CBSE BOARD EXAMINATION 2025–26

Subject: Mathematics Subject Code: 041 Class: XII Time Allowed: 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 Multiple Choice Questions (MCQs) and 2 Assertion-Reason (A-R) 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 Case Study Based questions of 4 marks each. Internal choice is provided in 2 marks questions only.
There is no overall choice. However, internal choice has been provided in 2 questions in Section B, 2 questions in Section C, all 4 questions in Section D and one sub-part each in the 3 questions of Section E.
Draw neat figures wherever required. Take π = 22/7 wherever required if not stated.
Use of calculators is not permitted.

SECTION A

This section comprises Multiple Choice Questions (MCQs) of 1 mark each and Assertion-Reason based questions of 1 mark each.

Q.1. Let f : ℝ → ℝ be defined by f(x) = 3x + 2. Which of the following is true?

(A) f is injective but not surjective
(B) f is surjective but not injective
(C) f is bijective
(D) f is neither injective nor surjective

Q.2. The principal value of sin⁻¹(−√3/2) is:

(A) π/3
(B) −π/3
(C) 2π/3
(D) −2π/3

Q.3. If A is a square matrix of order 3 and |A| = 5, then |adj A| is:

(A) 5
(B) 25
(C) 125
(D) 625

Q.4. If A = [[2, 3], [1, −1]] and B = [[1, 0], [2, 4]], then (AB)ᵀ is:

(A) [[8, −1], [12, −4]]
(B) [[8, 12], [−1, −4]]
(C) [[2, 6], [3, −4]]
(D) [[2, 3], [6, −4]]

Q.5. If f(x) = x² sin(1/x) for x ≠ 0 and f(0) = 0, then at x = 0, f is:

(A) Continuous but not differentiable
(B) Differentiable but not continuous
(C) Both continuous and differentiable
(D) Neither continuous nor differentiable

Q.6. The value of dy/dx at (1, 1) for the curve x² + y² = 2 is:

(A) 1
(B) −1
(C) 0
(D) 2

Q.7. The function f(x) = 2x³ − 3x² − 12x + 4 is strictly increasing on the interval:

(A) (−2, 1)
(B) (−∞, −1) ∪ (2, ∞)
(C) (−1, 2)
(D) (−∞, −2) ∪ (1, ∞)

Q.8. ∫ eˣ (sin x + cos x) dx is equal to:

(A) eˣ sin x + C
(B) eˣ cos x + C
(C) eˣ (sin x − cos x) + C
(D) − eˣ cos x + C

Q.9. ∫₀¹ x(1 − x)ⁿ dx is equal to:

(A) 1/[(n+1)(n+2)]
(B) 1/(n+2)
(C) n!/[(n+2)!]
(D) 1/(n+1)

Q.10. The area of the region bounded by the curve y = x² and the line y = 4 is:

(A) 32/3 sq. units
(B) 16/3 sq. units
(C) 8/3 sq. units
(D) 64/3 sq. units

Q.11. The differential equation dy/dx + y cot x = 2 cos x is a:

(A) Linear differential equation of first order
(B) Variable separable differential equation
(C) Second order differential equation
(D) Non-linear differential equation

Q.12. If a = 2î − ĵ + k̂ and b = î + 3ĵ − 2k̂, then a × b is:

(A) −î + 5ĵ + 7k̂
(B) î − 5ĵ + 7k̂
(C) î + 5ĵ − 7k̂
(D) −î − 5ĵ + 7k̂

Q.13. The angle between the lines r = (2î − ĵ) + λ(î + 2ĵ − 2k̂) and r = (î + 2ĵ) + μ(2î + ĵ + 2k̂) is:

(A) 0°
(B) 30°
(C) 60°
(D) 90°

Q.14. The direction cosines of a line making equal angles with all three coordinate axes are:

