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CBSE Class 12 Maths starter questions: weeks 32–46

30 short questions to open a lesson for CBSE Class 12 Mathematics, weeks 32–46 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 32

Question 1: Theorem of total probability
Riya travels to college by Metro Line A on 70% of days and by Line B on the rest. She is late on 10% of Line-A days and 20% of Line-B days. The probability that she is late on a given day is (a) 0.13 (b) 0.15 (c) 0.30 (d) 0.07
Question 2: Conditional probability
A and B are events with P(A ∩ B) ≠ 0 and P(A | B) = P(B | A). Then (a) P(A ∩ B) = P(A)P(B) (b) A ⊂ B (c) A and B are independent (d) P(A) = P(B)
Question 3: Feasible region and corner points
Shade the region where 2x + 3y ≤ 12, x ≥ 0 and y ≥ 0. Its vertices (corner points) are (a) (0, 0), (4, 0), (0, 6) (b) (6, 0), (0, 4) (c) (0, 0), (6, 0), (0, 4) (d) (0, 0), (6, 0), (0, 4), (6, 4)

Week 34

Question 1: Random variable and its probability distribution
Which of the following can be the probability distribution of a random variable taking three values? (a) 0.3, 0.3, 0.3 (b) 0.4, 0.4, 0.4 (c) 0.5, -0.1, 0.6 (d) 0.2, 0.5, 0.3
Question 2: Feasible region and corner points
Which of the following points lies in the feasible region of x + 2y ≤ 8, 3x + y ≤ 9, x ≥ 0, y ≥ 0? (a) (3, 1) (b) (2, 3) (c) (0, 5) (d) (4, 0)
Question 3: Direction cosines and direction ratios
For the vector 2i + 3j - 6k, the cosines of the angles it makes with the x-, y- and z-axes are (a) 2/7, 3/7, -6/7 (b) 2, 3, -6 (c) 2/49, 3/49, -6/49 (d) -2/7, -3/7, 6/7

Week 35

Question 1: Mean of a random variable
A random variable X takes the values -1, 0 and 2 with probabilities 0.3, 0.2 and 0.5. The mean of X is (a) 1 (b) 0.3 (c) 0.7 (d) 1/3
Question 2: Equation of a line (vector and Cartesian)
The vector equation of the line (x - 2)/3 = (1 - y)/2 = z/(-1) is (a) r = 2i - j + λ(3i + 2j - k) (b) r = 2i + j + λ(3i + 2j - k) (c) r = 2i + j + λ(3i - 2j - k) (d) r = 3i - 2j - k + λ(2i + j)
Question 3: Graphical solution: bounded region
The corner points of the bounded feasible region of an LPP are (0, 0), (5, 0), (3, 4) and (0, 6). The maximum value of Z = 3x + 2y is (a) 17 (b) 15 (c) 12 (d) 18

Week 36

Question 1: Random variable and its probability distribution
A random variable X has P(X = x) = k(x + 1) for x = 0, 1, 2, 3 (and 0 otherwise). Then P(X ≥ 2) equals (a) 7/10 (b) 3/10 (c) 1/2 (d) 4/10
Question 2: Vectors: magnitude, direction and types
A displacement is d = 2i - 3j + 6k metres. How far is the end point from the start, in metres? (a) 49 (b) 5 (c) √11 (d) 7
Question 3: Direction cosines and direction ratios of a line
Which of the following can be the angles that a line makes with the positive x-, y- and z-axes? (a) 45°, 60°, 60° (b) 30°, 45°, 60° (c) 60°, 60°, 60° (d) 90°, 45°, 30°

Week 39

Question 1: Special cases: multiple optima and infeasibility
The system of constraints y ≥ 2x + 3, 2x - y ≥ 1, x ≥ 0, y ≥ 0 has (a) an unbounded feasible region (b) exactly one feasible solution (c) a bounded feasible region (d) no feasible solution
Question 2: Multiplication theorem on probability
Riya takes two cards, one after the other and without putting the first back, from a shuffled pack of 52. What is the probability that the first is a king and the second is a queen? (a) 4/663 (b) 1/169 (c) 2/663 (d) 8/663
Question 3: Domain and range (principal value branches)
Which of the following inverse trigonometric values is not defined? (a) sec⁻¹(1/2) (b) tan⁻¹(100) (c) sin⁻¹(√3 - 1) (d) cos⁻¹(-1)

Week 40

Question 1: Matrix, order and construction
In the matrix A = 2 -1 5; 0 3 4, the element a_23 is (a) 3 (b) 5 (c) 4 (d) 0
Question 2: Area of a triangle using determinants
The area of the triangle with vertices (-2, 1), (3, 4) and (5, -1) is (a) 19/2 sq units (b) 31 sq units (c) 29/2 sq units (d) 31/2 sq units
Question 3: Second order derivatives
The second derivative of y = 5x⁴ - 2x³ + x with respect to x is (a) 60x² - 12x + 1 (b) 20x³ - 6x² + 1 (c) 60x² - 12x (d) 120x - 12

Week 43

Question 1: Maxima and minima (local)
The function f(x) = xe^(-x) has a local maximum value equal to (a) e (b) 1/e (c) 1 (d) -1/e
Question 2: Integration by substitution
/(xlog_e x) (for x > 1) equals (a) log|log x| + C (b) ((log x)²)/2 + C (c) 1/(log x) + C (d) xlog x + C
Question 3: Area using horizontal strips
Using horizontal strips, the area of the region bounded by the parabola x = y² and the line x = 4 is (a) 32/3 (b) 16/3 (c) 16 (d) 64/3

Week 44

Question 1: Order and degree
The sum of the order and the degree of the differential equation ((d³y)/(dx³))² + ((d²y)/(dx²))⁵ + dy/dx = sin x is (a) 7 (b) 8 (c) 5 (d) 6
Question 2: Vectors: magnitude, direction and types
For non-zero vectors a and b, |a + b| = |a| + |b| holds if and only if (a) a and b are perpendicular (b) a and b have the same direction (c) a and b have opposite directions (d) |a| = |b|
Question 3: Angle between two lines
The lines (x - 1)/2 = (y + 2)/k = z/3 and x/k = (y - 1)/1 = (z + 4)/(-2) are perpendicular for k = (a) 3/2 (b) -2 (c) 6 (d) 2

Week 45

Question 1: Graphical solution: unbounded region
The minimum value of Z = x + y subject to x + 2y ≥ 8, 3x + 2y ≥ 12, x ≥ 0, y ≥ 0 is (a) 8 (b) 6 (c) 5 (d) does not exist
Question 2: Multiplication theorem on probability
Three penalty takers score independently with probabilities 0.9, 0.8 and 0.5. The probability that exactly two of them score is (a) 0.41 (b) 0.36 (c) 0.49 (d) 0.58
Question 3: Principal values
The value of cot⁻¹(-1) + sin⁻¹(-(√3)/2) is (a) (π)/12 (b) (11π)/12 (c) -(π)/12 (d) (5π)/12

Week 46

Question 1: Types of matrices and equality
If 2x - y 3; x + y z = 5 3; 7 2z - 4, then x + y + z equals (a) 9 (b) 11 (c) 15 (d) 7
Question 2: Results on adjoint, inverse and |kA|
A and B are square matrices of order 3 with |A| = 2 and |B| = -3. Then |2A⁻¹B^T| equals (a) -12 (b) -3 (c) 12 (d) -48
Question 3: Implicit differentiation
Differentiating the circle x² + y² = 25 implicitly gives dy/dx = (a) (25 - x²)/y (b) x/y (c) -y/x (d) -x/y

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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