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CBSE Class 12 Maths Question of the Day: chapters 11–13

30 CBSE Class 12 Mathematics puzzles from the Question of the Day rotation, chapters 11–13. Each puzzle is three board-style questions on one chapter, shown as an image with the question text written out below it; the answers, hints and step-marked solutions are on the daily puzzle page.

Chapter 11: Three Dimensional Geometry

Chapter 11: Three Dimensional Geometry
Three board-style questions on Three Dimensional Geometry (CBSE Class 12 Mathematics). No calculator.
1. The direction cosines of the line through A(1, -2, 3) and B(3, 1, -3), directed from A to B, are Options: A) 2/(√13), 3/(√13), -6/(√13); B) 2, 3, -6; C) -27, -37, 67; D) 27, 37, -67
2. The Cartesian equations of the line through (2, -1, 4) parallel to the y-axis are Options: A) x/2 = y/(-1) = z/4; B) (x - 2)/1 = (y + 1)/0 = (z - 4)/1; C) (x + 2)/0 = (y - 1)/1 = (z + 4)/0; D) (x - 2)/0 = (y + 1)/1 = (z - 4)/0
3. In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. Assertion (A): The lines with direction ratios 1, -2, 2 and 2, 2, 1 are perpendicular. Reason (R): Two lines are perpendicular if and only if their direction ratios are proportional. Options: A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).; B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).; C) Assertion (A) is true but Reason (R) is false.; D) Assertion (A) is false but Reason (R) is true.
Chapter 11: Three Dimensional Geometry · Angles between lines
Three board-style questions on Three Dimensional Geometry (CBSE Class 12 Mathematics). No calculator.
1. The cosine of the angle between lines with direction ratios 2, 2, 1 and 4, 1, 8 is Options: A) 1/3; B) 2/9; C) 4/9; D) 2/3
2. Lines with direction ratios 1, -2, 1 and 2, k, 4 are perpendicular when k equals Options: A) -3; B) 2; C) 3; D) 6
3. The angle between lines with direction ratios 2, -4, 6 and 1, -2, 3 is Options: A) (π)/2; B) (π)/3; C) (π)/4; D) 0
Chapter 11: Three Dimensional Geometry · Points on a line
Three board-style questions on Three Dimensional Geometry (CBSE Class 12 Mathematics). No calculator. Consider the line r = (i - 2 k) + λ(2 i + j + 2 k).
1. The point with λ = 2 is Options: A) (3, 1, 0); B) (5, 2, 2); C) (4, 2, 4); D) (5, 2, 6)
2. The distance of that point from (1, 0, -2) is Options: A) 9; B) 2 √3; C) 3; D) 6
3. Does the point (7, 3, 4) lie on the line? Options: A) No; B) Yes, with λ = 2; C) Yes, with λ = 3; D) Yes, with λ = 4
Chapter 11: Three Dimensional Geometry · Intersecting lines
Three board-style questions on Three Dimensional Geometry (CBSE Class 12 Mathematics). No calculator. L_1: r = (i + j) + λ(i + k) and L_2: r = (2 i + j + k) + μ(j + k).
1. Equating the x- and y-coordinates gives λ equal to Options: A) 1; B) 2; C) -1; D) 0
2. The lines meet at Options: A) (1, 1, 0); B) (2, 1, 1); C) (2, 2, 2); D) they do not meet
3. The shortest distance between L_1 and L_2 is Options: A) 0; B) (√3)/3; C) 1; D) √2
Chapter 11: Three Dimensional Geometry · Skew lines
Three board-style questions on Three Dimensional Geometry (CBSE Class 12 Mathematics). No calculator. L_1: r = i + λ(i + j) and L_2: r = 2 k + μ(i - j).
1. b_1 × b_2 equals Options: A) 0; B) 2 k; C) -2 k; D) 2 i - 2 j
2. (a_2 - a_1) · (b_1 × b_2) equals Options: A) 4; B) -4; C) 2; D) 0
3. The shortest distance between the lines is Options: A) 2; B) 2 √2; C) 4; D) 1
Chapter 11: Three Dimensional Geometry · Parallel lines
Three board-style questions on Three Dimensional Geometry (CBSE Class 12 Mathematics). No calculator. L_1: r = (i + 2 j + 3 k) + λ(2 i + 3 j + 6 k) and L_2: r = (3 i + 3 j + 5 k) + μ(2 i + 3 j + 6 k).
