📋 PREDICTED QUESTION PAPER

MATHEMATICS — CODE 041

Class XII | Board Examination 2025–26

📌 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 MCQs and 2 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 Case Study / Source-based questions of 4 marks each.
7. There is no overall choice. However, internal choices are provided in:
   • 2 questions of Section B
   • 3 questions of Section C
   • 2 questions of Section D
   • 2 sub-parts of Section E
8. Use of calculators is NOT allowed.

SECTION A

Multiple Choice Questions — 1 Mark Each

Select the most appropriate answer from the given options.

Q1. The value of sin⁻¹( sin(3π/5) ) is:

• (A) 3π/5
• (B) 2π/5
• (C) π/5
• (D) −3π/5

Q2. If A is a square matrix of order 3 such that |A| = 5, then |adj(A)| is:

• (A) 5
• (B) 25
• (C) 125
• (D) 625

Q3. If A = | 2 3 |, then A⁻¹ equals: | 5 −2 |

• (A) (1/19) × | −2 −3 | | −5 2 |
• (B) (1/19) × | −2 −3 | | −5 −2 |
• (C) (1/19) × | −2 3 | | 5 2 |
• (D) Does not exist

Q4. The function f(x) = x³ − 3x is strictly increasing on:

• (A) (−1, 1)
• (B) (−∞, −1) ∪ (1, ∞)
• (C) (−∞, ∞)
• (D) (0, ∞)

Q5. ∫ dx / [ x(xⁿ + 1) ] equals:

• (A) (1/n) · log| xⁿ / (xⁿ+1) | + C
• (B) log| xⁿ / (xⁿ+1) | + C
• (C) (1/n) · log| (xⁿ+1) / xⁿ | + C
• (D) None of these

Q6. The order and degree of the differential equation:

( d²y/dx² )³ + ( dy/dx )² + sin( dy/dx ) + 1 = 0

are respectively:

• (A) 2 and 3
• (B) 2 and not defined
• (C) 1 and not defined
• (D) 3 and 2

Q7. If a and b are unit vectors and θ is the angle between them, then |a − b| is:

• (A) cos(θ/2)
• (B) 2 cos(θ/2)
• (C) 2 sin(θ/2)
• (D) sin(θ/2)

Q8. The direction ratios of the line joining A(2, 3, −1) and B(3, −2, 1) are proportional to:

• (A) 1, −5, 2
• (B) −1, 5, −2
• (C) 1, 5, −2
• (D) 5, −1, 2

Q9. The corner points of the feasible region of an LPP are (0,0), (4,0), (3,3) and (0,4). If objective function is Z = 3x + 4y, the maximum value of Z is:

• (A) 16
• (B) 12
• (C) 21
• (D) 18

Q10. If P(A) = 3/8, P(B) = 1/2 and P(A∩B) = 1/4, then P(A' | B') is:

• (A) 3/5
• (B) 5/8
• (C) 1/4
• (D) 3/8

Q11. ∫₀^π sin²x dx equals:

• (A) π
• (B) π/2
• (C) 0
• (D) π/4

Q12. If y = e^(tan⁻¹x), then (1 + x²)(d²y/dx²) + (2x − 1)(dy/dx) equals:

• (A) 0
• (B) 1
• (C) y
• (D) 2y

Q13. The number of all possible matrices of order 3×3 with each entry 0 or 1 is:

• (A) 27
• (B) 18
• (C) 512
• (D) 81

Q14. The value of λ for which the vectors 2î − 3ĵ + 4k̂ and λî + 6ĵ − 8k̂ are collinear is:

• (A) 2
• (B) −2
• (C) 4
• (D) −4

Q15. The area of the region bounded by y = x² and y = x is:

• (A) 1/6 sq. units
• (B) 1/3 sq. units
• (C) 1/2 sq. units
• (D) 1/4 sq. units

Q16. If A and B are symmetric matrices of the same order, then AB − BA is:

• (A) Symmetric matrix
• (B) Skew-symmetric matrix
• (C) Zero matrix
• (D) Identity matrix

Q17. The general solution of the differential equation dy/dx = e^(x+y) is:

• (A) eˣ + e⁻ʸ = C
• (B) eˣ − e⁻ʸ = C
• (C) e⁻ˣ + eʸ = C
• (D) eˣ + eʸ = C

Q18. The value of the determinant:

| 1    ω    ω² |
| ω    ω²   1  |
| ω²   1    ω  |

where ω is a cube root of unity, is:

• (A) 0
• (B) 1
• (C) ω
• (D) ω²

Assertion–Reason Questions

In Q19 and Q20, choose the correct option:

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

Q19.

Assertion (A): The relation R = { (a, b) : |a − b| is divisible by 4 } on set A = {1, 2, 3, ..., 10} is an equivalence relation.

Reason (R): A relation is an equivalence relation if it is reflexive, symmetric and transitive.

Q20.

Assertion (A): ∫₀^(2π) |sin x| dx = 4

Reason (R): |sin x| is an even function, so ∫₀^(2π)|sin x| dx = 2 × ∫₀^π |sin x| dx

SECTION B

Very Short Answer — 2 Marks Each

Q21. Find the principal value of:

tan⁻¹(1) + cos⁻¹(−1/2) + sin⁻¹(−1/2)

Q22. Find the interval(s) in which f(x) = sin x + cos x, 0 ≤ x ≤ 2π, is strictly decreasing.

