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CBSE Class 12 Maths Question of the Day: all the puzzles

Every CBSE Class 12 Mathematics puzzle in the Question of the Day rotation, as images you can open, save or share. Each puzzle is three board-style questions on one chapter; the question text is written out below each image, and the answers, hints and step-marked solutions are on the daily puzzle page.

Every puzzle in the rotation

Chapter 1: Relations and Functions
Three board-style questions on Relations and Functions (CBSE Class 12 Mathematics). No calculator.
1. The function f: ℤ → ℤ, f(x) = 5 - x, is Options: A) one-one but not onto; B) neither one-one nor onto; C) bijective; D) onto but not one-one
2. The function f: ℝ → [-1, 1], f(x) = sin x, is Options: A) one-one but not onto; B) neither one-one nor onto; C) bijective; D) onto but not one-one
3. The relation R = (1, 1), (2, 2), (1, 2), (2, 1), (2, 3) on the set 1, 2, 3 is Options: A) reflexive and symmetric but not transitive; B) symmetric and transitive but not reflexive; C) neither reflexive, nor symmetric, nor transitive; D) reflexive only
Chapter 2: Inverse Trigonometric Functions
Three board-style questions on Inverse Trigonometric Functions (CBSE Class 12 Mathematics). No calculator.
1. The principal value of sec⁻¹(-2/(√3)) is Options: A) -(π)/6; B) (5π)/6; C) (7π)/6; D) (2π)/3
2. The range of the principal value branch of sec⁻¹x is Options: A) [-(π)/2, (π)/2]; B) [0, π] - (π)/2; C) (0, π); D) [-(π)/2, (π)/2] - 0
3. The value of cot⁻¹(-1) + sin⁻¹(-(√3)/2) is Options: A) (π)/12; B) (11π)/12; C) -(π)/12; D) (5π)/12
Chapter 3: Matrices
Three board-style questions on Matrices (CBSE Class 12 Mathematics). No calculator.
1. The 3 × 3 matrix A = [a_ij] with a_ij = i² - j² is Options: A) a skew-symmetric matrix; B) a symmetric matrix; C) a diagonal matrix; D) a scalar matrix
2. A, B, C are matrices of orders 2 × 3, 3 × 2 and 2 × 2 respectively. Which of the following is defined? Options: A) AC; B) BA + C; C) AB + C; D) CB
3. If 2x - y 3; x + y z = 5 3; 7 2z - 4, then x + y + z equals Options: A) 9; B) 11; C) 15; D) 7
Chapter 4: Determinants
Three board-style questions on Determinants (CBSE Class 12 Mathematics). No calculator.
1. If A is a square matrix of order 3 with |A| = 5, then |-2A^T| equals Options: A) -40; B) 40; C) -10; D) -80
2. The inverse of A = 2 3; 1 4 is Options: A) 1/11 4 -3; -1 2; B) 15 4 3; 1 2; C) 15 2 -3; -1 4; D) 15 4 -3; -1 2
3. Using determinants, the equation of the line joining (2, 3) and (-1, 5) is Options: A) 3x + 2y = 12; B) 2x + 3y = 13; C) 2x - 3y = -5; D) 3x - 2y = 0
Chapter 5: Continuity and Differentiability
Three board-style questions on Continuity and Differentiability (CBSE Class 12 Mathematics). No calculator.
1. The function f(x) = (sin 5x + sin x)/3x, x ≠ 0; k, x = 0 is continuous at x = 0. The value of k is Options: A) 13; B) 53; C) 2; D) 6
2. d/dx[sin² (3x)] equals Options: A) sin 6x; B) 2sin 3x; C) 6sin 6x; D) 3sin 6x
3. d/dx[log_e(sec x + tan x)], where sec x + tan x > 0, equals Options: A) sec x tan x; B) tan x; C) sec x; D) 1/(sec x + tan x)
Chapter 6: Application of Derivatives
Three board-style questions on Application of Derivatives (CBSE Class 12 Mathematics). No calculator.
1. The area of a circular oil patch on water is increasing at 48π cm^2/s. At the instant when its radius is 8 cm, the radius is increasing at Options: A) 6 cm/s; B) 3 cm/s; C) 3/2 cm/s; D) 12 cm/s
2. A shop’s revenue (in ₹) from selling x units of a product is R(x) = 30x - 0.01x^2. The marginal revenue when x = 500 is Options: A) ₹20; B) ₹25; C) ₹10; D) ₹12 500
3. The function f(x) = x³ - 12x is strictly decreasing on Options: A) (-∞, 2); B) (-∞, -2); C) (2, ∞); D) (-2, 2)
Chapter 7: Integrals
Three board-style questions on Integrals (CBSE Class 12 Mathematics). No calculator.
1. ∫ e^(2log_e x) dx (for x > 0) equals Options: A) 2x + C; B) (e^(2log x))/2 + C; C) x² + C; D) (x³)/3 + C
2. /(xlog_e x) (for x > 1) equals Options: A) log|log x| + C; B) ((log x)²)/2 + C; C) 1/(log x) + C; D) xlog x + C
3. /(√(9 - 4x²)) equals Options: A) 12sin⁻¹3x/2 + C; B) sin⁻¹2x/3 + C; C) 13sin⁻¹2x/3 + C; D) 12sin⁻¹2x/3 + C
Chapter 8: Application of Integrals
Three board-style questions on Application of Integrals (CBSE Class 12 Mathematics). No calculator.
1. The area of the region bounded by the parabola y² = 8x and the line x = 8 is Options: A) 256/3 sq units; B) 128/3 sq units; C) 64/3 sq units; D) 512/3 sq units
2. If the area of the region bounded by the line y = kx (k > 0), the x-axis and the line x = 3 is 18 sq units, then k equals Options: A) 2; B) 4; C) 6; D) 4/3
3. The area of the region (x, y): 0 ≤ y ≤ √(9 - x²) is Options: A) 3π sq units; B) 9π sq units; C) (9π)/4 sq units; D) (9π)/2 sq units
Chapter 9: Differential Equations
Three board-style questions on Differential Equations (CBSE Class 12 Mathematics). No calculator.
1. The sum of the order and the degree of the differential equation ((d³y)/(dx³))² + ((d²y)/(dx²))⁵ + dy/dx = sin x is Options: A) 7; B) 8; C) 5; D) 6
2. The number of arbitrary constants in the general solution of a differential equation of order 4 is Options: A) 0; B) 2; C) 3; D) 4
3. Which of the following is a solution of (d²y)/(dx²) + 9y = 0? Options: A) y = sin 9x; B) y = e^(3x); C) y = 2cos 3x - sin 3x; D) y = x² + 9
Chapter 10: Vector Algebra
Three board-style questions on Vector Algebra (CBSE Class 12 Mathematics). No calculator.
1. A vector of magnitude 14 in the direction opposite to 2i - 3j + 6k is Options: A) 4i - 6j + 12k; B) -4i + 6j - 12k; C) -28i + 42j - 84k; D) -2i + 3j - 6k
2. The position vectors of A and B are i + 2j - k and 6i - 3j + 4k. The position vector of the point dividing AB internally in the ratio 2: 3 is Options: A) 72i - 12j + 32k; B) 4i - j + 2k; C) 3i + k; D) 3i + 2k
3. (2i - j) × (i + 3k) equals Options: A) -3i + 6j + k; B) 3i + 6j - k; C) -3i - 6j + k; D) 2i - 3k
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 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 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 1: Relations and Functions · One-one and onto
Three board-style questions on Relations and Functions (CBSE Class 12 Mathematics). No calculator.