(A) 1/√3, 1/√3, 1/√3
(B) 1/√2, 1/√2, 0
(C) 1, 1, 1
(D) 1/3, 1/3, 1/3

Q.15. The corner points of the feasible region determined by the system of linear constraints are (0, 0), (4, 0), (3, 2), and (0, 5). If the objective function is Z = 3x + 4y, the maximum value of Z is:

(A) 20
(B) 17
(C) 12
(D) 21

Q.16. If P(A) = 2/5, P(B) = 1/3, and P(A ∩ B) = 1/5, then P(A′ | B′) is:

(A) 5/6
(B) 3/4
(C) 2/5
(D) 7/12

Q.17. The value of tan⁻¹(1) + tan⁻¹(√3) is:

(A) 5π/12
(B) 7π/12
(C) π/2
(D) 7π/6

Q.18. If f(x) = |x − 2|, then at x = 2, f is:

(A) Differentiable
(B) Continuous but not differentiable
(C) Discontinuous
(D) Neither continuous nor differentiable

The following questions (Q.19 and Q.20) are Assertion-Reason based. Read both Assertion (A) and Reason (R) carefully and select the correct answer from the options below:

(A) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
(C) Assertion (A) is true but Reason (R) is false.
(D) Assertion (A) is false but Reason (R) is true.

Q.19.

Assertion (A): The function f(x) = |x| is not differentiable at x = 0.

Reason (R): A function is differentiable at a point if and only if the left-hand derivative and right-hand derivative at that point are equal.

Q.20.

Assertion (A): Every square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

Reason (R): For any square matrix A, the matrix (A + Aᵀ)/2 is symmetric and (A − Aᵀ)/2 is skew-symmetric.

SECTION B

This section comprises Very Short Answer (VSA) type questions of 2 marks each.

Q.21. Check whether the relation R = {(a, b) : a − b is divisible by 5} defined on the set A = {1, 3, 5, 7, 9, 11} is an equivalence relation or not. Justify briefly.

Q.22. Find the value of sin⁻¹(sin 5π/6) + cos⁻¹(cos 7π/6).

OR

Prove that: tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3) = π.

Q.23. If y = (sin x)ˣ, find dy/dx.

Q.24. Evaluate: ∫ x/(x² + 1) dx.

OR

Evaluate: ∫₀^(π/2) log(sin x) − log(cos x) dx.

Q.25. Find the unit vector in the direction of the vector a + b, where a = î + 2ĵ − k̂ and b = 2î − 3ĵ + 5k̂.

SECTION C

This section comprises Short Answer (SA) type questions of 3 marks each.

Q.26. Show that the relation R on ℝ defined as R = {(a, b) : a ≤ b²} is neither reflexive nor transitive.

Q.27. Differentiate the following with respect to x:

y = sin⁻¹(2x√(1 − x²)), where −1/√2 < x < 1/√2.

OR

If x = a(θ − sin θ) and y = a(1 − cos θ), find d²y/dx².

Q.28. Evaluate: ∫ (x + 1)/√(2x² + 4x + 3) dx.

Q.29. Find the general solution of the differential equation:

(1 + x²) dy/dx + 2xy = (1/(1 + x²)).

OR

Solve the differential equation: dy/dx = y/x + sin(y/x).

Q.30. Find the probability distribution of the number of heads obtained when three fair coins are tossed simultaneously. Also find the mean of the distribution.

Q.31. Maximise: Z = 4x + 3y

Subject to the constraints:

3x + 4y ≤ 24
8x + 6y ≤ 48
x ≥ 0, y ≥ 0

Find the maximum value of Z and the point at which it is attained.

SECTION D

This section comprises Long Answer (LA) type questions of 5 marks each.

Q.32. Using integration, find the area of the region bounded by the parabola y = x², the line y = x + 2, and the x-axis.

OR

Using integration, find the area of the circle x² + y² = 16, which is exterior to the parabola y² = 6x.