OR

Find the local maximum and local minimum values of f(x) = sin x − cos x, 0 < x < 2π.

Q23. Evaluate:

∫₀^(π/4)  tan²x  dx

Q24. If a = î + ĵ + k̂ and b = ĵ − k̂, find a vector c such that:

**a** × **c** = **b**   and   **a** · **c** = 3

OR

If a + b + c = 0 and |a| = 3, |b| = 5, |c| = 7, find the angle between a and b.

Q25. A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.

SECTION C

Short Answer — 3 Marks Each

Q26. Show that the relation R defined on ℝ by:

R = { (a, b) : a ≤ b² }

is neither reflexive, nor symmetric, nor transitive.

OR

Let A = {1, 2, 3}. Write all equivalence relations on A that contain the pair (1, 2).

Q27. If x = a sin t and y = a( cos t + log tan(t/2) ), find d²y/dx².

OR

If y = (sin x)^x + sin⁻¹(√x), find dy/dx.

Q28. Evaluate:

∫  (x² + 1) / [ (x² + 2)(x² + 3) ]  dx

Q29. Solve the differential equation:

x · (dy/dx) + y − x + xy·cot x = 0,    x ≠ 0

OR

Solve the differential equation:

(x² − y²) dx + 2xy dy = 0,    y(1) = 1

Q30. Evaluate:

∫₀^π   (x · sin x) / (1 + cos²x)   dx

Q31. An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted, and it is returned to the urn. Moreover, 2 additional balls of the colour drawn are added to the urn, and then a second ball is drawn at random.

Find the probability that the second ball drawn is red.

SECTION D

Long Answer — 5 Marks Each

Q32. Using the matrix method, solve the system of equations:

x  −  y  + 2z  =   7
3x + 4y  − 5z  =  −5
2x −  y  + 3z  =  12

OR

If A = | 1 2 −3 | find A⁻¹. Hence solve: | 2 3 2 | | 3 −3 −4 |

x + 2y − 3z = −4
2x + 3y + 2z =  2
3x − 3y − 4z = 11

Q33. Find the area of the region bounded by the ellipse:

x²/4  +  y²/9  =  1

using integration.

OR

Sketch the graph of y = |x + 3| and evaluate ∫₋₆^0 |x + 3| dx.

Hence, find the area of the region bounded by the curve y = |x + 3|, the x-axis, and the lines x = −6 and x = 0.

Q34. Find the shortest distance between the lines:

**r** = (î + 2ĵ + k̂) + λ(î − ĵ + k̂)

**r** = (2î − ĵ − k̂) + μ(2î + ĵ + 2k̂)

Also, find the equation of the line passing through the point (1, 2, −4) and perpendicular to both the above lines.

Q35. A manufacturer produces two types of steel trunks. He has two machines A and B.

Formulate the LPP and find — by graphical method — how many trunks of each type must be made daily to earn maximum profit.

SECTION E

Case Study Based Questions — 4 Marks Each

Q36 | Case Study 1 — Rate of Change

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 m/s.

Based on the above information, answer the following:

(i) (1 mark) If x is the distance of the foot of the ladder from the wall and y is the height of the top of the ladder on the wall, express y in terms of x.

(ii) (1 mark) Find the rate at which the height of the top of the ladder is decreasing when the foot of the ladder is 4 m away from the wall.

(iii) (2 marks) Find the rate at which the area of the triangle formed by the ladder, wall and ground is changing when the foot is 4 m from the wall.

OR (for iii)

Find the rate at which the angle θ made by the ladder with the ground is changing when the foot of the ladder is 4 m from the wall.

Q37 | Case Study 2 — Probability & Bayes' Theorem

In a factory, machines X, Y and Z manufacture bolts in the ratio:

A bolt is drawn at random and is found to be defective.

Based on the above information, answer the following:

(i) (2 marks) Find the total probability that a randomly drawn bolt is defective.

(ii) (1 mark) What is the probability that the defective bolt was manufactured by machine X?

(iii) (1 mark) What is the probability that the defective bolt was manufactured by machine Y or machine Z?

OR (for iii)

If defective bolts are discarded, what is the probability that a non-defective bolt was made by machine Z?

Q38 | Case Study 3 — Application of Derivatives

A farmer wants to build a rectangular garden along one wall of his barn and needs to fence the other three sides. He has 120 metres of fencing available.

Let the side of the garden along the barn wall = 2x metres and width = y metres.

Based on the above information, answer the following:

(i) (1 mark) Express the area A of the garden as a function of x.

(ii) (1 mark) Find the value of x for which the area is maximum.

(iii) (2 marks) Find the maximum area of the garden. Also verify it is indeed a maximum using the second derivative test.

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Paper Code: MATH / XII / 2025–26 / PREDICTED Prepared by: Chief Paper Setter & Senior Examiner — CBSE Trend Analysis Cell Based on: 11-Year Deep Trend Analysis (2014–2025 | Delhi + All India + Compartment Sets)