1. f: ℝ → ℝ, f(x) = 3x + 2 is Options: A) one-one and onto; B) one-one but not onto; C) onto but not one-one; D) neither one-one nor onto
2. f: ℕ → ℕ, f(n) = n² is Options: A) one-one and onto; B) one-one but not onto; C) onto but not one-one; D) neither one-one nor onto
3. f: ℝ → ℝ, f(x) = x² is Options: A) one-one and onto; B) one-one but not onto; C) onto but not one-one; D) neither one-one nor onto
Chapter 2: Inverse Trigonometric Functions · Inverse of a function of an angle
Three board-style questions on Inverse Trigonometric Functions (CBSE Class 12 Mathematics). No calculator.
1. sin⁻¹(sin(2π)/3) equals Options: A) - (π)/3; B) (π)/3; C) (π)/6; D) (2 π)/3
2. cos⁻¹(cos(7π)/6) equals Options: A) (7 π)/6; B) (5 π)/6; C) (π)/6; D) - (π)/6
3. tan⁻¹(tan(3π)/4) equals Options: A) (3 π)/4; B) (π)/4; C) - (3 π)/4; D) - (π)/4
Chapter 3: Matrices · Symmetric and skew-symmetric parts
Three board-style questions on Matrices (CBSE Class 12 Mathematics). No calculator. Let A = 2 5; 1 3.
1. A + A' equals Options: A) 0 4; -4 0; B) 4 5; 1 6; C) 4 6; 6 6; D) 4 10; 2 6
2. A - A' equals Options: A) 0 5; -1 0; B) 0 -4; 4 0; C) 4 6; 6 6; D) 0 4; -4 0
3. Writing A = P + Q with P symmetric and Q skew-symmetric, P equals Options: A) 0 2; -2 0; B) 2 3; 3 3; C) 4 6; 6 6; D) 2 5; 5 3
Chapter 4: Determinants · Consistency
Three board-style questions on Determinants (CBSE Class 12 Mathematics). No calculator. Consider the system x + 2y = 3, 2x + 4y = 7, written as AX = B.
1. |A| equals Options: A) 8; B) -8; C) 2; D) 0
2. The system is Options: A) consistent with a unique solution; B) consistent with infinitely many solutions; C) inconsistent; D) consistent with exactly two solutions
3. The system kx + 3y = 1, 3x + ky = 2 fails to have a unique solution exactly when k equals Options: A) ± 9; B) 0; C) 3 only; D) ± 3
Chapter 5: Continuity and Differentiability · Logarithmic differentiation
Three board-style questions on Continuity and Differentiability (CBSE Class 12 Mathematics). No calculator.
1. If y = x^(sin x) (x > 0), then dy/dx equals Options: A) sin x · x^(sin x - 1); B) x^(sin x)(cos xlog x + (sin x)/x); C) x^(sin x)cos xlog x; D) x^(sin x)(sin xlog x + (cos x)/x)
2. d/dx(2^x) equals Options: A) x · 2^(x - 1); B) 2^x; C) (2^x)/(log 2); D) 2^xlog 2
3. If y = (log x)^x (x > 1), then dy/dx equals Options: A) (log x)^x(log(log x) + 1/(log x)); B) x(log x)^(x - 1); C) (log x)^xlog(log x); D) (log x)^x(log x + 1x)
Chapter 6: Application of Derivatives · Related rates
Three board-style questions on Application of Derivatives (CBSE Class 12 Mathematics). No calculator.
1. A 5 m ladder leans against a vertical wall. Its foot slides away from the wall at 2 m/s. When the foot is 3 m from the wall, the top slides down at Options: A) 8/3 m/s; B) 1.2 m/s; C) 1.5 m/s; D) 2 m/s
2. The volume of a cube increases at 9 cm³/s. When its edge is 10 cm, its surface area increases at Options: A) 3.6 cm²/s; B) 0.36 cm²/s; C) 36 cm²/s; D) 1.8 cm²/s
3. A stone dropped into a pond makes a circular wave whose radius grows at 3 cm/s. When the radius is 10 cm, the enclosed area grows at Options: A) 300π cm²/s; B) 30π cm²/s; C) 60π cm²/s; D) 20π cm²/s
Chapter 7: Integrals · Evaluating definite integrals
Three board-style questions on Integrals (CBSE Class 12 Mathematics). No calculator.
1. ∫_0¹ x e^x dx equals Options: A) 1; B) e; C) 1 + e; D) -1 + e
2. ∫_0^(π/4) tan² x dx equals Options: A) 1 - (π)/4; B) (π)/4; C) (π)/4 + 1; D) -1 + (π)/4
3. ∫_1² log x dx equals Options: A) -1 + 2 log 2; B) 1 - log 2; C) log 2; D) 2 log 2
Chapter 8: Application of Integrals · Area and the sign of an integral
Three board-style questions on Application of Integrals (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): The area bounded by y = x³, the x-axis and the lines x = -1 and x = 1 is 12 square unit. Reason (R): ∫_-1¹ x³ dx = 0. 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 area bounded by y = 2x, the x-axis and the line x = 3 is Options: A) 9; B) 3; C) 6; D) 18
3. The area of the triangle bounded by y = x, y = 4 - x and the x-axis is Options: A) 16; B) 4; C) 8; D) 2
Chapter 9: Differential Equations · Order and degree
Three board-style questions on Differential Equations (CBSE Class 12 Mathematics). No calculator. Consider ((d²y)/(dx²))³ + (dy/dx)² + y = 0.
1. Its order is Options: A) 1; B) 2; C) 3; D) 5
2. Its degree is Options: A) 6; B) 2; C) 1; D) 3
3. The general solution of a differential equation of order 3 contains this many arbitrary constants: Options: A) 2; B) 0; C) 3; D) 1
Chapter 10: Vector Algebra · Scalar product
Three board-style questions on Vector Algebra (CBSE Class 12 Mathematics). No calculator. Let a = i + 2 j - 3 k and b = 3 i - j + 2 k.
1. a · b equals Options: A) -1; B) -5; C) 5; D) 11
2. | b| equals Options: A) 14; B) 4; C) √14; D) √6
3. The projection of a on b is Options: A) (5 √14)/14; B) - 5/14; C) - (5 √14)/14; D) -5
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 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 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 1: Relations and Functions · Congruence modulo 5
Three board-style questions on Relations and Functions (CBSE Class 12 Mathematics). No calculator. On ℤ, define a R b if a - b is divisible by 5.
1. R is Options: A) reflexive and symmetric but not transitive; B) symmetric only; C) an equivalence relation; D) reflexive only
2. The equivalence class [2] is Options: A) 5k: k ∈ ℤ; B) 5k + 2: k ∈ ℤ; C) 2, 7, 12, … only; D) 2k + 5: k ∈ ℤ
3. The number of distinct equivalence classes is Options: A) 5; B) 2; C) 4; D) ∞
Chapter 2: Inverse Trigonometric Functions · Combining principal values
Three board-style questions on Inverse Trigonometric Functions (CBSE Class 12 Mathematics). No calculator.
1. sin⁻¹(1) + cos⁻¹(0) equals Options: A) 0; B) π; C) (3 π)/2; D) (π)/2
2. tan⁻¹√3 - sec⁻¹(-2) equals Options: A) (π)/3; B) - (2 π)/3; C) π; D) - (π)/3
3. cosec⁻¹(2) + cot⁻¹(1) equals Options: A) (7 π)/12; B) (π)/3; C) (5 π)/12; D) (π)/2
Chapter 3: Matrices · Equal matrices
Three board-style questions on Matrices (CBSE Class 12 Mathematics). No calculator. x + y 2; 5 x - y = 6 2; 5 2.
1. x equals Options: A) 4; B) 3; C) 6; D) 2
2. y equals Options: A) 2; B) 3; C) 4; D) 1
3. x + 2y equals Options: A) 8; B) 6; C) 10; D) 12
Chapter 4: Determinants · Matrix method for two equations
Three board-style questions on Determinants (CBSE Class 12 Mathematics). No calculator. Solve 2x + y = 5, 3x + 2y = 8 by writing it as AX = B.
