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*Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e.
MATHEMATICS – Code No. 041
SAMPLE QUESTION PAPER
CLASS - XII (2025-26)
Maximum Marks: 80 Time: 3 hours
General Instructions:
Read the following instructions very carefully and strictly follow them:
- This Question paper contains 38 questions. All questions are compulsory.
- This Question paper is divided into five Sections - A, B, C, D and E.
- In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) with only one
correct option and Questions no. 19 and 20 are Assertion-Reason based questions of 1
mark each.
- In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying
2 marks each.
- In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3
marks each.
6. In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5
marks each.
- In Section E, Questions no. 36 to 38 are Case study-based questions, carrying 4 marks
each.
- There is no overall choice. However, an internal choice has been provided in 2 questions
in Section B, 3 questions in Section C, 2 questions in Section D and one subpart each in 2
questions of Section E.
- Use of calculator is not allowed.
SECTION-A
This section comprises of multiple choice questions (MCQs) of 1 mark each.
Select the correct option (Question 1 - Question 18)
Q.No. Questions Marks
Identify the function shown in the grap
(A) sin
− 1
𝑥 (B) sin
− 1
( 2 𝑥) (C) sin
− 1
𝑥
2
) (D) 2 sin
− 1
For Visually Impaired:
Inverse Trigonometric Function, whose domain is [−
1
3
1
3
] , is …
(A) cos
− 1
𝑥 (B) cos
− 1
𝑥
3
(C) cos
− 1
( 3 𝑥) (D) 3 cos
− 1
h 1
*Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e.
If for three matrices 𝐴 = [𝑎
𝑖𝑗
]
𝑚× 4
, B = [𝑏
𝑖𝑗
]
𝑛× 3
𝑎𝑛𝑑 C = [𝑐
𝑖𝑗
]
𝑝×𝑞
products 𝐴𝐵 and
𝐴𝐶 both are defined and are square matrices of same order, then value of 𝑚, 𝑛, 𝑝
and 𝑞 are:
(A) 𝑚 = 𝑞 = 3 𝑎𝑛𝑑 𝑛 = 𝑝 = 4 (B) 𝑚 = 2 , 𝑞 = 3 𝑎𝑛𝑑 𝑛 = 𝑝 = 4
(C) 𝑚 = 𝑞 = 4 𝑎𝑛𝑑 𝑛 = 𝑝 = 3 (D) 𝑚 = 4 , 𝑝 = 2 𝑎𝑛𝑑 𝑛 = 𝑞 = 3
If the matrix 𝐴 = [
] is skew-symmetric, then value of
𝑞+𝑡
𝑝+𝑟
is….
(A)− 2 (B) 0 (C) 1 (D) 2
- If 𝐴 is a square matrix of order 4 and |𝑎𝑑𝑗 𝐴| = 27 , then 𝐴 (𝑎𝑑𝑗 𝐴) is equal to
(A) 3 (B) 9 (C) 3 𝐼 (D) 9 𝐼
The inverse of the matrix [
] is…
(A) [
] (B)
[
1
3
1
2
1
5
]
(C)
[
1
3
1
2
1
5
]
(D) [
]
Value of the determinant |
cos 67
𝑜
sin 67
𝑜
sin 23
𝑜
cos 23
𝑜
| is
(A) 0 (B)
1
2
(C)
√ 3
2
(D) 1
If a function defined by 𝑓(𝑥) = {
cos 𝑥 , 𝑥 > 𝜋
is continuous at 𝑥 = 𝜋, then the value of 𝑘 is
(A) 𝜋 (B)
− 1
𝜋
(C) 0 (D)
− 2
𝜋
- If 𝑓(𝑥) = 𝑥 tan
− 1
𝑥 , then 𝑓
′
( 1 )is equal to
(A)
𝜋
4
1
2
(B)
𝜋
4
1
2
(C) −
𝜋
4
1
2
(D) −
𝜋
4
1
2
A function 𝑓
2
is increasing on the interval
(A) (−∞, −
1
4
] (B) (−∞,
1
4
) (C) [−
1
4
, ∞) (D)[−
1
4
1
4
]
- The solution of the differential equation 𝑥𝑑𝑥 + 𝑦𝑑𝑦 = 0 represents a family of
(A) straight lines (B) parabolas (C) Circles (D) Ellipses
*Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e.
