NBSE Class 12 Question Paper 2022 Mathematics

Total number of printed pages : 5 NB/XII/MAT/1

2022
MATHEMATICS
Full marks: 80 Time: 3 hours

General instructions:
i) Approximately 15 minutes is allotted to read the question paper and revise the
answers.
ii) The question paper consists of 18 questions. All questions are compulsory.
iii) Marks are indicated against each question.
iv) Internal and general choice has been provided in some questions.
v) Use of simple calculators (non-scientific and non-programmable) only is permitted.
N.B: Check to ensure that all pages of the question paper is complete as indicated on the
top left side.

Section – A

1. Choose the correct answer from the given alternatives:

(a) Number of binary operation on the set a, b are 1
(i) 10 (ii) 16 (iii) 20 (iv) 8

 3
(b) The principal value of sin 1   is
 1
 2 
   
(i) (ii) (iii) (iv)
6 4 3 2

dy
(c) If y  e cos x , then is equal to 1
dx
(i) e cos x (ii) e sin x (iii)  cos x e sin x (iv)  sin xe cos x

(d) The rate of change of the area of a circle with respect to its radius at r = 6 cm is 1
(i) 10π (ii) 12π (iii) 8π (iv) 11π

3
dx
(e)  is equal to 1
1
1 x2
 2  
(i) (ii) (iii) (iv)
3 3 6 12
     
(f) If a is an unit vector and  x  a .  x  a   8 , then x is 1
(i) 2 (ii) 3 (iii) 4 (iv) 5

-2- NB/XII/MAT/1

(g) The planes: 2 x  y  4 z  5 and 5 x  2.5 y  10 z  6 are 1
(i) Perpendicular (ii) Parallel
 5
(iii) Intersect y-axis (iv) Passes through  0,0, 
 4

B
(h) If P(A) = 0.8, P(B) = 0.5 and P   = 0.4, then P(A  B) is 1
A
(i) 0.32 (ii) 0.20 (iii) 0.40 (iv) None of these

Section – B

2 7 1
2. Prove: tan 1  tan 1  tan 1 2
11 24 2

3. Find the value of k so that the given function
kx  1, if x  5
f x    is continuous at x = 5 2
3 x  5, if x  5

4. Find the points at which the tangent to the curve y  x 3  3x 2  9 x  7 is parallel
to the x-axis. 2

5. Evaluate 
log x 
3
dx 2
x

x y
6. Form the differential equation representing the family of curves   1 , where
a b
a and b are arbitrary constants. 2

7. Find a vector in the direction of the vector a  5iˆ  ˆj  2kˆ , which has magnitude 8
units. 2

x5 y 2 z x y z
8. Show that the lines   and   are perpendicular to each
7 5 1 1 2 3
other. 2

1 3
9. Given that the events A and B are such that P(A) = , P(A  B) = and
2 5
P(B) = p. Find p if they are (i) mutually exclusive (ii) independent 2

-3- NB/XII/MAT/1

Section – C
10. Case study based question:
Students of class-XII, planned to plant saplings along straight lines, parallel to
each other to one side of the playground ensuring that they had enough play area.
Let us assume that they planted one of the rows of the saplings along the line 𝑦 =
𝑥 − 4. Let L be the set of all lines which are parallel on the ground and R be a
relation on L. 4

Answer the following using the above information.
a). Let relation R be defined by R = {(L1, L2) : L1 // L2 where L1, L2  L}, then R
is _______ relation.
(i) Equivalence (ii) Only reflexive
(iii) Not reflexive (iv) symmetric but not transitive
b). The function f : R  R defined by f(x) = x – 4 is
(i) bijective (ii) 1-1 but not onto
(iii) onto but not 1-1 (iv) neither 1-1 nor onto
c). Let f : R  R be defined by f(x) = x – 4, then the range of f(x) is
(i) R (ii) Z
(iii) W (iv) Q
d). Let R = {(L1, L2) : L1 // L2 and L1 : y = x – 4}, then which of the following can be
taken as L2?
(i) 2 x  2 y  5  0 (ii) 2 x  y  5
(iii) 2 x  2 y  7  0 (iv) x  y  7
11. a. Express the following matrix as the sum of a symmetric and a skew symmetric
 3 3  1
matrix  2 2 1 