1. A⁻¹ equals Options: A) -2 1; 3 -2; B) 2 -3; -1 2; C) 2 -1; -3 2; D) 2 1; 3 2
2. x equals Options: A) 1; B) 3; C) 2; D) -2
3. y equals Options: A) 3; B) 1; C) 2; D) -1
Chapter 5: Continuity and Differentiability · Making a function differentiable
Three board-style questions on Continuity and Differentiability (CBSE Class 12 Mathematics). No calculator. f(x) = x², x ≤ 1; ax + b, x > 1 is differentiable at x = 1.
1. a equals Options: A) 1; B) 2; C) -1; D) 3
2. b equals Options: A) 0; B) 2; C) -1; D) 1
3. The number of points at which g(x) = |x| + |x - 1| is NOT differentiable is Options: A) 2; B) 3; C) 0; D) 1
Chapter 6: Application of Derivatives · Monotonicity and extremes
Three board-style questions on Application of Derivatives (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): f(x) = e^x is strictly increasing on ℝ. Reason (R): If f'(x) > 0 for every x in an interval, then f is strictly increasing on that interval. 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. For x > 0, the minimum value of x + 1x is Options: A) 1/2; B) 1; C) 2; D) 0
3. The maximum value of sin x + cos x on [0, (π)/2] is Options: A) 1; B) 2; C) √2; D) √3
Chapter 7: Integrals · Harder integrals
Three board-style questions on Integrals (CBSE Class 12 Mathematics). No calculator.
1. ∫ e^x(sin x + cos x) dx equals Options: A) e^xcos x + C; B) e^xsin x + C; C) e^x(sin x - cos x) + C; D) -e^xcos x + C
2. ∫ dx/(x² + 4x + 8) equals Options: A) tan⁻¹(x + 2)/2 + C; B) 12tan⁻¹(x + 2) + C; C) 14tan⁻¹(x + 2)/2 + C; D) 12tan⁻¹(x + 2)/2 + C
3. ∫_0^(π) xsin x dx equals Options: A) (π)/2; B) π; C) 2 π; D) 0
Chapter 8: Application of Integrals · Area under a curve
Three board-style questions on Application of Integrals (CBSE Class 12 Mathematics). No calculator.
1. The area bounded by y = x², the x-axis and the lines x = 0 and x = 2, in square units, is Options: A) 4; B) 8/3; C) 8; D) 4/3
2. The area bounded by y = 2x + 1, the x-axis and the lines x = 0 and x = 3 is Options: A) 15; B) 12; C) 9; D) 10
3. The area bounded by y = x³, the x-axis and the lines x = 0 and x = 1 is Options: A) 1/4; B) 1/2; C) 1; D) 1/3
Chapter 9: Differential Equations · Separating the variables
Three board-style questions on Differential Equations (CBSE Class 12 Mathematics). No calculator.
1. The general solution of dy/dx = 3x² is Options: A) y = 3x³ + C; B) y = x³ + C; C) y = 6x + C; D) y = Cx³
2. The solution of dy/dx = 3x² with y = 3 when x = 1 has y(2) equal to Options: A) 10; B) 8; C) 9; D) 12
3. The general solution of dy/dx = y is Options: A) y = x + C; B) y = Cx; C) y = e^x + C; D) y = Ce^x
Chapter 10: Vector Algebra · Angles and perpendicular vectors
Three board-style questions on Vector Algebra (CBSE Class 12 Mathematics). No calculator.
1. (i + j) · (j + k) equals Options: A) 0; B) 2; C) 1; D) -1
2. The angle between i + j and j + k is Options: A) (π)/2; B) (π)/6; C) (π)/3; D) (π)/4
3. 2 i + λ j + k is perpendicular to i - 2 j + 3 k when λ equals Options: A) 5/2; B) 5; C) 2/5; D) - 5/2
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 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 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 1: Relations and Functions · Counting relations and functions
Three board-style questions on Relations and Functions (CBSE Class 12 Mathematics). No calculator. Let A = 1, 2, 3.
1. The number of relations on A is Options: A) 64; B) 512; C) 81; D) 9
2. The number of reflexive relations on A is Options: A) 27; B) 64; C) 8; D) 512
3. The number of one-one functions from A to A is Options: A) 6; B) 3; C) 9; D) 27
Chapter 2: Inverse Trigonometric Functions · Domain of a composite expression
Three board-style questions on Inverse Trigonometric Functions (CBSE Class 12 Mathematics). No calculator.
1. The domain of f(x) = sin⁻¹(2x - 1) is Options: A) [0, 12]; B) [-1, 0]; C) [-1, 1]; D) [0, 1]
2. The domain of cos⁻¹(x/3) is Options: A) [-13, 13]; B) [-1, 1]; C) [0, 3]; D) [-3, 3]
3. The range of f(x) = 2sin⁻¹x is Options: A) [-(π)/2, (π)/2]; B) [0, 2π]; C) [-2, 2]; D) [-π, π]
Chapter 3: Matrices · A matrix equation
Three board-style questions on Matrices (CBSE Class 12 Mathematics). No calculator. Let A = 1 2; -1 3 and let I be the 2 × 2 identity matrix.
1. A² - 4A equals Options: A) O; B) -5I; C) 5I; D) -5 8; 0 -5
2. The value of k for which A² - 4A + kI = O is Options: A) 1; B) -5; C) 4; D) 5
3. Using A² - 4A + 5I = O, A⁻¹ equals Options: A) 3 -2; 1 1; B) 15 1 -2; 1 3; C) 15 3 2; -1 1; D) 15 3 -2; 1 1
Chapter 4: Determinants · A 3 by 3 determinant
Three board-style questions on Determinants (CBSE Class 12 Mathematics). No calculator. Let A = 1 1 1; 1 2 3; 1 4 9.
1. |A| equals Options: A) 0; B) 6; C) 2; D) 12
2. The cofactor A_31 equals Options: A) -5; B) -1; C) 5; D) 1
3. |adj A| equals Options: A) 4; B) 6; C) 2; D) 8
Chapter 5: Continuity and Differentiability · Reasoning about derivatives
Three board-style questions on Continuity and Differentiability (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): f(x) = |x - 3| is continuous at x = 3 but not differentiable there. Reason (R): Every continuous function is differentiable. 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. If y = log(cos e^x), then dy/dx equals Options: A) -e^xtan e^x; B) tan e^x; C) -tan e^x; D) e^xcot e^x
3. If y = e^xsin x, then (d²y)/(dx²) - 2dy/dx + 2y equals Options: A) - 2 e^x sin x; B) 0; C) 2 e^x cos x; D) e^x
Chapter 6: Application of Derivatives · A quartic
Three board-style questions on Application of Derivatives (CBSE Class 12 Mathematics). No calculator. Let f(x) = x⁴ - 2x^2.
1. f is strictly increasing on Options: A) (-∞, -1) ∪ (0, 1); B) (-1, 0) ∪ (1, ∞); C) (0, ∞); D) (-1, 1)
2. The local minimum value of f is Options: A) 0; B) -2; C) 1; D) -1
3. The number of local extreme points (maxima and minima together) of f is Options: A) 2; B) 1; C) 3; D) 4
Chapter 7: Integrals · Basic antiderivatives
Three board-style questions on Integrals (CBSE Class 12 Mathematics). No calculator.
1. ∫ (4x³ - 6x) dx equals Options: A) 12x² - 6 + C; B) x⁴ - 3x² + C; C) 4x⁴ - 6x² + C; D) x⁴ - 6x² + C
2. ∫ e^(-2x) dx equals Options: A) -2e^(-2x) + C; B) e^(-2x) + C; C) -(e^(-2x))/2 + C; D) (e^(-2x))/2 + C
3. ∫ sec² x dx equals Options: A) tan x + C; B) sec x tan x + C; C) (sec³ x)/3 + C; D) -cot x + C
Chapter 8: Application of Integrals · Circles and ellipses
Three board-style questions on Application of Integrals (CBSE Class 12 Mathematics). No calculator.
