CBSE Class 12 Mathematics 2023 — Previous-Year Question Paper

If A = 0 1 \-1 0 and (3I + 4A)(3I - 4A) = x²I, then the value(s) x is/are : (A) ± √(7) (B) 0 (C) ± 5 (D) 25

If for a square matrix A, A² - 3A + I = O and A⁻¹ = xA + yI, then the value of x+y is : (A) -2 (B) 2 (C) 3 (D) -3

If for a 2 × 2 matrix A, |A| = 2, then |4A⁻¹| is equal to : (A) 4 (B) 2 (C) 8 (D) (1)/(32)

Let A be a 3 × 3 matrix such that |adj A| = 64. Then |A| is equal to : (A) Only 8 (B) Only -8 (C) 64 (D) 8 or -8

The general solution of the differential equation x dy - (1 + x²) dx = dx is : (A) y = 2x + (x³)/(3) + C (B) y = 2 log x + (x³)/(3) + C (C) y = (x²)/(2) + C (D) y = 2 log x + (x²)/(2) + C

If f(x) = a(x - cos x), x ∈ mathbbR is a strictly decreasing function, then 'a' lies in which of the following intervals? (A) \0\ (B) (0, ∞) (C) (-∞, 0) (D) (-∞, ∞)

The corner points of the feasible region in the graphical representation of a linear programming problem are (2, 72), (15, 20) and (40, 15). If z = 18x + 9y is the objective function, then : (A) z is maximum at (2, 72) and minimum at (15, 20). (B) z is maximum at (15, 20) and minimum at (40, 15). (C) z is maximum at (40, 15) and minimum at (15, 20). (D) z is maximum at (40, 15) and minimum at (2, 72).

The number of corner points of the feasible region formed by the constraints x - y ≥ 0, 2y ≤ x + 2, x ≥ 0, y ≥ 0 is : (A) 2 (B) 3 (C) 4 (D) 5 Questions number 19 and 20 are Assertion and Reason based questions and each question carries 1 mark. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer to these questions from the codes (a), (b), (c) and (d) as given below : (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (c) Assertion (A) is true but Reason (R) is false. (d) Assertion (A) is false but Reason (R) is true.

If (d)/(dx)(f(x)) = log x, then f(x) equals: (a) -(1)/(x) + C (b) x(log x - 1) + C (c) x(log x + x) + C (d) (1)/(x) + C

∫₀π/6 sec²(x - (π)/(6)) dx is equal to: (a) frac1√(3) (b) -frac1√(3) (c) √(3) (d) -√(3)

Direction cosines of the line (x-1)/(2) = (1-y)/(3) = (2z-1)/(12) are: (a) (2)/(7), (3)/(7), (6)/(7) (b) frac2√(157), -frac3√(157), frac12√(157) (c) (2)/(7), -(3)/(7), -(6)/(7) (d) (2)/(7), -(3)/(7), (6)/(7)

If P((A)/(B)) = 0.3, P(A) = 0.4 and P(B) = 0.8, then P((B)/(A)) is equal to: (a) 0.6 (b) 0.3 (c) 0.06 (d) 0.4

Assertion (A): The range of the function f(x) = 2sin⁻¹ x + (3π)/(2), where x ∈ [-1, 1], is [(π)/(2), (5π)/(2)]. Reason (R): The range of the principal value branch of sin⁻¹(x) is [0, π]. (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (c) Assertion (A) is true and Reason (R) is false. (d) Assertion (A) is false and Reason (R) is true.

Assertion (A): Equation of a line passing through the points (1, 2, 3) and (3, -1, 3) is (x-3)/(2) = (y+1)/(3) = (z-3)/(0). Reason (R): Equation of a line passing through points (x₁, y₁, z₁), (x₂, y₂, z₂) is given by (x-x₁)/(x₂-x₁) = (y-y₁)/(y₂-y₁) = (z-z₁)/(z₂-z₁). (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (c) Assertion (A) is true and Reason (R) is false. (d) Assertion (A) is false and Reason (R) is true.

Find all vectors of magnitude 3√(3) which are collinear with the vector hati + hatj + hatk.

