Class 12 Maths · 3 hours · 80 marks

CBSE Class 12 Maths Sample Paper 1

Set in the pattern of CBSE's 2026-27 sample question papers. Write it in 3 hours, then check with the marking scheme and solutions.

This question paper contains 38 questions. All questions are compulsory.
Section A has 20 questions of 1 mark each: 18 multiple-choice and 2 assertion-reason. Section B has 5 very short answer questions of 2 marks, Section C 6 short answer questions of 3 marks, Section D 4 long answer questions of 5 marks and Section E 3 case study-based questions of 4 marks.
There is no overall choice. Internal choice is given in 2 questions in Section B, 3 in Section C, 2 in Section D and one part each of 2 questions in Section E; attempt only one of the alternatives.
Calculators are not allowed.
For the assertion-reason questions, choose: (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.

Section A (20 × 1 mark)

Question 1 · 1 mark

The vector 12î + yĵ − 4k̂, with y > 0, has magnitude 13. The value of y is

(a) 5(b) 9(c) 1(d) 3

Question 2 · 1 mark

In a linear programming problem with decision variables x and y, which of the following is a non-negative restriction?

(a) x + y ≤ 12(b) x − 2y ≥ 0(c) Z = 5x + 3y(d) y ≥ 0

Question 3 · 1 mark

∫₁³ (2x + 1) dx equals

(a) 9(b) 8(c) 12(d) 10

Question 4 · 1 mark

For a value of x at which sin x ≠ 0, the expression (sin x)⁻¹ means

(a) −sin x(b) sin⁻¹ x(c) cosec x(d) sin(−x)

Question 5 · 1 mark

Let R be the feasible region of a linear programming problem and Z = ax + by the objective function. If R is bounded, then

(a) Z has a maximum value on R but may have no minimum value(b) the maximum of Z occurs only at a point inside R, not on its boundary(c) Z may have neither a maximum nor a minimum value on R(d) Z has both a maximum and a minimum value on R, each occurring at a corner point of R

Question 6 · 1 mark

The principal value branch of which of the following functions has the open interval (0, π) as its range?

(a) cos⁻¹(b) cot⁻¹(c) sec⁻¹(d) tan⁻¹

Question 7 · 1 mark

If x + y32x − y = 9321, then xy equals

(a) 1(b) 9(c) 20(d) 16

Question 8 · 1 mark

The area of the region bounded by y = |x − 2|, the x-axis and the ordinates x = 0 and x = k, where k > 2, is 10 square units. Then k equals

(a) 6(b) 4(c) 2 + 2√5(d) 10

Question 9 · 1 mark

The degree of the differential equation (y″)² + (y′)³ − 7y = 0 is

(a) 1(b) 3(c) 2(d) 5

Question 10 · 1 mark

If (p − 2)î + 3ĵ − qk̂ = 5î + (r + 1)ĵ + 4k̂, then p + q + r equals

(a) 9(b) 13(c) 5(d) 1

Question 11 · 1 mark

A and B are mutually exclusive events with P(A) = 0.3 and P(B) = 0.4. Which of the following is true?

(a) A and B are independent(b) P(A ∩ B) = 0.12(c) P(A|B) = 0(d) P(A ∪ B) = 0.58

Question 12 · 1 mark

The number of relations in the set {1, 2, 3} that are both reflexive and symmetric is

(a) 6(b) 64(c) 8(d) 512

Question 13 · 1 mark

The line joining the origin O to the point P(1, 1, √2) makes with the positive direction of the z-axis an angle of

(a) 60°(b) 45°(c) 30°(d) 90°

Question 14 · 1 mark

The function f(x) = x³ − 8x − 2 for x ≠ 2, and f(2) = k, is continuous at x = 2. The value of k is

(a) 4(b) 6(c) 12(d) 0

Question 15 · 1 mark

A jar at a school fair holds n coupons, exactly 2 of which win a prize. Two coupons are drawn one after the other without replacement. If the probability that both coupons win is 115, then n is