Q.33. Solve the following system of equations using matrices (Matrix method):

x + y + z = 6
x − y + z = 2
2x + y − z = 1

OR

If A = [[1, 2, 3], [3, −2, 1], [4, 2, 1]], show that A³ − 23A − 40I = 0, where I is the identity matrix of order 3. Hence find A⁻¹.

Q.34. Find the equation of the plane passing through the points A(1, 1, 0), B(1, 2, 1), and C(−2, 2, −1). Also find the perpendicular distance from the point D(2, −1, 3) to this plane.

OR

Find the equations of the line passing through the point (2, 1, −1) and perpendicular to each of the lines:

(x − 1)/1 = (y − 2)/2 = (z − 3)/3 and (x + 2)/−3 = (y − 1)/2 = z/−1.

Q.35. A manufacturer produces two types of articles P and Q. Each unit of P requires 3 hours of machine work and 1 hour of skilled labour. Each unit of Q requires 1 hour of machine work and 3 hours of skilled labour. The total available machine hours are 12 and the total available skilled labour hours are 12 per week. Profits are ₹200 per unit of P and ₹300 per unit of Q. Formulate the LPP and solve graphically to determine the number of units of each type to be produced per week for maximum profit, and find the maximum profit.

OR

Solve graphically:

Minimise: Z = 5x + 10y

Subject to:
x + 2y ≤ 120
x + y ≥ 60
x − 2y ≥ 0
x ≥ 0, y ≥ 0

SECTION E

This section comprises Case Study Based questions. Each question is of 4 marks. Internal choice is provided in sub-part (iii) of each question.

Q.36. Case Study — Relations and Functions

A school teacher is conducting a Mathematics workshop on Functions Mapping. She draws two sets A = {1, 2, 3, 4} and B = {a, b, c, d} on the board and asks students to define various functions and verify their properties.

Based on the above situation, answer the following:

(i) How many one-one functions can be defined from A to B? [1 mark]

(ii) Is the function f: A → B defined by f(1) = a, f(2) = b, f(3) = c, f(4) = d a bijection? Give reason. [1 mark]

(iii) Define a relation R on A as R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1)}. Verify whether R is an equivalence relation on A. [2 marks]

OR

(iii) Let g : A → A be defined by g = {(1,2),(2,1),(3,4),(4,3)}. Verify whether g is invertible and find g⁻¹ if it exists.

Q.37. Case Study — Applications of Derivatives

An engineer is designing a cylindrical water tank of fixed volume 1000π cm³ to be fabricated from a thin sheet of metal. To reduce material cost, the engineer wants to minimise the total surface area of the tank (including both circular bases and the curved surface).

Based on the above, answer the following:

(i) Express the total surface area S of the tank in terms of the radius r alone. [1 mark]

(ii) Find dS/dr. [1 mark]

(iii) Find the radius and height of the tank for minimum surface area and verify that it is indeed a minimum. [2 marks]

OR

(iii) A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/s. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?

Q.38. Case Study — Probability and Bayes' Theorem

In a factory, Machine I produces 40% of the total output, Machine II produces 35%, and Machine III produces 25%. Of the output produced: Machine I has a 2% defective rate, Machine II has a 3% defective rate, and Machine III has a 2% defective rate. An item is drawn at random from the day's production and found to be defective.

Based on this information, answer the following:

(i) What is the total probability that a randomly selected item is defective? [1 mark]

(ii) What is the probability that the defective item was produced by Machine II? [1 mark]

(iii) Find the probability that the defective item was produced by Machine I or Machine III. [2 marks]

OR

(iii) Given that the defective item was produced by Machine III, what is the conditional probability that a second independently drawn item (from the full production) is also defective?

— End of Question Paper —

Note to Examinees: Read all instructions carefully before attempting the paper. Write your Roll Number clearly in the space provided on the answer sheet.

Subject: Mathematics (Code: 041) | Class: XII | Maximum Marks: 80 | Time: 3 Hours