1. The lines are Options: A) intersecting; B) skew; C) parallel and distinct; D) the same line
2. | b| equals Options: A) 7; B) 49; C) √11; D) 11
3. The distance between the lines is Options: A) 4 √5; B) 3; C) (4 √5)/49; D) (4 √5)/7
Chapter 11: Three Dimensional Geometry · Foot of a perpendicular
Three board-style questions on Three Dimensional Geometry (CBSE Class 12 Mathematics). No calculator. P is (1, 2, 3) and L is the line r = λ(i + j + k).
1. The foot of the perpendicular from P to L is Options: A) (2, 2, 2); B) (3, 3, 3); C) (0, 0, 0); D) (1, 1, 1)
2. The distance of P from L is Options: A) 2; B) √2; C) √14; D) 1
3. The image (reflection) of P in L is Options: A) (4, 4, 4); B) (3, 2, 3); C) (3, 2, 1); D) (1, 2, 3)
Chapter 11: Three Dimensional Geometry · Recognising skew lines
Three board-style questions on Three Dimensional Geometry (CBSE Class 12 Mathematics). No calculator. L_1: (x - 1)/2 = (y - 2)/3 = (z - 3)/4 and L_2: (x - 2)/3 = (y - 4)/4 = (z - 5)/5.
1. In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. Assertion (A): L_1 and L_2 are skew lines. Reason (R): Two lines in space that are neither parallel nor intersecting are skew. Options: A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).; B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).; C) Assertion (A) is true but Reason (R) is false.; D) Assertion (A) is false but Reason (R) is true.
2. The shortest distance between L_1 and L_2 is Options: A) √6; B) 1; C) (√6)/3; D) (√6)/6
3. The direction cosines of L_2 are Options: A) 3/(√12), 4/(√12), 5/(√12); B) 3/50, 4/50, 5/50; C) 3/(5√2), 4/(5√2), 1/(√2); D) 35, 45, 1
Chapter 11: Three Dimensional Geometry · A line through two points
Three board-style questions on Three Dimensional Geometry (CBSE Class 12 Mathematics). No calculator. A line passes through A(1, 2, 3) and B(3, 5, 9).
1. Its direction ratios are Options: A) 1, 2, 3; B) 3, 5, 9; C) 2, 3, 6; D) 4, 7, 12
2. Its direction cosines are Options: A) 2, 3, 6; B) 27, 37, 67; C) 211, 311, 611; D) 2/(√7), 3/(√7), 6/(√7)
3. Its Cartesian equation is Options: A) (x - 3)/1 = (y - 5)/2 = (z - 9)/3; B) (x - 1)/2 = (y - 2)/3 = (z - 3)/6; C) (x + 1)/2 = (y + 2)/3 = (z + 3)/6; D) (x - 2)/1 = (y - 3)/2 = (z - 6)/3
Chapter 11: Three Dimensional Geometry · Reading a line's equation
Three board-style questions on Three Dimensional Geometry (CBSE Class 12 Mathematics). No calculator. Consider the line (x - 2)/3 = (y + 1)/2 = (z - 4)/(-6).
1. A point on the line is Options: A) (2, 1, 4); B) (3, 2, -6); C) (-2, 1, -4); D) (2, -1, 4)
2. Its direction ratios are Options: A) 2, -1, 4; B) 3, 2, -6; C) 3, 2, 6; D) -3, -2, -6
3. Its vector equation is Options: A) r = (3 i + 2 j - 6 k) + λ(2 i - j + 4 k); B) r = (2 i - j + 4 k) + λ(3 i + 2 j - 6 k); C) r = (2 i + j + 4 k) + λ(3 i + 2 j - 6 k); D) r = (-2 i + j - 4 k) + λ(3 i + 2 j + 6 k)

Chapter 12: Linear Programming

Chapter 12: Linear Programming
Three board-style questions on Linear Programming (CBSE Class 12 Mathematics). No calculator.