- For Visually Impaired:
If 𝑍 = 𝑎𝑥 + 𝑏𝑦 + 𝑐, where 𝑎, 𝑏, 𝑐 > 0 , attains its maximum value at two of its
corner points (4,0) and (0,3) of the feasible region determined by the system of
linear inequalities, then
(A) 4 𝑎 = 3 𝑏 (B) 3 𝑎 = 4 𝑏 (C) 4 𝑎 + 𝑐 = 3 𝑏 (D) 3 𝑎 + 𝑐 = 4 𝑏
- The feasible region of a linear programming problem is bounded but the
objective function attains its minimum value at more than one point. One of the
points is (5,0).
Then one of the other possible points at which the objective function attains its
minimum value is
(A) (2,9) (B) (6,6) (C) (4,7) (D) (0,0)
For Visually Impaired:
The graph of the inequality 3 𝑥 + 5 𝑦 < 10 is the
(A) Entire 𝑋𝑌 −plane
(B) Open Half plane that doesn’t contain origin
(C) Open Half plane that contains origin, but not the points of the line 3 𝑥 +
(D) Half plane that contains origin and the points of the line 3 𝑥 + 5 𝑦 = 10
- A person observed the first 4 digits of your 6-digit PIN. What is the probability
that the person can guess your PIN?
(A)
1
81
(B)
1
100
(C)
1
90
(D) 1
*Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e.
ASSERTION-REASON BASED QUESTIONS
(Question numbers 19 and 20 are Assertion-Reason based questions
carrying 1 mark each. Two statements are given, one labelled Assertion (A)
and the other labelled Reason (R). Select the correct answer from the
options (A), (B), (C) and (D) as given below.)
(A) Both (A) and (R) are true and (R) is the correct explanation of (A).
(B) Both (A) and (R) are true but (R) is not the correct explanation of (A).
(C) (A) is true but (R) is false.
(D) (A) is false but (R) is true.
Assertion (A): Value of the expression sin
− 1
√ 3
2
) + tan
− 1
1 − sec
− 1
2 ) is
𝜋
4
Reason (R): Principal value branch of sin
− 1
𝑥 is [−
𝜋
2
𝜋
2
] and that of s𝑒𝑐
− 1
is [ 0 , 𝜋] − {
𝜋
2
Assertion(A): Given two non-zero vectors 𝑎⃗ and 𝑏
. If 𝑟⃗ is another non-zero
vector such that 𝑟⃗ × (𝑎⃗ + 𝑏
. Then 𝑟⃗ is perpendicular to 𝑎⃗ × 𝑏
Reason (R): The vector (𝑎⃗ + 𝑏
) is perpendicular to the plane of 𝑎⃗ and 𝑏
SECTION B
This section comprises of 5 very short answer (VSA) type questions of 2 marks each.
21A
21B
Evaluate tan (tan
− 1
𝜋
3
OR
Find the domain of cos
− 1
If 𝑦 = log tan (
𝜋
4
𝑥
2
), then prove that
𝒅𝒚
𝒅𝒙
23A
23B
Find: ∫
( 𝑥− 3
)
( 𝑥− 1
)
3
𝑥
OR
Find out the area of shaded region in the enclosed figure.
*Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e.
Solve graphically:
Maximise 𝑍 = 2 𝑥 + 𝑦 subject to
𝑥
2
For Visually Impaired:
The objective function 𝑍 = 3 𝑥 + 2 𝑦 of a linear programming problem under
some constraints is to be maximized and minimized. The corner points of the
feasible region are 𝐴( 600 , 0 ), 𝐵( 1200 , 0 ), 𝐶( 800 , 400 ) and 𝐷( 400 , 200 ). Find the
point at which 𝑍 is maximum and the point at which 𝑍 is minimum. Also, find the
corresponding maximum and minimum values of 𝑍.)
