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SAMPLE QUESTION PAPER
Class:-XII
Session 2023-
Mathematics (Code-041)
Time: 3 hours Maximum marks: 80
General Instructions:
- This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However, there are
internal choices in some questions.
- Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.
- Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.
- Section C has 6 Short Answer (SA)-type questions of 3 marks each.
- Section D has 4 Long Answer (LA)-type questions of 5 marks each.
- Section E has 3 source based/case based/passage based/integrated units of assessment of 4 marks each with
sub-parts.
___________________________________________________________________________________________
Section –A
(Multiple Choice Questions)
Each question carries 1 mark
Q1. If
ij
A a
is a square matrix of order 2 such that
1, when
0, when
ij
i j
a
i j
, then
2
A is
(a)
2 2
(b)
2 2
(c)
2 2
(d)
2 2
Q2. If A and B are invertible square matrices of the same order, then which of the following is not correct?
(a)
AB
| B |
(b)
| | |B|
AB
A
(c)
1
1 1
AB B A
(d)
1
1 1
A B B A
Q3. 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) 3 (c) -9 (d) 6
Q4. If
, if 0
3, if 0
kx
x
x f x
x
is continuous at x 0 , then the value of k is
(a) −3 (b) 0 (c) 3 (d) any real number
Q5. The lines
r i j k 2 i 3 j 6 k
and
r 2 i j k 6 i 9 j 18 k
; (where & are
scalars) are
(a) coincident (b) skew (c) intersecting (d) parallel
Q6. The degree of the differential equation
3 2
2 2
2
dy d y
is
dx dx
(a) 4 (b)
(c) 2 (d) Not defined
Q7. 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,
is
(a) p 2 q (b)
q
p (c) p 3 q (d) p q
Q8. ABCD is a rhombus whose diagonals intersect at E. Then EA EB EC ED
equals to
(a) 0
(b) AD
(c) 2 BD
(d) 2 AD
Q9. For any integer n, the value of
2
cos x 3
Sin (2n + 1) x dx
e
is
(a) -1 (b) 0 (c) 1 (d) 2
Q10. The value of A, if
1 2 0 2 , where ,
x x
A x x x
x x
is
(a)
2
2 x 1 (b) 0 (c)
3
2 x 1 (d)
2
2 x 1
Q11. 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 2 y 10 (c) x y 1 (d) x y 1
Q20. 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.
Section –B
[This section comprises of very short answer type questions (VSA) of 2 marks each]
Q21. Find the value of
1
sin cos.
OR
Find the domain of
1 2
sin x 4.
Q22. Find the interval/s in which the function f : defined by
x
f x x e is increasing.
Q23. If
2
f x x
x x
, then find the maximum value of
f x.
OR
Find the maximum profit that a company can make, if the profit function is given by
2
P x 72 42 x x ,where x
is the number of units and P
is the profit in rupees.
Q24. Evaluate :
1
1
log.
x
dx
x
Q25. Check whether the function f :
defined by
3
f x x x,has any critical point/s or not?
If yes, then find the point/s.
Section – C
[This section comprises of short answer type questions (SA) of 3 marks each]
Q26. Find :
2
2 2
x
dx x
x x
Q27. The random variable X has a probability distribution
P X of the following form, where ' k 'is some
real number:
, if 0
2 , if 1
3 , if 2
0, otherwise
k x
k x
P X
k x
(i) Determine the value of k.
(ii) Find P X 2 .
(iii) Find
P X .
Q28. Find :
3
x
dx x
x
OR
Evaluate:
4
0
log 1 tan x dx.
Q29. Solve the differential equation:
2
x x
y y
ye dx xe y dy y
OR
Solve the differential equation:
2
cos tan ; 0.
dy
x y x x
dx
Q30. Solve the following Linear Programming Problem graphically:
Minimize: z x 2 y,
subject to the constraints: x 2 y 100, 2 x y 0, 2 x y 200, x y, 0.
OR
Solve the following Linear Programming Problem graphically:
Maximize: z x 2 y,
subject to the constraints: x 3, x y 5, x 2 y 6, y0.
Q31. If
y
x
a bx e x then prove that
2
2
2
d y a
x
dx a bx
Section –D
[This section comprises of long answer type questions (LA) of 5 marks each]
Q32. Make a rough sketch of the region
2
x y , : 0 y x 1, 0 y x 1, 0 x 2 and find the
area of the region, using the method of integration.
Q33. Let be the set of all natural numbers and R be a relation on defined by
a b R c d, , ad bcfor all
a b, , c d, . Show that R is an equivalence relation on
. Also, find the equivalence class of
2,6 , i.e.,
OR
Show that the function f : x : 1 x 1
defined by ,
x
f x x
x
is one-one and
onto function.
Q34. Using the matrix method, solve the following system of linear equations :
(iii) Let E be the event of committing an error in processing the form and let
1 2
E ,E and
3
E be the
events that Jayant, Sonia and Oliver processed the form. Find the value of
3
1
i
i
P E E
Q37. Read the following passage and answer the questions given below:
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
1
F 6 i 0 j kN,
Team B pulls with force
2
F 4 i 4 j kN,
Team C pulls with force
3
ˆ ˆ
F 3 i 3 j kN,
(i) What is the magnitude of the force of Team A?
(ii) Which team will win the game?
(iii) Find the magnitude of the resultant force exerted by the teams.
OR
(iii) In what direction is the ring getting pulled?
Q38. Read the following passage and answer the questions given below:
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
2
y x x , where ' x 'is the number of days exposed to the
sunlight, for x 3.
(i) Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
(ii) 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?