1. By integration, the area enclosed by x² + y² = 16 is Options: A) 32π square units; B) 4π square units; C) 8π square units; D) 16π square units
2. The area of the part of that circle in the first quadrant is Options: A) 2π square units; B) 8π square units; C) π square units; D) 4π square units
3. The area enclosed by the ellipse (x²)/9 + (y²)/4 = 1 is Options: A) 36π square units; B) 13π square units; C) 5π square units; D) 6π square units
Chapter 9: Differential Equations · Separable equations
Three board-style questions on Differential Equations (CBSE Class 12 Mathematics). No calculator.
1. For y ≠ 0, the general solution of dy/dx = 2x/y is Options: A) y² - 2x² = C; B) y² + 2x² = C; C) y = x² + C; D) y² = Cx²
2. For the curve satisfying dy/dx = 2x/y through (2, 3), the constant y² - 2x² equals Options: A) 5; B) -1; C) 17; D) 1
3. The general solution of dy/dx = e^(x + y) is Options: A) e^x + e^(-y) = C; B) e^x - e^(-y) = C; C) e^(x + y) = C; D) e^y = e^x + C
Chapter 10: Vector Algebra · Vector product
Three board-style questions on Vector Algebra (CBSE Class 12 Mathematics). No calculator. Let a = i + 2 j + 3 k and b = 2 i - j + k.
1. a × b equals Options: A) 5 i + 5 j - 5 k; B) -5 i - 5 j + 5 k; C) 5 i - 5 j - 5 k; D) i + 5 j - 5 k
2. | a × b| equals Options: A) 5; B) 5 √3; C) 15; D) (5 √3)/2
3. The area of the triangle with adjacent sides a and b is Options: A) 15/2; B) 5/2; C) 5 √3; D) (5 √3)/2
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 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 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 1: Relations and Functions · Building up properties
Three board-style questions on Relations and Functions (CBSE Class 12 Mathematics). No calculator. Let A = 1, 2, 3 and R = (1, 2), (2, 3).
1. R is Options: A) reflexive; B) symmetric; C) transitive; D) none of reflexive, symmetric, transitive
2. The least number of ordered pairs to add to R to make it transitive is Options: A) 1; B) 2; C) 0; D) 3
3. The least number of ordered pairs to add to R to make it symmetric is Options: A) 2; B) 4; C) 1; D) 3
Chapter 2: Inverse Trigonometric Functions · Trigonometric ratios of inverse values
Three board-style questions on Inverse Trigonometric Functions (CBSE Class 12 Mathematics). No calculator.
1. sin(cos⁻¹35) equals Options: A) 4/5; B) - 4/5; C) 5/4; D) 3/5
2. tan(sin⁻¹5/13) equals Options: A) 5/12; B) 12/5; C) 5/13; D) 12/13
3. cos(tan⁻¹1/(√3)) equals Options: A) (√2)/2; B) 1/2; C) √3; D) (√3)/2
Chapter 3: Matrices · Skew-symmetric matrices
Three board-style questions on Matrices (CBSE Class 12 Mathematics). No calculator.
1. If 0 a 3; 2 0 b; c -1 0 is skew-symmetric, then a + b + c equals Options: A) -4; B) 4; C) -2; D) 0
2. Which statement is true for EVERY skew-symmetric matrix? Options: A) all its entries are zero; B) all its diagonal entries are zero; C) it equals its transpose; D) its diagonal entries are all 1
3. For every square matrix A, the matrix A - A' is Options: A) symmetric; B) the zero matrix; C) skew-symmetric; D) the identity matrix
Chapter 4: Determinants · Invertibility
Three board-style questions on Determinants (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): A = 2 4; 1 2 is not invertible. Reason (R): A square matrix A is invertible if and only if |A| ≠ 0. 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. A is a 3 × 3 matrix with |A| = 5. Then |adj A| equals Options: A) 5; B) 25; C) 125; D) 15
3. With the same A, |A⁻¹| equals Options: A) -5; B) 1/25; C) 1/5; D) 5
Chapter 5: Continuity and Differentiability · Parametric curves with trigonometry
Three board-style questions on Continuity and Differentiability (CBSE Class 12 Mathematics). No calculator. A curve is given by x = acos³θ, y = asin³θ with a > 0 and 0 < θ < (π)/2.
1. dy/dx equals Options: A) -tan²θ; B) tanθ; C) -cotθ; D) -tanθ
2. At θ = (π)/4, dy/dx equals Options: A) 0; B) - √2; C) 1; D) -1
3. The derivative of cos⁻¹x at x = 12 equals Options: A) -2; B) - (2 √3)/3; C) (2 √3)/3; D) - (√3)/2
Chapter 6: Application of Derivatives · Rates of change
Three board-style questions on Application of Derivatives (CBSE Class 12 Mathematics). No calculator.
1. The radius of a circular ripple increases at 0.5 cm/s. When the radius is 4 cm, its area increases at Options: A) 16π cm^2/s; B) 2π cm^2/s; C) 4π cm^2/s; D) 8π cm^2/s
2. At the same moment its circumference increases at Options: A) 0.5π cm/s; B) 2π cm/s; C) 4π cm/s; D) π cm/s
3. The radius of a spherical balloon increases at 0.1 cm/s. When the radius is 3 cm, its volume increases at Options: A) 36π cm^3/s; B) 1.2π cm^3/s; C) 3.6π cm^3/s; D) 0.9π cm^3/s
Chapter 7: Integrals · Definite integrals
Three board-style questions on Integrals (CBSE Class 12 Mathematics). No calculator.
1. ∫_0¹ x² dx equals Options: A) 1; B) 2/3; C) 1/2; D) 1/3
2. ∫_0^(π/2) cos x dx equals Options: A) 0; B) -1; C) (π)/2; D) 1
3. ∫_0² (2x + 1) dx equals Options: A) 8; B) 6; C) 5; D) 4
Chapter 8: Application of Integrals · Parabolas and lines
Three board-style questions on Application of Integrals (CBSE Class 12 Mathematics). No calculator.
1. The area of the region bounded by the parabola y² = 4x and the line x = 4 is Options: A) 16; B) 16/3; C) 32/3; D) 64/3
2. The area in the first quadrant bounded by y² = 4x, the x-axis and the lines x = 1 and x = 4 is Options: A) 14/3; B) 7; C) 28/3; D) 32/3
3. The area of the region bounded by x² = 4y and the line y = 1 is Options: A) 4; B) 8/3; C) 16/3; D) 4/3
Chapter 9: Differential Equations · Linear equations
Three board-style questions on Differential Equations (CBSE Class 12 Mathematics). No calculator.
1. The integrating factor of dy/dx + y = e^(-x) is Options: A) e^(-x); B) x; C) e^x; D) e^(2x)
2. The general solution of dy/dx + y = e^(-x) is Options: A) y = (x + C)e^(-x); B) y = (x + C)e^x; C) y = Ce^(-x); D) y = xe^(-x)
3. For x > 0, the general solution of dy/dx + 2/xy = x is Options: A) yx = (x³)/3 + C; B) yx² = (x⁴)/4 + C; C) y = (x²)/4 + C; D) yx² = (x³)/3 + C
Chapter 10: Vector Algebra · Position vectors and section formula
Three board-style questions on Vector Algebra (CBSE Class 12 Mathematics). No calculator. P and Q have position vectors i + 2 j - k and 4 i - j + 5 k.