1. 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 Options: A) 17; B) 15; C) 12; D) 18
2. Which of the following points lies in the feasible region of x + 2y ≤ 8, 3x + y ≤ 9, x ≥ 0, y ≥ 0? Options: A) (3, 1); B) (2, 3); C) (0, 5); D) (4, 0)
3. The feasible region for the constraints x + y ≥ 6, x + y ≤ 4, x ≥ 0, y ≥ 0 is Options: A) empty; B) bounded; C) unbounded; D) a single point
Chapter 12: Linear Programming · An unbounded region
Three board-style questions on Linear Programming (CBSE Class 12 Mathematics). No calculator. Minimise Z = 2x + 3y subject to x + y ≥ 4, x ≥ 0, y ≥ 0.
1. The corner points of the feasible region are Options: A) (0, 0), (4, 0), (0, 4); B) (4, 0), (0, 4); C) (4, 4); D) (2, 2)
2. The minimum value of Z is Options: A) 8; B) 0; C) 10; D) 12
3. The maximum value of Z on this region Options: A) is 12; B) is 8; C) does not exist; D) is 0
Chapter 12: Linear Programming · Three constraints
Three board-style questions on Linear Programming (CBSE Class 12 Mathematics). No calculator. The feasible region is given by x + y ≤ 6, x ≤ 4, y ≤ 5, x ≥ 0, y ≥ 0.
1. The number of corner points of the feasible region is Options: A) 4; B) 6; C) 5; D) 3
2. The maximum of Z = 3x + 4y is Options: A) 24; B) 26; C) 20; D) 23
3. For Z = x + y, the maximum value 6 is attained at Options: A) (4, 2) only; B) (1, 5) only; C) every point of the segment joining (4, 2) and (1, 5); D) (0, 0)
Chapter 12: Linear Programming · Chairs and tables
Three board-style questions on Linear Programming (CBSE Class 12 Mathematics). No calculator. A carpenter makes x chairs and y tables a day. A chair takes 2 hours and a table 3 hours; she works at most 12 hours and makes at most 5 items a day. Profit is ₹100 a chair and ₹180 a table.
1. The constraint on hours is Options: A) 2x + 3y ≤ 12; B) 3x + 2y ≤ 12; C) 2x + 3y ≥ 12; D) x + y ≤ 12
2. The lines 2x + 3y = 12 and x + y = 5 meet at the corner Options: A) (5, 0); B) (0, 4); C) (2, 3); D) (3, 2)
3. The maximum daily profit is Options: A) ₹660; B) ₹500; C) ₹720; D) ₹900
Chapter 12: Linear Programming · A minimum on an unbounded region
Three board-style questions on Linear Programming (CBSE Class 12 Mathematics). No calculator. Minimise Z = 3x + 5y subject to x + 3y ≥ 3, x + y ≥ 2, x ≥ 0, y ≥ 0.
1. The lines x + 3y = 3 and x + y = 2 meet at Options: A) (3, 0); B) (32, 12); C) (12, 32); D) (1, 1)
2. The minimum value of Z is Options: A) 9; B) 10; C) 7; D) 6
3. In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. Assertion (A): The minimum of Z exists here even though the feasible region is unbounded. Reason (R): On an unbounded region, if the open half-plane Z < m (with m the smallest corner value) has no point in common with the region, then m is the minimum. Options: A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).; B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).; C) Assertion (A) is true but Reason (R) is false.; D) Assertion (A) is false but Reason (R) is true.
Chapter 12: Linear Programming · Feasibility
Three board-style questions on Linear Programming (CBSE Class 12 Mathematics). No calculator.
1. The region given by x + y ≥ 2, x + y ≤ 1, x ≥ 0, y ≥ 0 is Options: A) a bounded triangle; B) unbounded; C) empty (no feasible solution); D) a single point
2. Which point is feasible for x + 2y ≤ 10, 2x + y ≤ 14, x ≥ 0, y ≥ 0? Options: A) (5, 4); B) (6, 2); C) (7, 1); D) (2, 5)
3. The maximum of Z = x + y on that region is Options: A) 10; B) 8; C) 7; D) 5
Chapter 12: Linear Programming · Changing the objective
Three board-style questions on Linear Programming (CBSE Class 12 Mathematics). No calculator. The corner points of a bounded feasible region are (0, 0), (6, 0), (4, 3) and (0, 5).