(a) 5(b) 15(c) 6(d) 30

Question 16 · 1 mark

A and B are independent events with P(A) = 0.3 and P(A ∩ B) = 0.12. Then P(B′) is

(a) 0.6(b) 0.4(c) 0.88(d) 0.036

Question 17 · 1 mark

The critical points of f(x) = x³ − 27x are

(a) x = 3 only(b) x = −3 and x = 3(c) x = 0 and x = 3(d) x = 0 only

Question 18 · 1 mark

Two lines in space that are neither parallel nor intersecting are called

(a) coplanar lines(b) skew lines(c) concurrent lines(d) perpendicular lines

Question 19 · 1 mark

Assertion (A): The area of the region bounded by y = −x², the x-axis and the ordinates x = 0 and x = 3 is 9 square units. Reason (R): If f(x) ≤ 0 on [a, b], the area bounded by y = f(x), the x-axis and the ordinates x = a and x = b is |∫_a^b f(x) dx|.

(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.

Question 20 · 1 mark

Assertion (A): The differential equation (y‴)² + log (y′) = x has order 3, and its degree is not defined. Reason (R): The degree of a differential equation is defined only when it is a polynomial equation in its derivatives.

(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.

Section B (5 × 2 marks)

Question 21 · 2 marks

Find the general solution of the differential equation 3y² dydx = sec² x.

Or Find the general solution of the differential equation dydx = x e^−2y.

Question 22 · 2 marks

Let f : R → R, f(x) = x² − 4x, and let R = {(a, b) : f(a) = f(b)} in R, an equivalence relation. Find the equivalence class [1].

Or Let g : Z → Z, g(x) = |x − 2|, and let R = {(a, b) : g(a) = g(b)} in Z, an equivalence relation. Find the equivalence class [5].

Question 23 · 2 marks

For events A and B, P(A) = 0.5, P(B) = 0.4 and P(A ∪ B) = 0.7. Find P(A′|B′).

Question 24 · 2 marks

A line through the origin O makes angles of 120° and 60° with the positive directions of the x-axis and the y-axis, and an acute angle with the positive direction of the z-axis. Find the coordinates of the point P on this line, on the side of that acute angle, with OP = 4 units.

Question 25 · 2 marks

Find the values of x for which x62x = 45−12.

Section C (6 × 3 marks)

Question 26 · 3 marks

Find the value of sin(2 tan⁻¹(1/3)) + cos(tan⁻¹(2√2)).

Or Find the value of sin(2 tan⁻¹(1/4)) − cos(tan⁻¹ √15).

Question 27 · 3 marks

Maximise Z = 4x + 3y subject to x + y ≤ 8, x ≤ 5, y ≤ 6, x ≥ 0, y ≥ 0, by the corner point method. By how much would the maximum value increase if the constraint x ≤ 5 were removed?

Or Maximise Z = 2x + 5y subject to x + y ≤ 9, x ≤ 7, y ≤ 6, x ≥ 0, y ≥ 0, by the corner point method. By how much would the maximum value increase if the constraint y ≤ 6 were removed?

Question 28 · 3 marks

Show that f(x) = x² for x ≤ 1 and f(x) = 2x − 1 for x > 1 is differentiable at x = 1.

Or Show that f(x) = x² for x ≤ 1 and f(x) = 3x − 2 for x > 1 is continuous but not differentiable at x = 1.

Question 29 · 3 marks

Using vectors, prove that the diagonals of a rhombus are perpendicular to each other.

Question 30 · 3 marks

A is a symmetric matrix and B a skew symmetric matrix of the same order. Prove that AB + BA is skew symmetric and AB − BA is symmetric.

Question 31 · 3 marks

Arjun remembers every digit of his phone's PIN except the last one. He tries digits from 0 to 9 at random for the last place, never repeating a digit he has tried. Find the probability that he unlocks the phone within his first three attempts.

