CBSE Class 12 Mathematics Exam 2025 Most Repeated Questions

Chapter -1 Relations and Functions

1

Let A = {1, 2, 3, 4} and define a relation R on A by R = {(a, b) : a – b is even}. Prove that R is an equivalence relation.

2

In the set R – {-1}, define the operation * by a * b = a + b + ab. Determine the identity element and the inverse of an element a under this operation.

3

Given functions f: R → R defined by f(x) = 2x + 3 and g: R → R defined by g(x) = x², find (g ∘ f)(x) and (f ∘ g)(x)

Chapter -2 Inverse Trigonometric Functions

4

Simplify the expression: sin⁻¹(3/5) + cos⁻¹(4/5).

5

Solve for x: sin⁻¹(x) + sin⁻¹(2x) = π/2.

6

Prove that sin⁻¹(x) = 2 tan⁻¹(x / (1 + √(1 – x²))) for -1 < x < 1.

Chapter -3 Matrices

7

If A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]], find AB and BA. Are they equal?

8

Let A and B be two invertible matrices of the same order. Show that (AB)⁻¹ = B⁻¹ A⁻¹.

9

Find the inverse of A = [[1, 2], [3, 4]] using the adjoint method.

Chapter -4 Determinants

10

Find the area of a triangle whose vertices are (1, 2), (3, 4), and (5, 6) using determinants.

11

If A is a square matrix such that |A| ≠ 0, prove that A⁻¹ = adj(A) / |A|.

12

Prove that the determinant of a skew-symmetric matrix of odd order is always zero.

Chapter -5 Continuity and Differentiability.

13

Prove that the function f(x) = x³ - 6x² + 9x is continuous for all x.

14

Find the derivative of f(x) = sin(x²) using the chain rule.

15

Find the derivative of f(x) = tan⁻¹(√(1 + x²) – 1) with respect to x.

Chapter -6 Applications of Derivatives.

16

Find the equation of the tangent to the curve y = x² + 3x – 5 at x = 2.

17

A rectangular sheet of paper of dimensions 12 cm × 8 cm is rolled to form a cylinder. Find the maximum volume of the cylinder that can be formed.

18

Find the approximate change in the volume of a sphere when its radius increases from 5 cm to 5.1 cm.

Chapter -7 Integrals

19

Evaluate: ∫ e^(2x) sin(3x) dx using integration by parts.

20

Find the area bounded by the curve y = x² and the x-axis from x = 0 to x = 2.

21

Evaluate: ∫ (sec²x – cosec²x) dx.

Chapter -8 Application of Integrals.

22

Find the area bounded by the curve y = x² and the x-axis between x = 0 and x = 3.

23

Determine the area of the region enclosed between the lines y = 2x + 3, y = -x + 5, and the x-axis.

24

Find the area of the region enclosed between the curves y = x³ and y = x.

Chapter -9 Differential Equations

25

Solve the differential equation: (dy/dx) = (x + y + 1)².

26

Find the particular solution of dy/dx = x² - y², given that y(0) = 1.

27

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

Chapter -10 Vector algebra

28

If a = (2, -1, 3) and b = (-3, 4, 2), find a unit vector perpendicular to both a and b.

29

Find a vector of magnitude 7 in the direction of a = (2, -1, 2).

30

.Find the value of λ if the vectors a = (λ, 2, 3) and b = (4, 6, -1) are perpendicular.

Chapter -11 Three dimensional geometry

31

If the equation of a plane is x + 2y – 2z + 5 = 0, find the direction cosines of its normal.

32

Find the equation of a plane that passes through the intersection of the planes x– y + 2z – 3 = 0 and 2x + y – z + 5 = 0, and also passes through the point (1, 1, 1).

33

Find the equation of the plane containing the line (x – 2)/3 = (y + 1)/-2 = (z – 4)/1 and parallel to the vector (1, -1, 2).

Chapter -12 Linear Programming

34

A factory produces two types of products, A and B. Each unit of A requires 2 hours of labor and 3 units of raw material, while each unit of B requires 3 hours of labor and 2 units of raw material. The total available labor is 18 hours, and
the total available raw material is 12 units. Find the maximum number of products that can be produced using linear programming.

35

Solve graphically the following LPP:
Maximize Z = 3x + 2y
Subject to:
X + 2y ≤ 8
3x + y ≤ 6
X, y ≥ 0

36

A company produces two products, P and Q. The profit per unit of P is ₹30, and for Q is ₹40. The company can manufacture at most 50 units in total, and the available budget is ₹5000. The cost of producing one unit of P is ₹50, and one unit of Q is ₹70. Formulate the problem as an LPP.

Chapter -13 Probability

37

A bag contains 3 red, 4 blue, and 5 green balls. Two balls are drawn randomly. Find the probability that both are of the same color.

38

In a factory, machines A, B, and C produce 30%, 45%, and 25% of the total products, respectively. The probability of a defective product from these machines is 0.02, 0.03, and 0.05, respectively. If a product is randomly selected and found to be defective, find the probability that it was produced by machine B.

39

A box contains 8 defective and 12 non-defective bulbs. If two bulbs are drawn at random, find the probability that at least one bulb is defective.