1. The point R dividing PQ internally in the ratio 2: 1 has position vector Options: A) 2 i + j + k; B) 3 i + 3 k; C) 52 i + 12 j + 2 k; D) 3 i + j + 3 k
2. The mid-point of PQ has position vector Options: A) 32 i - 32 j + 3 k; B) 52 i + 12 j + 2 k; C) 3 i - 3 j + 6 k; D) 5 i + j + 4 k
3. |PQ| equals Options: A) 54; B) 3 √6; C) 6; D) 3 √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 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 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 1: Relations and Functions · Functions between finite sets
Three board-style questions on Relations and Functions (CBSE Class 12 Mathematics). No calculator. Let A = 1, 2, 3, 4 and B = p, q, r.
1. The number of functions from A to B is Options: A) 64; B) 81; C) 7; D) 12
2. The number of one-one functions from A to B is Options: A) 0; B) 24; C) 6; D) 4
3. The number of onto functions from A to B is Options: A) 45; B) 81; C) 24; D) 36
Chapter 2: Inverse Trigonometric Functions · Outside the principal branch
Three board-style questions on Inverse Trigonometric Functions (CBSE Class 12 Mathematics). No calculator.
1. cos⁻¹(cos(-(π)/3)) equals Options: A) (π)/3; B) (2 π)/3; C) (π)/6; D) - (π)/3
2. sin⁻¹(sin(5π)/4) equals Options: A) - (π)/4; B) - (3 π)/4; C) (5 π)/4; D) (π)/4
3. sin(2sin⁻¹12) equals Options: A) 1; B) 1/2; C) (√3)/2; D) √3
Chapter 3: Matrices · Products that vanish
Three board-style questions on Matrices (CBSE Class 12 Mathematics). No calculator. Let A = 1 1; 1 1 and B = 1 -1; -1 1.
1. AB equals Options: A) 1 -1; 1 -1; B) I; C) O; D) 2 0; 0 2
2. In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. Assertion (A): For these matrices, AB = O although A ≠ O and B ≠ O. Reason (R): Matrix multiplication is commutative. 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.
3. BA equals Options: A) 2I; B) O; C) A; D) B
Chapter 4: Determinants · Matrix method for three equations
Three board-style questions on Determinants (CBSE Class 12 Mathematics). No calculator. Solve x + y + z = 6, x - y + z = 2, 2x + y - z = 1 as AX = B.
1. |A| equals Options: A) 0; B) 6; C) 2; D) -6
2. x equals Options: A) 0; B) 2; C) 1; D) 3
3. z equals Options: A) 1; B) 4; C) 3; D) 2
Chapter 5: Continuity and Differentiability · Chain rule
Three board-style questions on Continuity and Differentiability (CBSE Class 12 Mathematics). No calculator.
1. d/dxsin(3x²) equals Options: A) 3x²cos(6x); B) -6xcos(3x²); C) cos(3x²); D) 6xcos(3x²)
2. d/dx(e^(2x)log x) equals Options: A) (2e^(2x))/x; B) 2e^(2x)log x; C) e^(2x)(2log x + 1x); D) e^(2x)(log x + 2x)
3. d/dxtan⁻¹(3x) equals Options: A) 1/(1 + 9x²); B) 3/(1 + 3x²); C) 3/(1 + x²); D) 3/(1 + 9x²)
Chapter 6: Application of Derivatives · A quadratic function
Three board-style questions on Application of Derivatives (CBSE Class 12 Mathematics). No calculator. Let f(x) = x² - 6x + 5.
1. f'(x) equals Options: A) 2x - 6; B) 2x + 5; C) x - 6; D) 2x - 6x
2. f is strictly increasing on Options: A) (1, 5); B) (-∞, 3); C) (3, ∞); D) (-∞, ∞)
3. The minimum value of f is Options: A) 5; B) 3; C) -4; D) -9
Chapter 7: Integrals · Integration by substitution
Three board-style questions on Integrals (CBSE Class 12 Mathematics). No calculator.
1. ∫ 2x/(1 + x²) dx equals Options: A) tan⁻¹x + C; B) log(1 + x²) + C; C) 2tan⁻¹x + C; D) 2log(1 + x²) + C
2. ∫ sin xcos x dx equals Options: A) cos² x + C; B) sin x + cos x + C; C) (sin² x)/2 + C; D) -sin² x + C
3. ∫ x e^(x²) dx equals Options: A) e^(x²) + C; B) (e^(x²))/2 + C; C) 2e^(x²) + C; D) (x² e^(x²))/2 + C
Chapter 8: Application of Integrals · Modulus functions
Three board-style questions on Application of Integrals (CBSE Class 12 Mathematics). No calculator.
1. The area bounded by y = |x - 1|, the x-axis and the lines x = -1 and x = 3 is Options: A) 4; B) 3; C) 8; D) 2
2. The area bounded by y = |x|, the x-axis and the lines x = -2 and x = 1 is Options: A) 2; B) 3/2; C) 3; D) 5/2
3. The area of the region bounded by y = 3 - |x| and the x-axis is Options: A) 6; B) 9/2; C) 9; D) 18
Chapter 9: Differential Equations · Homogeneous equations
Three board-style questions on Differential Equations (CBSE Class 12 Mathematics). No calculator. Consider dy/dx = (x + y)/x for x > 0.
1. Putting y = vx gives Options: A) xdv/dx = 1; B) xdv/dx = v; C) dv/dx = 1 + v; D) xdv/dx = 1 + 2v
2. The general solution is Options: A) y = x + C; B) y = xlog x + Cx; C) y = log x + C; D) y = Cxlog x
3. Which of these differential equations is homogeneous? Options: A) dy/dx = x + y²; B) dy/dx = (x² + y²)/xy; C) dy/dx = (x + 1)/y; D) dy/dx = y/x + x
Chapter 10: Vector Algebra · Parallel and perpendicular
Three board-style questions on Vector Algebra (CBSE Class 12 Mathematics). No calculator.
1. 2 i + 3 j - k and 4 i + λ j - 2 k are parallel when λ equals Options: A) 3; B) -6; C) 6; D) 12
2. Which of these is a vector of length 1 at right angles to both i + j + k and i - j + k? Options: A) 1/(√2)(i + k); B) 1/(√2)(i - k); C) 1/(√3)(i + j - k); D) j
3. If | a + b| = | a - b| for non-zero vectors a, b, then Options: A) a is parallel to b; B) a is perpendicular to b; C) | a| = | b|; D) a = b
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 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 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 1: Relations and Functions · Harder functions
Three board-style questions on Relations and Functions (CBSE Class 12 Mathematics). No calculator.
1. f: ℝ → ℝ, f(x) = x|x| is Options: A) one-one and onto; B) one-one but not onto; C) onto but not one-one; D) neither one-one nor onto
2. f: ℝ → ℝ, f(x) = |x| + x is Options: A) one-one and onto; B) one-one but not onto; C) onto but not one-one; D) neither one-one nor onto
3. f: ℝ - 2 → ℝ - 1, f(x) = (x - 1)/(x - 2) is Options: A) one-one and onto; B) one-one but not onto; C) onto but not one-one; D) neither one-one nor onto
Chapter 2: Inverse Trigonometric Functions · Principal branches in reasoning
Three board-style questions on Inverse Trigonometric Functions (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): sin⁻¹(sin(3π)/4) = (π)/4. Reason (R): The principal value branch of sin⁻¹ is [-(π)/2, (π)/2]. 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. cot(sin⁻¹12) equals Options: A) √3; B) 2; C) (√3)/2; D) (√3)/3
3. sin⁻¹(cos(π)/3) equals Options: A) (π)/2; B) - (π)/6; C) (π)/3; D) (π)/6
Chapter 3: Matrices · A rotation matrix
Three board-style questions on Matrices (CBSE Class 12 Mathematics). No calculator. Let A = cosθ -sinθ; sinθ cosθ.