1. The maximum of Z = 2x + 3y is Options: A) 12; B) 17; C) 18; D) 15
2. If Z = x + qy is maximum at both (6, 0) and (4, 3), then q equals Options: A) 2; B) 1/3; C) 3/2; D) 2/3
3. The minimum of Z = x - y on the region is Options: A) 1; B) -3; C) 0; D) -5
Chapter 12: Linear Programming · A triangle of feasible points
Three board-style questions on Linear Programming (CBSE Class 12 Mathematics). No calculator. Maximise and minimise Z = 3x + 2y subject to x + y ≤ 4, x ≥ 0, y ≥ 0.
1. The corner points of the feasible region are Options: A) (0, 0), (4, 0), (0, 4); B) (0, 0), (4, 4); C) (4, 0), (0, 4) only; D) (0, 0), (3, 0), (0, 2)
2. The maximum value of Z is Options: A) 8; B) 12; C) 20; D) 10
3. The minimum value of Z is Options: A) 8; B) 4; C) 12; D) 0
Chapter 12: Linear Programming · Using corner values
Three board-style questions on Linear Programming (CBSE Class 12 Mathematics). No calculator. The corner points of a bounded feasible region are (0, 0), (0, 4), (3, 2) and (5, 0).
1. For Z = 5x + 3y, the value at (3, 2) is Options: A) 21; B) 13; C) 17; D) 19
2. The maximum of Z = 5x + 3y on the region is Options: A) 21; B) 12; C) 25; D) 30
3. If Z = px + 3y takes its maximum value at both (3, 2) and (5, 0), then p equals Options: A) 6; B) 2; C) 5; D) 3
Chapter 12: Linear Programming · Two constraints
Three board-style questions on Linear Programming (CBSE Class 12 Mathematics). No calculator. The feasible region is given by x + 2y ≤ 8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0.
1. The two lines x + 2y = 8 and 3x + 2y = 12 meet at Options: A) (4, 2); B) (2, 4); C) (2, 3); D) (3, 2)
2. The maximum of Z = x + y is Options: A) 6; B) 5; C) 8; D) 4
3. The maximum of Z = 2x + 5y is Options: A) 23; B) 19; C) 20; D) 8

Chapter 13: Probability

Chapter 13: Probability
Three board-style questions on Probability (CBSE Class 12 Mathematics). No calculator.
1. If P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2, then P(A′ | B) equals Options: A) 0.2; B) 0.4; C) 0.5; D) 0.8
2. 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 Options: A) 7/10; B) 3/10; C) 12; D) 4/10
3. 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 Options: A) 1; B) 0.3; C) 0.7; D) 13
Chapter 13: Probability · Two bags
Three board-style questions on Probability (CBSE Class 12 Mathematics). No calculator. Bag I has 3 red and 2 black balls; Bag II has 2 red and 4 black balls. A bag is chosen at random and one ball is drawn from it.
1. P(the ball is red) equals Options: A) 5/11; B) 7/15; C) 1/2; D) 8/15
2. Given that the ball is red, the probability that it came from Bag I is Options: A) 9/14; B) 5/14; C) 1/2; D) 3/5
3. P(the ball is black) equals Options: A) 8/15; B) 7/15; C) 1/2; D) 3/5
Chapter 13: Probability · Random variables
Three board-style questions on Probability (CBSE Class 12 Mathematics). No calculator.
1. A fair coin is tossed twice and X is the number of heads. P(X = 1) equals Options: A) 1/4; B) 1/3; C) 1/2; D) 3/4
2. For the same X, the mean E(X) equals Options: A) 1; B) 3/2; C) 2; D) 1/2
3. A random variable has P(X = x) = kx for x = 1, 2, 3, 4 (and 0 otherwise). Then k equals Options: A) 1/6; B) 1/4; C) 1/10; D) 1/24
Chapter 13: Probability · A screening test
Three board-style questions on Probability (CBSE Class 12 Mathematics). No calculator. 1% of a population has a condition. A test is positive for 90% of people with the condition and for 5% of people without it. A person is chosen at random and tested.
