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CBSE Class 12 Maths starter questions: weeks 1–18

33 short questions to open a lesson for CBSE Class 12 Mathematics, weeks 1–18 of the teaching year, three or four a week. Each question is shown as an image with its text underneath; the answers and mark schemes are free on the free weekly starters page, to show or project in class.

Week 1

Question 1: Counting relations and functions
The number of onto functions from 1, 2, 3 to p, q is (a) 6 (b) 8 (c) 2 (d) 9
Question 2: Types of relations
On the set ℝ of real numbers, the relation R = (a, b): a ≤ b + 1 is (a) reflexive only (b) symmetric only (c) an equivalence relation (d) transitive only
Question 3: One-one and onto functions
The function f: ℝ → [-1, 1], f(x) = sin x, is (a) one-one but not onto (b) neither one-one nor onto (c) bijective (d) onto but not one-one

Week 2

Question 1: Counting relations and functions
The number of reflexive relations on a set with 3 elements is (a) 8 (b) 64 (c) 512 (d) 27
Question 2: Types of relations
Let A = 1, 2, 3 and let R = (1, 1), (2, 2), (3, 3), (1, 3), (3, 1) be a relation on A. Which statement about R is correct? (a) R is reflexive but not symmetric (b) R is an equivalence relation (c) R is symmetric but not transitive (d) R is transitive but not reflexive
Question 3: One-one and onto functions
The function f: ℤ → ℤ, f(x) = 5 - x, is (a) one-one but not onto (b) neither one-one nor onto (c) bijective (d) onto but not one-one

Week 3

Question 1: Compositions and simplification
The value of tan(cos⁻¹3/5) is (a) 3/4 (b) 4/5 (c) 4/3 (d) 5/3
Question 2: Domain and range (principal value branches)
The range of the principal value branch of sec⁻¹x is (a) [-(π)/2, (π)/2] (b) [0, π] - (π)/2 (c) (0, π) (d) [-(π)/2, (π)/2] - 0
Question 3: One-one and onto functions
The function f: ℕ → ℕ, f(n) = n + 3, is (a) one-one but not onto (b) onto but not one-one (c) both one-one and onto (d) neither one-one nor onto

Week 4

Question 1: Matrix multiplication and its properties
A, B, C are matrices of orders 2 × 3, 3 × 2 and 2 × 2 respectively. Which of the following is defined? (a) AC (b) BA + C (c) AB + C (d) CB
Question 2: Compositions and simplification
The value of sin⁻¹(sin(4π)/5) is (a) (4π)/5 (b) -(π)/5 (c) (3π)/5 (d) (π)/5
Question 3: One-one and onto functions
Let A have 5 elements and B have 4 elements. If f: A → B is onto, then f (a) cannot be one-one (b) must be one-one (c) is bijective (d) cannot exist

Week 5

Question 1: Invertible matrices
If A is an invertible square matrix such that A² = A, then A equals (a) -I (b) O (c) I (d) 2I
Question 2: Matrix multiplication and its properties
A is a square matrix with A² = A and B = I - A. Then AB equals (a) B (b) I (c) A (d) O
Question 3: Principal values
The principal value of sec⁻¹(-2/(√3)) is (a) -(π)/6 (b) (5π)/6 (c) (7π)/6 (d) (2π)/3

Week 6

Question 1: Adjoint and inverse of a matrix
The inverse of A = 2 3; 1 4 is (a) 1/11 4 -3; -1 2 (b) 1/5 4 3; 1 2 (c) 1/5 2 -3; -1 4 (d) 1/5 4 -3; -1 2
Question 2: Evaluating determinants
The value of 3 -2; 4 5 is (a) 23 (b) 7 (c) -23 (d) 15
Question 3: Transpose of a matrix
If A is a matrix of order 3 × 4, the order of A^T is (a) 4 × 3 (b) 3 × 4 (c) 3 × 3 (d) 4 × 4

Week 13

Question 1: Consistency of a system of linear equations
How many solutions does the pair of equations 3x - y = 4, 6x - 2y = 8 have? (a) exactly two (b) exactly one (c) none (d) infinitely many
Question 2: Results on adjoint, inverse and |kA|
If A is a square matrix of order 3 with |A| = 5, then |-2A^T| equals (a) -40 (b) 40 (c) -10 (d) -80
Question 3: Compositions and simplification
The value of cos⁻¹(cos(7π)/4) is (a) (π)/4 (b) -(π)/4 (c) (3π)/4 (d) (7π)/4

Week 14

Question 1: Chain rule and composite functions
d/dx[sin² (3x)] equals (a) sin 6x (b) 2sin 3x (c) 6sin 6x (d) 3sin 6x
Question 2: Continuity
The function f(x) = (sin 5x + sin x)/3x, x ≠ 0; k, x = 0 is continuous at x = 0. The value of k is (a) 1/3 (b) 5/3 (c) 2 (d) 6
Question 3: Types of matrices and equality
The 3 × 3 matrix A = [a_ij] with a_ij = i² - j² is (a) a skew-symmetric matrix (b) a symmetric matrix (c) a diagonal matrix (d) a scalar matrix

Week 16

Question 1: Logarithmic differentiation
The derivative of x^(2x) (x > 0) with respect to x is (a) 2x · x^(2x - 1) (b) 2x^(2x)(1 + log x) (c) x^(2x)(1 + log x) (d) 2x^(2x)log x
Question 2: Derivatives of inverse trigonometric functions
Using the chain rule, differentiate tan⁻¹(3x) with respect to x. The result is (a) 3/(√(1 - 9x²)) (b) 1/(1 + 9x²) (c) 3/(1 + 3x²) (d) 3/(1 + 9x²)
Question 3: Area of a triangle using determinants
Using determinants, the equation of the line joining (2, 3) and (-1, 5) is (a) 3x + 2y = 12 (b) 2x + 3y = 13 (c) 2x - 3y = -5 (d) 3x - 2y = 0

Week 17

Question 1: Increasing and decreasing functions
The function f(x) = x³ - 12x is strictly decreasing on (a) (-∞, 2) (b) (-∞, -2) (c) (2, ∞) (d) (-2, 2)
Question 2: Parametric differentiation
A curve is given by x = t³, y = 3t² (t ≠ 0). Its gradient dy/dx is (a) 2t (b) 2/t (c) t/2 (d) 6/t
Question 3: Domain and range (principal value branches)
The domain of f(x) = sin⁻¹x + cos⁻¹(x - 1) + tan⁻¹x is (a) [-1, 1] (b) [-1, 2] (c) [0, 1] (d) [0, 2]

Week 18

Question 1: Optimisation problems
For x > 0, the least value of f(x) = 4x + 9/x is (a) 3/2 (b) 13 (c) 6 (d) 12
Question 2: Increasing and decreasing functions
The function f(x) = log_e x - x is strictly increasing on (a) (0, ∞) (b) (1, ∞) (c) (0, 1) (d) (-∞, 1)
Question 3: Matrix, order and construction
A matrix has 12 elements. The number of different orders it can have is (a) 6 (b) 4 (c) 3 (d) 12

More starter questions

Other weeks: Weeks 1–18 · Weeks 19–31 · Weeks 32–46. Answers and mark schemes, free for everyone, are on the free weekly starters page.

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