1. AA' equals Options: A) 2I; B) A; C) O; D) I
2. If A + A' = I and 0 < θ < (π)/2, then θ equals Options: A) (π)/4; B) (π)/3; C) (π)/12; D) (π)/6
3. For θ = (π)/4, A² equals Options: A) 0 1; -1 0; B) 0 -1; 1 0; C) I; D) 1 0; 0 -1
Chapter 4: Determinants · Evaluating determinants
Three board-style questions on Determinants (CBSE Class 12 Mathematics). No calculator.
1. 3 -2; 4 5 equals Options: A) 23; B) -23; C) 15; D) 7
2. 2 0 1; 1 3 -1; 0 2 4 equals Options: A) 34; B) 28; C) 26; D) 30
3. If x 2; 8 x = 0, then x equals Options: A) ± 16; B) ± 2; C) 4 only; D) ± 4
Chapter 5: Continuity and Differentiability · Continuity
Three board-style questions on Continuity and Differentiability (CBSE Class 12 Mathematics). No calculator.
1. f(x) = kx + 1, x ≤ 2; 3x - 1, x > 2 is continuous at x = 2. Then k equals Options: A) 1; B) 3; C) 2; D) 5/2
2. At x = 0, the function f(x) = |x| is Options: A) continuous and differentiable; B) continuous but not differentiable; C) differentiable but not continuous; D) neither continuous nor differentiable
3. f(x) = (x² - 9)/(x - 3) for x ≠ 3 and f(3) = k. If f is continuous at x = 3, then k equals Options: A) 3; B) 9; C) 6; D) 0
Chapter 6: Application of Derivatives · Turning points of a cubic
Three board-style questions on Application of Derivatives (CBSE Class 12 Mathematics). No calculator. Let f(x) = x³ - 3x² - 9x + 2.
1. The critical points of f are Options: A) x = -1, 3; B) x = 1, -3; C) x = -1, -3; D) x = 1, 3
2. The local maximum value of f is Options: A) -25; B) 2; C) 9; D) 7
3. f is strictly decreasing on Options: A) (-1, 3); B) (-3, 1); C) (-∞, -1); D) (3, ∞)
Chapter 7: Integrals · Integration by parts
Three board-style questions on Integrals (CBSE Class 12 Mathematics). No calculator.
1. ∫ x e^x dx equals Options: A) x e^x + C; B) e^x(x + 1) + C; C) (x²)/2e^x + C; D) e^x(x - 1) + C
2. ∫ log x dx equals Options: A) 1x + C; B) xlog x - x + C; C) xlog x + C; D) ((log x)²)/2 + C
3. ∫ xcos x dx equals Options: A) xsin x + cos x + C; B) xsin x - cos x + C; C) -xsin x + cos x + C; D) (x²)/2sin x + C
Chapter 8: Application of Integrals · A quarter ellipse and a chord
Three board-style questions on Application of Integrals (CBSE Class 12 Mathematics). No calculator. Consider the ellipse (x²)/16 + (y²)/9 = 1.
1. The area of the part of the ellipse in the first quadrant is Options: A) 6π; B) (3π)/2; C) 12π; D) 3π
2. The area of the smaller region bounded by the ellipse and the line x/4 + y/3 = 1 is Options: A) 3π - 12; B) 6 - (3π)/2; C) 3π - 6; D) 6π - 6
3. The area enclosed by the whole ellipse is Options: A) 6π; B) 7π; C) 12π; D) 24π
Chapter 9: Differential Equations · Checking solutions
Three board-style questions on Differential Equations (CBSE Class 12 Mathematics). No calculator.
1. y = e^(2x) is a solution of Options: A) (d²y)/(dx²) + 4y = 0; B) dy/dx = y; C) (d²y)/(dx²) - 4y = 0; D) (d²y)/(dx²) = 2y
2. y = Asin x + Bcos x (A, B arbitrary) is the general solution of (d²y)/(dx²) + y = 0. The number of arbitrary constants is Options: A) 1; B) 2; C) 0; D) 3
3. y = x² + 2x satisfies Options: A) xdy/dx - y = x²; B) xdy/dx + y = x²; C) dy/dx = y; D) xdy/dx - y = 2x
Chapter 10: Vector Algebra · Direction cosines
Three board-style questions on Vector Algebra (CBSE Class 12 Mathematics). No calculator.
1. A line makes angles 60^∘ and 45^∘ with the x- and y-axes and an acute angle θ with the z-axis. Then θ equals Options: A) 45^∘; B) 60^∘; C) 90^∘; D) 30^∘
2. The direction cosines of a line with direction ratios 2, -1, 2 are Options: A) 2, -1, 2; B) 23, 13, 23; C) 23, -13, 23; D) 29, -19, 29
3. The vector of magnitude 10 in the direction of 3 i + 4 k is Options: A) 8 i + 6 k; B) 6 i + 8 k; C) 30 i + 40 k; D) 3 i + 4 k
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 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 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 1: Relations and Functions · Equal squares
Three board-style questions on Relations and Functions (CBSE Class 12 Mathematics). No calculator. On ℤ, define a R b if a² = b^2.
1. R is Options: A) reflexive only; B) symmetric and transitive but not reflexive; C) an equivalence relation; D) not symmetric
2. The equivalence class [-4] is Options: A) -4; B) 4; C) -4, 4; D) 16
3. The number of elements in the equivalence class [0] is Options: A) 0; B) 1; C) ∞; D) 2
Chapter 2: Inverse Trigonometric Functions · Graphs of inverse functions
Three board-style questions on Inverse Trigonometric Functions (CBSE Class 12 Mathematics). No calculator.
1. The graph of y = sin⁻¹x passes through Options: A) ((π)/2, 1); B) (1, (π)/2); C) (-1, (π)/2); D) (0, π)
2. On [-1, 1] the function y = cos⁻¹x is Options: A) increasing; B) decreasing; C) constant; D) increasing then decreasing
3. The greatest value of tan⁻¹x for -1 ≤ x ≤ √3 is Options: A) (π)/3; B) - (π)/4; C) (π)/4; D) (π)/2
Chapter 3: Matrices · Adding and scaling
Three board-style questions on Matrices (CBSE Class 12 Mathematics). No calculator. Let A = 2 -1; 3 4 and B = 1 0; -2 5.
1. A + B equals Options: A) 3 -1; 5 9; B) 2 0; -6 20; C) 1 -1; 5 -1; D) 3 -1; 1 9
2. 2A - B equals Options: A) 3 -2; 8 3; B) 3 -1; 8 3; C) 3 -2; 4 3; D) 5 -2; 4 13
3. The number of possible orders of a matrix with 6 elements is Options: A) 6; B) 3; C) 2; D) 4
Chapter 4: Determinants · Minors and cofactors
Three board-style questions on Determinants (CBSE Class 12 Mathematics). No calculator. Let A = 2 -3 1; 4 0 5; -1 2 3.
1. The minor M_11 equals Options: A) -6; B) 10; C) -10; D) 6
2. The cofactor A_12 equals Options: A) -17; B) -7; C) 17; D) 7
3. The cofactor A_23 equals Options: A) 1; B) -1; C) 7; D) -7
Chapter 5: Continuity and Differentiability · Implicit differentiation
Three board-style questions on Continuity and Differentiability (CBSE Class 12 Mathematics). No calculator. The curve x² + xy + y² = 7 passes through (1, 2).
1. dy/dx equals Options: A) -(x + 2y)/(2x + y); B) -2x/2y; C) -(2x + y)/(x + 2y); D) (2x + y)/(x + 2y)
2. At (1, 2), dy/dx equals Options: A) 4/5; B) - 5/4; C) - 1/2; D) - 4/5
3. d/dx(x^x), for x > 0, equals Options: A) x^x(1 + log x); B) x^x(x + log x); C) x · x^(x - 1); D) x^x log x
Chapter 6: Application of Derivatives · Absolute extremes on an interval
Three board-style questions on Application of Derivatives (CBSE Class 12 Mathematics). No calculator. Let f(x) = 2x³ - 15x² + 36x + 1 on [1, 5].