(a) Position vectors of the points A, B and C as shown in the figure below are veca, vecb and vecc respectively. If vecAC = (5)/(4) vecAB, express vecc in terms of veca and vecb. OR (b) Determine whether the lines whose equations are x = 2λ + 2, y = 7λ + 1, z = -3λ - 3 and x = -μ - 2, y = 2μ + 8, z = 4μ + 5 are perpendicular or not.

If y = (x + √(x² - 1))², then show that (x² - 1) ((dy)/(dx))² = 4y².

Show that the function f(x) = (16 sin x)/(4 + cos x) - x is strictly decreasing in ((π)/(2), π).

(a) A function f : A → B defined as f(x) = 2x is both one-one and onto. If A = \1, 2, 3, 4\, then find the set B. OR (b) Evaluate: sin⁻¹(sin(3π)/(4)) + cos⁻¹(cos(3π)/(4)) + tan⁻¹(1)

Evaluate: ∫₀π/2 [log (sin x) - log (2 cos x)] dx

Find: ∫ frac1√(x)(√(x) + 1)(√(x) + 2) dx

Solve the following linear programming problem graphically: Maximize z = 6x + 3y subject to the constraints: 4x + y ≥ 80 3x + 2y ≤ 150 x + 5y ≥ 115 x ≥ 0, y ≥ 0

(a) Find the particular solution of the differential equation (dy)/(dx) + sec² x · y = tan x · sec² x, given that y(0) = 0. OR (b) Solve the differential equation given by x dy - y dx - √(x² + y²) dx = 0.

(a) The probability distribution of a random variable X is given below: X: 1, 2, 3 with P(X): (k)/(2), (k)/(3), (k)/(6) respectively. (i) Find the value of k. (ii) Find P(1 ≤ X < 3). (iii) Find E(X), the mean of X. OR (b) A and B are independent events such that P(A ∩ barB) = (1)/(4) and P(barA ∩ B) = (1)/(6). Find P(A) and P(B).

(a) Evaluate: ∫₀π/2 ex sin x dx OR (b) Find: ∫ (1)/(cos(x-a)cos(x-b)) dx

A relation R is defined on the set of real numbers mathbbR as R = \(x, y) : x · y is an irrational number\. Check whether R is reflexive, symmetric or transitive.

(a) If A = 1 2 -2 \-1 3 0 \0 -2 1 and B⁻¹ = 3 -1 1 \-15 6 -5 \5 -2 2 , find (AB)⁻¹. OR (b) Solve the following system of equations by matrix method: x + 2y + 3z = 6, 2x - y + z = 2, 3x + 2y - 2z = 3.

(a) Find the vector and the Cartesian equations of a line passing through the point (1, 2, -4) and parallel to the line joining the points A(3, 3, -5) and B(1, 0, -11). Hence, find the distance between the two lines. OR (b) Find the equations of the line passing through the points A(1, 2, 3) and B(3, 5, 9). Hence, find the coordinates of the points on this line which are at a distance of 14 units from point B.

A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A conical tank, whose conical part is filled with water, has a tap fitted and water is dripping from the tap at a rate of 2 cm³/s. The semi-vertical angle of the conical tank is 45^°. Based on the above information, answer the following questions: (i) Express the volume of water in the tank in terms of radius r. (ii) At what rate is the radius changing when r = 2√(2) cm? (iii) (क) At what rate is the wet surface area of the conical tank decreasing when r = 2√(2) cm? OR (iii) (ख) When the slant height is 4 cm, find the rate of change of height 'h'. CASE STUDY - 3

The equation of the path traced by a roller-coaster is given by the polynomial f(x) = a(x + 9)(x + 1)(x - 3). If the roller-coaster crosses y-axis at a point (0, -1), answer the following : (i) Find the value of 'a'. (ii) Find f''(x) at x = 1.

There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam. The Venn diagram below represents the probabilities of three different types of Yoga, A, B and C performed by the people of a society. Further, it is given that the probability of a member performing type C Yoga is 0.44. (In the Venn diagram, the region of only A is 0.32, A ∩ B is 0.09, only B is y, B ∩ C is x, only C is 0.21, and the region outside all three is 0.11.) On the basis of the above information, answer the following questions: (i) Find the value of x. (ii) Find the value of y. (iii)(a) Find P((C)/(B)). OR (iii)(b) Find the probability that a randomly selected person of the society does Yoga of type A or B but not C.