1. P(the test is positive) equals Options: A) 0.095; B) 0.009; C) 0.0495; D) 0.0585
2. Given a positive test, the probability that the person has the condition is Options: A) 1/100; B) 2/13; C) 11/13; D) 9/10
3. P(the person has the condition AND tests positive) equals Options: A) 0.0585; B) 0.01; C) 0.9; D) 0.009
Chapter 13: Probability · A probability distribution
Three board-style questions on Probability (CBSE Class 12 Mathematics). No calculator. X takes the values 0, 1, 2, 3 with probabilities 0.1, 0.3, 0.4, 0.2.
1. E(X) equals Options: A) 1.5; B) 1.7; C) 2; D) 1.4
2. P(X ≥ 2) equals Options: A) 0.9; B) 0.6; C) 0.4; D) 0.2
3. P(X ≤ 1 | X ≤ 2) equals Options: A) 2/5; B) 4/5; C) 1/4; D) 1/2
Chapter 13: Probability · Independence and dice
Three board-style questions on Probability (CBSE Class 12 Mathematics). No calculator.
1. In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. Assertion (A): If P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.3, then A and B are not independent. Reason (R): A and B are independent if and only if P(A ∩ B) = P(A)P(B). Options: A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).; B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).; C) Assertion (A) is true but Reason (R) is false.; D) Assertion (A) is false but Reason (R) is true.
2. Two fair dice are thrown. P(the sum is 8 | the first die shows 3) equals Options: A) 5/36; B) 1/6; C) 1/36; D) 1/3
3. Two fair dice are thrown. P(at least one die shows 6 | the sum is 9) equals Options: A) 1/2; B) 11/36; C) 2/9; D) 1/4
Chapter 13: Probability · Conditional probability from a table
Three board-style questions on Probability (CBSE Class 12 Mathematics). No calculator. P(A) = 0.6, P(B) = 0.5 and P(A ∩ B) = 0.3.
1. P(A | B) equals Options: A) 0.3; B) 0.6; C) 0.18; D) 0.5
2. P(B | A) equals Options: A) 0.5; B) 0.6; C) 0.3; D) 0.9
3. Are A and B independent? Options: A) Yes, because P(A ∩ B) = P(A)P(B); B) No, because P(A ∩ B) ≠ 0; C) No, because P(A) ≠ P(B); D) Yes, because P(A) + P(B) > 1
Chapter 13: Probability · A die
Three board-style questions on Probability (CBSE Class 12 Mathematics). No calculator. A fair die is thrown. E: the number is even. F: the number is at most 4.
1. P(E | F) equals Options: A) 1/3; B) 1/4; C) 1/2; D) 2/3
2. P(F | E) equals Options: A) 1/2; B) 1/3; C) 2/3; D) 3/4
3. P(E ∩ F) equals Options: A) 1/3; B) 2/3; C) 1/6; D) 1/2
Chapter 13: Probability · Drawing two cards
Three board-style questions on Probability (CBSE Class 12 Mathematics). No calculator. Two cards are drawn one after the other from a well-shuffled pack of 52.
1. Without replacement, P(both are aces) is Options: A) 3/52; B) 1/169; C) 1/221; D) 1/26
2. Without replacement, P(second is an ace | first is an ace) is Options: A) 3/52; B) 1/17; C) 1/13; D) 4/51
3. With replacement, P(both are aces) is Options: A) 1/169; B) 1/26; C) 1/221; D) 2/13
Chapter 13: Probability · Independent events
Three board-style questions on Probability (CBSE Class 12 Mathematics). No calculator. A and B are independent with P(A) = 12 and P(B) = 13.
1. P(A ∩ B) equals Options: A) 1/6; B) 1/5; C) 1/3; D) 5/6
2. P(A ∪ B) equals Options: A) 5/6; B) 1/6; C) 2/3; D) 1/2
3. P(neither A nor B) equals Options: A) 1/6; B) 2/3; C) 1/2; D) 1/3

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Other chapters: Chapters 1–4 · Chapters 5–7 · Chapters 8–10 · Chapters 11–13. See all CBSE Maths daily and weekly questions.