Section D (4 × 5 marks)

Question 32 · 5 marks

(a) Find where the curve y = 2 cos x − 1 meets the x-axis for 0 ≤ x ≤ π. (b) Find the area of the region bounded by the curve, the x-axis and the ordinates x = 0 and x = π. (c) Show that ∫₀^π (2 cos x − 1) dx = −π and say what this number measures.

Or (a) Find where the curve y = 1 − 2 sin x meets the x-axis for 0 ≤ x ≤ π. (b) Find the area of the region bounded by the curve, the x-axis and the ordinates x = 0 and x = π. (c) Show that ∫₀^π (1 − 2 sin x) dx = π − 4 and say what this number measures.

Question 33 · 5 marks

(a) Using ∫₀^a f(x) dx = ∫₀^a f(a − x) dx, show that ∫₀^π x g(sin x) dx = π2 ∫₀^π g(sin x) dx for a continuous function g. (b) Hence evaluate ∫₀^π x sin x cos² x dx.

Or Evaluate ∫₀^π/2 sin² xsin x + cos x dx.

Question 34 · 5 marks

For f(x) = x⁴ − 4x³ − 8x² + 48x + 1: (a) find the intervals in which f is increasing and in which it is decreasing; (b) find the points of local maxima and local minima and the local maximum and minimum values.

Question 35 · 5 marks

A cube has one vertex at the origin O and the edges OA, OC and OD along the positive x, y and z-axes, each of length 2 units. F is the vertex opposite O.
(a) Find the angle between the diagonal OF and the face diagonal AC.
(b) Find the shortest distance between OF and the edge through A parallel to OD.
(c) Find the shortest distance between OF and AC.

Section E (3 × 4 marks)

Question 36 · 4 marks

A dairy cooperative near Anand sells milk, curd and ghee at two outlets. Daily sales are recorded in the matrix S, rows for Outlet 1 and Outlet 2, columns for milk (litres), curd (kg) and ghee (kg). The selling prices per unit form the column matrix P and the cost prices per unit the column matrix C, in rupees, in the same order.

Matrix	Entries
S	[[120, 40, 10], [90, 60, 15]]
P	[[56], [80], [600]]
C	[[48], [65], [520]]

(a) The order of the revenue matrix SP is [1]

(a) order 2 × 1(b) order 1 × 2(c) order 3 × 1(d) order 2 × 3

(b) The daily revenue of Outlet 2 is [1]

(a) ₹18,840(b) ₹15,920(c) ₹34,760(d) ₹2,820

Or Find the daily cost matrix SC, and use it with SP to find the profit of each outlet. Which law of matrix multiplication shows this equals S(P − C)?

Question 37 · 4 marks

A housing society in Jaipur fills its overhead tank with a pump whose output varies as the motor warms up. The caretaker notes that t minutes after switching on, water enters at the rate r(t) = 3t² − 12t + 15 litres per minute, for 0 ≤ t ≤ 6. The volume added in the first x minutes is the area function V(x) = ∫₀^x r(t) dt, and the volume added between two times is the definite integral of r over that interval.

(a) V′(2) equals [1]

(a) 3(b) 14(c) 0(d) 15

(b) The volume added in the first 3 minutes, in litres, is [1]

(a) 18(b) 27(c) 12(d) 45

Or Find the average rate of inflow over the first 6 minutes, 16 ∫₀⁶ r(t) dt.

Question 38 · 4 marks

A carpenter in Jodhpur is making a wooden gate for an arched opening in a haveli wall. Taking the left foot of the opening as the origin and measuring in metres, the floor is the x-axis and the top edge of the opening follows y = 3 sin πx6 for 0 ≤ x ≤ 6, as in the figure. The opening is the region bounded by this curve and the x-axis. To estimate the timber, he treats the opening as a large number of thin vertical strips and adds their areas.

(a) The width of the opening at a height of 1.5 m, in metres, is [1]

(a) 6(b) 3(c) 4(d) 2

(b) The elementary area of a vertical strip of width dx at a distance x metres from the origin is [1]

(a) x dy(b) 3 sin πx6 dx(c) πx6 dx(d) 3x dx