 4 5 2
Or 4

b. Using properties of determinants, show that
 a b c 2a 2a 
 2b bca 2b   a  b  c 3
 
 2c 2c ca b 

-4- NB/XII/MAT/1

12. Answer any three (3) questions.
a. The length x of a rectangle is decreasing at the rate of 5 cm/min and the
width y is increasing at the rate of 4 cm/min. When x = 8 cm and y = 6 cm,
find the rates of change of (a) the perimeter and (b) the area of the rectangle. 4
x
b. Evaluate:  dx 4
x 1x  2x  3
a
x
c. Evaluate:  dx 4
0 x ax

 x
 x
 x
d. Solve the differential equation: 1  e  dx  e y 1  dy  0
y
4
   y
 
dy
e. Find the general solution of the differential equation x  2 y  x 2 log x 4
dx
13. Answer any two (2) questions.
a. Find x such that the four points A(3, 2, 1) , B(4, x, 5), C(4, 2,  2) and
D(6, 5,  1) are coplanar. 4
b. Find the shortest distance between the lines.

  
r  iˆ  2 ˆj  kˆ   iˆ  ˆj  kˆ  4
and

  
r  2iˆ  ˆj  kˆ   2iˆ  ˆj  2kˆ 
c. Find the vector equation of the plane passing through the intersection of the planes

  
 
r . 2iˆ  2 ˆj  3kˆ  7, r . 2iˆ  5 ˆj  3kˆ  9 and through the point (2, 1, 3). 4
14. a. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000
truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively.
One of the insured persons meets with an accident. What is the probability that he
is a scooter driver?
Or 4

b. The probability that a bulb produced by a factory will fuse after 150 days of use
is 0.05. find the probability that out of 5 such bulbs (i) none (ii) not more than one
(iii) more than one
will fuse after 150 days of use.

Section – D
15. a. Solve the system of linear equations by using matrix method.
x  y  2z  7
3 x  4 y  5 z  5
2 x  y  3 z  12

-5- NB/XII/MAT/1

Or 6
b. Using elementary transformation, find the inverse of the matrix
2 0 1
A 5 1 0
0 1 3

16. a. Show that of all the rectangles inscribed in a given fixed circle, the square
has the maximum area.
Or 6
b. A wire of length 28m is to be cut into two pieces. One of the pieces is to be
made into a square and the other into a circle. What should be the length of
the two pieces so that the combined area of the square and the circle is
minimum?
17. a. Find the area of the region in the first quadrant enclosed by x-axis, line
x  3 y and the circle x 2  y 2  4 .

Or 6
b. Using the method of integration, find the area of region bounded by the
triangle whose vertices are (  1, 0), (1, 3) and (3, 2).
18. a. A company manufactures two types of novelty souvenirs made of plywood.
Souvenirs of type A require 5 minutes each for cutting and 10 minutes each
for assembling. Souvenirs of type B require 8 minutes each for cutting and
8 minutes each for assembling. There are 3 hours 20 minutes available for
cutting and 4 hours for assembling. The profit is 5 each for type A and 6
each for type B souvenirs. How many souvenirs of each type should the
company manufacture in order to maximize the profit?
Or 6
b. There are two types of fertilizer F1 and F2. F1 consists of 10% nitrogen and
6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric
acid. After testing the soil conditions, a farmer finds that she needs at least
14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F 1 costs
6/kg and F2 costs 5/kg, determine how much of each type of fertilizer
should be used so that nutrient requirements are met at a minimum cost.
What is the minimum cost?

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