 9)
1
SECTION D
This section comprises of long answer -type question (LA) of 5 marks each)
Q3 Make a rough sketch of the region {( x , y ) : 0 ≤ y ≤ x 2+ 1,0≤ y ≤ x +1,0 ≤ x ≤ 2 } and find the area of
2 the region using integration.
Q3 Let A={1,2,3 , … , 9 } and R be the relation in A × A defined by (a, b) R (c, d)
3 if a+ d = b+ c, for (a , b),(c , d) in A × A . Prove that R is an equivalence relation and also
obtain the equivalence class [(2,5)].
OR
Consider f:R+→[-9,∞ ¿ given by f(x) =5x2 +6x-9. Prove that f is bijective.

Q3 x−3 y−5 z−7

An insect is crawling along the line 1 = −2 = 1 and another insect is crawling along
4
the line
 x+1 y +1 z +1
= = . At what points on the lines should they reach so that the distance
7 −6 1

between them is the shortest ? Find the shortest possible distance between them.
OR
The equation of motion of a rocket are:
X = 2t, y = -4t, z = 4t ,where the time t is given in seconds, and the coordinates of a
moving point in km. What is the path of the rocket ?At what distances will the rocket be
from the starting point O(0,0,0) and from the following line in 10 seconds?
^
r⃗ =20 i−10 ^j+40 k^ + μ ¿

Q3
5

Given two matrices A = and B= verify that BA=6I.Use the result to

solve the system x - y = 3, 2x + 3y + 4z = 17, y + 2z =7.

SECTION E
This section comprises of 3 case -study /passage -based questions of 4 marks each with
two sub-parts.First two case study questions have three sub-parts(i),(ii),(iii) of marks
1,1,2 respectively. The third case study question has two sub-parts of 2 marks each.

Q3 Case – Study 1: Read the following passage and answer the questions given below:
6
Some young entrepreneurs started an industry “Young achievers” for casting metal into
various shapes. They put up an advertisement online stating the same and expecting
order to cast method for toys, sculptures, decorative pieces and more.
A group of friends wanted to make innovative toys and hence contacted the “Young
achievers” to order them to cast metal into solid half cylinders with a rectangular base
and semi circular ends.
 (i) If r,hand V are radius, length and volume respectively casted half cylinder, then find the
total surface area function S of the casted half cylinder in terms of V and r.
(ii) ) For the given volume V, Find the condition for the total surface area S to be
minimum.
(iii) Use second derivative test to prove that Surface area is minimum for given
volume.
OR
(iii) Find the ratio h: 2r for S to be minimum.

Q3 Dr. Rohan residing in Delhi went to see an apartment of 3BHK in Noida. The window of
7 the house in the form of a rectangle surrounded by a semicircular opening having a
perimeter of the window 10 m as shown in the figure

(i) If x and y represents the length and breadth of the rectangular region, then
what is the relation between the variables.
(ii) Dr. Rohan is interested in maximize the area of the whole window.For this to
happen what should be value of x?
 OR
(ii) For maximum value of area ,find the breadth of the rectangular part of the
window.
(iii) Find the maximum area of window.

Q3 Mahindra Tractors is India’s leading farm equipment manufacturer. It is the largest tractor
8 selling factory in the world. This factory has two machine A and B . Past record shows that
machine A produced 60% and machine B produced 40% of the output(tractors). Further
2% of the tractors produced by machine A and 1% produced by machine B were defective.
All the tractors are put into one big store hall and one tractor is chosen at random.

(i) Find the total probability of chosen tractor (at random) is defective.
(ii) If in random choosing, chosen tractor is defective ,then find the probability that
the chosen tractor is produced by machine ‘A’