- Two students Mehul and Rashi are seeking admission in a college. The
probability that Mehul is selected is 0.4 and the probability of selection of exactly
one of the them is 0.5. Chances of selection of them is independent of each
other. Find the chances of selection of Rashi. Also find the probability of selection
of at least one of them.
SECTION D
This section comprises of 4 long answer (LA) type questions of 5 marks each
For two matrices 𝐴 = [
] and 𝐵 = [
], find the product 𝐴𝐵
and hence solve the system of equations:
33A
33B
Evaluate: ∫
log( 1 +𝑥)
1 +𝑥
2
1
0
OR
Find ∫
( 3 sin 𝜃− 2 ) cos 𝜃
5 −𝑐𝑜𝑠
2
𝜃− 4 sin 𝜃
34A
34B
Solve the differential equation: 𝑦 +
𝑑
𝑑𝑥
( 𝑥𝑦 ) = 𝑥 (sin 𝑥 + 𝑥)
OR
Find the particular solution of the differential equation:
𝑥
𝑦
⁄
𝑥
𝑦
⁄
) 𝑑𝑦 = 0 given that 𝑦
*Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e.
The two lines
𝑥− 1
3
= −𝑦 , 𝑧 + 1 = 0 and
−𝑥
2
𝑦+ 1
2
= 𝑧 + 2 intersect at a point
whose y-coordinate is 1. Find the co-ordinates of their point of intersection. Find
the vector equation of a line perpendicular to both the given lines and passing
through this point of intersection.
SECTION- E
This section comprises of 3 case-study/passage-based questions of 4 marks each with
subparts. The first two case study questions have three subparts (I), (II), (III) of marks 1,
1, 2 respectively. The third case study question has two subparts of 2 marks each
- Case Study - 1
A city’s traffic management department is planning to optimize traffic flow by
analyzing the connectivity between various traffic signals. The city has five major
spots labelled 𝐴, 𝐵, 𝐶, 𝐷, 𝑎𝑛𝑑 𝐸.
The department has collected the following data regarding one-way traffic flow
between spots:
- Traffic flows from 𝐴 to 𝐵, 𝐴 to 𝐶, and 𝐴 to 𝐷.
- Traffic flows from 𝐵 to 𝐶 and 𝐵 to 𝐸.
- Traffic flows from 𝐶 to 𝐸.
- Traffic flows from 𝐷 to 𝐸 and 𝐷 to 𝐶.
The department wants to represent and analyze this data using relations and
functions. Use the given data to answer the following questions:
I. Is the traffic flow reflexive? Justify. [1]
II. Is the traffic flow transitive? Justify. [1]
III A. Represent the relation describing the traffic flow as a set of ordered pairs.
Also state the domain and range of the relation.
OR
III B. Does the traffic flow represent a function? Justify your answer. [2]
*Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e.
- Case Study - 3
Excessive use of screens can result in vision problems, obesity, sleep disorders,
anxiety, low retention problems and can impede social and emotional
comprehension and expression. It is essential to be mindful of the amount of
time we spend on screens and to reduce our screen-time by taking regular
breaks, setting time limits, and engaging in non-screen-based activities.
In a class of students of the age group 14 to 17, the students were categorised
into three groups according to a feedback form filled by them. The first group
constituted of the students who spent more than 4 hours per day on the mobile
screen or the gaming screens, while the second group spent 2 to 4 hours /day
on the same activities. The third group spent less than 2 hours /day on the same.
The first group with the high screen time is 60% of all the students, whereas the
second group with moderate screen time is 30% and the third group with low
screen time is only 10% of the total number of students. It was observed that
80% students of first group faced severe anxiety and low retention issues, with
70% of second group, and 30% of third group having the same symptoms.
I. What is the total percentage of students who suffer from anxiety and low
retention issues in the class? [2]
II. A student is selected at random, and he is found to suffer from anxiety
and low retention issues. What is the probability that he/she spends
screen time more than 4 hours per day? [2]