1. The absolute maximum value of f on [1, 5] is Options: A) 56; B) 24; C) 29; D) 28
2. The absolute minimum value of f on [1, 5] is Options: A) 29; B) 24; C) 28; D) 1
3. The local maximum value of f is Options: A) 24; B) 28; C) 56; D) 29
$CBSE Class 12 Maths Question of the Day on Integrals · Partial fractions, a 3-part multiple-choice question: Three board-style questions on Integrals (CBSE Class 12 Mathematics). No calculator. Write 1/((x - 1)(x + 2)) = A/(x - 1) + B/(x + 2). (a)…$
Chapter 7: Integrals · Partial fractions
Three board-style questions on Integrals (CBSE Class 12 Mathematics). No calculator. Write 1/((x - 1)(x + 2)) = A/(x - 1) + B/(x + 2).
1. A equals Options: A) 1/3; B) 1; C) 1/2; D) - 1/3
2. ∫ dx/((x - 1)(x + 2)) equals Options: A) log|(x - 1)(x + 2)| + C; B) 13log|(x + 2)/(x - 1)| + C; C) 13log|(x - 1)/(x + 2)| + C; D) 3log|(x - 1)/(x + 2)| + C
3. ∫ x/((x + 1)(x + 2)) dx equals Options: A) log|x + 1| + 2log|x + 2| + C; B) -log|x + 1| - 2log|x + 2| + C; C) 2log|x + 1| - log|x + 2| + C; D) -log|x + 1| + 2log|x + 2| + C
Chapter 8: Application of Integrals · Sine and cosine curves
Three board-style questions on Application of Integrals (CBSE Class 12 Mathematics). No calculator.
1. The area bounded by y = sin x and the x-axis for 0 ≤ x ≤ π is Options: A) 2; B) 0; C) π; D) 1
2. The area bounded by y = cos x, the x-axis and 0 ≤ x ≤ (π)/2 is Options: A) 1/2; B) (π)/2; C) 1; D) 2
3. The area bounded by y = cos x and the x-axis for 0 ≤ x ≤ π is Options: A) 1; B) π; C) 2; D) 0
Chapter 9: Differential Equations · An initial value problem
Three board-style questions on Differential Equations (CBSE Class 12 Mathematics). No calculator. Consider dy/dx + ytan x = sec x for 0 ≤ x < (π)/2, with y = 1 when x = 0.
1. The integrating factor is Options: A) tan x; B) e^(tan x); C) cos x; D) sec x
2. The general solution is Options: A) y = sin x + Ccos x; B) y = cos x + Csin x; C) y = tan x + C; D) ycos x = tan x + C
3. The particular solution is Options: A) y = sin x; B) y = cos x; C) y = sin x + cos x; D) y = 1 + sin x
Chapter 10: Vector Algebra · Products from magnitudes
Three board-style questions on Vector Algebra (CBSE Class 12 Mathematics). No calculator. | a| = 3, | b| = 4 and a · b = 6.
1. The angle between a and b is Options: A) (π)/2; B) (π)/6; C) (π)/4; D) (π)/3
2. | a × b| equals Options: A) 6 √2; B) 12; C) 6 √3; D) 6
3. | a - b| equals Options: A) 1; B) 5; C) √13; D) √37
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 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 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 1: Relations and Functions · Reasoning about relations and functions
Three board-style questions on Relations and Functions (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): The function f: ℤ → ℤ, f(x) = x + 5 is onto. Reason (R): For every y ∈ ℤ, x = y - 5 is in ℤ and f(x) = y. 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. On A = 1, 2, 3, 4, R = (x, y): x + y is even is an equivalence relation. The number of equivalence classes is Options: A) 2; B) 1; C) 3; D) 4
3. The number of ordered pairs in that relation R is Options: A) 16; B) 6; C) 8; D) 4
Chapter 2: Inverse Trigonometric Functions · Principal values
Three board-style questions on Inverse Trigonometric Functions (CBSE Class 12 Mathematics). No calculator.
1. The principal value of sin⁻¹(12) is Options: A) - (π)/6; B) (π)/6; C) (π)/3; D) (5 π)/6
2. The principal value of cos⁻¹(-12) is Options: A) (π)/3; B) (2 π)/3; C) (4 π)/3; D) - (π)/3
3. The principal value of tan⁻¹(-1) is Options: A) (π)/4; B) - (3 π)/4; C) (3 π)/4; D) - (π)/4
Chapter 3: Matrices · Order and entries
Three board-style questions on Matrices (CBSE Class 12 Mathematics). No calculator.
1. A product PQ is formed from a 2 × 3 matrix P and a 3 × 5 matrix Q. The order of PQ is Options: A) 3 × 3; B) 2 × 5; C) 5 × 2; D) PQ is not defined
2. The 2 × 2 matrix A = [a_ij] with a_ij = i + 2j is Options: A) 3 6; 4 8; B) 3 4; 5 6; C) 3 5; 4 6; D) 2 4; 3 5
3. If P = 1 2 3; 4 5 6, the entry in row 1, column 2 of P' (the transpose) is Options: A) 2; B) 4; C) 5; D) 3
Chapter 4: Determinants · Area of a triangle
Three board-style questions on Determinants (CBSE Class 12 Mathematics). No calculator.
1. The area of the triangle with vertices (1, 2), (4, 6) and (7, 2), in square units, is Options: A) 12; B) 24; C) 6; D) 18
2. The points (2, 3), (4, k) and (6, -3) are collinear when k equals Options: A) 3; B) -3; C) 6; D) 0
3. The triangle with vertices (k, 0), (4, 0) and (0, 2) has area 4 square units. Then k equals Options: A) 8 only; B) 0 or 8; C) ± 8; D) 0 only
Chapter 5: Continuity and Differentiability · Parametric differentiation
Three board-style questions on Continuity and Differentiability (CBSE Class 12 Mathematics). No calculator. A curve is given by x = 2t², y = 4t (t ≠ 0).
1. dy/dx equals Options: A) t; B) 1t; C) 2/t; D) 4t
2. At t = 2, dy/dx equals Options: A) 1/2; B) 8; C) 2; D) 1/4
3. (d²y)/(dx²) equals Options: A) 1/(4t³); B) -1/(4t²); C) -1/(t²); D) -1/(4t³)
Chapter 6: Application of Derivatives · Best products and sums
Three board-style questions on Application of Derivatives (CBSE Class 12 Mathematics). No calculator.
1. Two positive numbers have sum 20. The greatest possible value of their product is Options: A) 400; B) 96; C) 100; D) 99
2. Two positive numbers have sum 20. The least possible value of the sum of their squares is Options: A) 100; B) 400; C) 202; D) 200
3. Two positive numbers have product 36. The least possible value of their sum is Options: A) 18; B) 13; C) 37; D) 12
Chapter 7: Integrals · Standard integrals
Three board-style questions on Integrals (CBSE Class 12 Mathematics). No calculator.
1. ∫ dx/(x² + 9) equals Options: A) tan⁻¹x/3 + C; B) 13tan⁻¹x/3 + C; C) 3tan⁻¹x/3 + C; D) 19tan⁻¹x + C
2. ∫ dx/(√(16 - x²)) equals Options: A) 14sin⁻¹x/4 + C; B) sin⁻¹x + C; C) sin⁻¹x/4 + C; D) 4sin⁻¹x/4 + C
3. ∫ dx/(x² - 4) equals Options: A) 14log|(x - 2)/(x + 2)| + C; B) 12log|(x - 2)/(x + 2)| + C; C) 14log|(x + 2)/(x - 2)| + C; D) log|x² - 4| + C
Chapter 8: Application of Integrals · A segment of a circle
Three board-style questions on Application of Integrals (CBSE Class 12 Mathematics). No calculator. Consider the circle x² + y² = 4.
1. The area of the smaller region cut off from the circle by the line x = 1 is Options: A) (π)/3 - (√3)/2; B) (4π)/3 - √3; C) (2π)/3 - √3; D) (4π)/3 + √3
2. The area in the first quadrant between the circle, the y-axis, the x-axis and the line x = 1 is Options: A) √3 + (π)/6; B) (√3)/2 + (π)/6; C) (π)/3; D) (√3)/2 + (π)/3
3. The area enclosed by the circle is Options: A) 16π; B) 4π; C) 8π; D) 2π
Chapter 9: Differential Equations · Growth
Three board-style questions on Differential Equations (CBSE Class 12 Mathematics). No calculator. A population grows at a rate proportional to its size, dP/dt = kP with k > 0, and doubles in 10 years.
1. If P_0 is the population at t = 0, then P equals Options: A) P_0 + kt; B) P_0e^(kt); C) P_0kt; D) e^(kt) + P_0
2. k equals Options: A) 10/(log 2); B) (log 2)/10; C) 2/10; D) 10log 2
3. The population becomes four times as large after this many years: Options: A) 15; B) 40; C) 30; D) 20
Chapter 10: Vector Algebra · Reasoning with products
Three board-style questions on Vector Algebra (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): For a = i - j and b = i + j, a × b = 2 k. Reason (R): a × b is perpendicular to both a and 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. The area of the triangle with vertices A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1) is Options: A) (√14)/2; B) √21; C) (√21)/2; D) 21/2
3. i + λ j + 3 k is perpendicular to 2 i + j - k when λ equals Options: A) 3; B) -1; C) 5; D) 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 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 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 1: Relations and Functions · Types of relations
Three board-style questions on Relations and Functions (CBSE Class 12 Mathematics). No calculator. Let A = 1, 2, 3 and R = (1,1), (2,2), (3,3), (1,2), (2,1).
1. R is Options: A) reflexive but not symmetric; B) an equivalence relation; C) symmetric but not transitive; D) transitive but not reflexive
2. The equivalence class [1] is Options: A) 1, 2, 3; B) 1, 2; C) 2; D) 1
3. The least number of ordered pairs that must be added to (1, 2) to make a reflexive relation on A is Options: A) 1; B) 2; C) 4; D) 3
Chapter 2: Inverse Trigonometric Functions · Domains and ranges
Three board-style questions on Inverse Trigonometric Functions (CBSE Class 12 Mathematics). No calculator.
1. The range of the principal value branch of cos⁻¹x is Options: A) (0, π); B) [-1, 1]; C) [-(π)/2, (π)/2]; D) [0, π]
2. The range of the principal value branch of tan⁻¹x is Options: A) (0, π); B) (-(π)/2, (π)/2); C) ℝ; D) [-(π)/2, (π)/2]
3. The domain of sin⁻¹x is Options: A) [-(π)/2, (π)/2]; B) (-1, 1); C) [-1, 1]; D) ℝ
Chapter 3: Matrices · Multiplying matrices
Three board-style questions on Matrices (CBSE Class 12 Mathematics). No calculator. Let A = 1 2; 3 4 and B = 0 1; 1 0.
1. AB equals Options: A) 0 2; 3 0; B) 1 2; 3 4; C) 3 4; 1 2; D) 2 1; 4 3
2. BA equals Options: A) 3 4; 1 2; B) 2 1; 4 3; C) 4 3; 2 1; D) 1 2; 3 4
3. A² equals Options: A) 7 15; 10 22; B) 2 4; 6 8; C) 1 4; 9 16; D) 7 10; 15 22
Chapter 4: Determinants · Adjoint and inverse
Three board-style questions on Determinants (CBSE Class 12 Mathematics). No calculator. Let A = 4 2; 3 2.
1. adj A equals Options: A) 2 -3; -2 4; B) 2 -2; -3 4; C) 2 2; 3 4; D) -2 2; 3 -4
2. A⁻¹ equals Options: A) 12 4 -2; -3 2; B) 12 2 -2; -3 4; C) 14 2 -2; -3 4; D) 2 -2; -3 4
3. |adj A| equals Options: A) 4; B) 2; C) 1/2; D) 8
Chapter 5: Continuity and Differentiability · Second-order derivatives
Three board-style questions on Continuity and Differentiability (CBSE Class 12 Mathematics). No calculator.
1. If y = x³log x, then (d²y)/(dx²) equals Options: A) 6xlog x + 3x; B) 3x²log x + x²; C) 6xlog x; D) 6xlog x + 5x
2. If y = Acos 2x + Bsin 2x, then (d²y)/(dx²) + ky = 0 for every A, B when k equals Options: A) 4; B) 2; C) -4; D) 1
3. If y = sin⁻¹x, which equation holds for -1 < x < 1? Options: A) (1 - x²)y'' - xy' = 0; B) (1 - x²)y'' + xy' = 0; C) y'' = xy'; D) (1 + x²)y'' - xy' = 0
Chapter 6: Application of Derivatives · An open box
Three board-style questions on Application of Derivatives (CBSE Class 12 Mathematics). No calculator. An open box is made from a square sheet of side 18 cm by cutting equal squares of side x cm from each corner and folding up the sides.
1. The volume is greatest when x equals Options: A) 2; B) 3; C) 4; D) 6
2. The greatest volume is Options: A) 432 cm³; B) 486 cm³; C) 400 cm³; D) 324 cm³
3. At that value of x, (d²V)/(dx²) equals Options: A) -144; B) 72; C) -72; D) 0
Chapter 7: Integrals · Properties of definite integrals
Three board-style questions on Integrals (CBSE Class 12 Mathematics). No calculator.
1. ∫_-1¹ x³cos x dx equals Options: A) 1; B) 2 cos 1; C) 2 sin 1; D) 0
2. ∫_0^(π/2) (sin x)/(sin x + cos x) dx equals Options: A) (π)/2; B) (π)/4; C) 1; D) (π)/8
3. ∫_-2² |x| dx equals Options: A) 0; B) 2; C) 8; D) 4
Chapter 8: Application of Integrals · Parabola with a line
Three board-style questions on Application of Integrals (CBSE Class 12 Mathematics). No calculator.
1. The area of the region bounded by y = x² and the line y = 4 is Options: A) 16; B) 8; C) 16/3; D) 32/3
2. The area of the region bounded by y = x² and the line y = x is Options: A) 1/3; B) 1/2; C) 5/6; D) 1/6
3. The area of the region bounded by y² = x and the line x = 9 is Options: A) 27; B) 18; C) 54; D) 36
Chapter 9: Differential Equations · Recognising a linear equation
Three board-style questions on Differential Equations (CBSE Class 12 Mathematics). No calculator. Consider dy/dx + y/x = x² for x > 0.
1. In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. Assertion (A): This differential equation is linear. Reason (R): An equation of the form dy/dx + Py = Q, where P and Q are functions of x only, is linear. 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. Its integrating factor is Options: A) 1x; B) log x; C) x; D) x²
3. Its general solution is Options: A) y = (x³)/4 + C; B) xy = (x⁴)/4 + C; C) xy = (x³)/3 + C; D) y = x³ + Cx
Chapter 10: Vector Algebra · Magnitude and unit vectors
Three board-style questions on Vector Algebra (CBSE Class 12 Mathematics). No calculator. Let a = 2 i - j + 2 k and b = i + 3 j - k.
1. | a| equals Options: A) √5; B) 3; C) 9; D) 5
2. The unit vector in the direction of a is Options: A) i - j + k; B) 19(2 i - j + 2 k); C) 13(2 i - j + 2 k); D) 13(2 i + j + 2 k)
3. a + b equals Options: A) 3 i + 4 j + k; B) 2 i - 3 j - 2 k; C) 3 i + 2 j + k; D) i - 4 j + 3 k
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 · 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 · 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