CBSE Class 12 Mathematics 2024 — Previous-Year Question Paper

Let hata and hatb be two unit vectors and θ be the angle between them such that sin θ = (3)/(5). Then hata · hatb is equal to: (A) ± (3)/(5) (B) ± (3)/(4) (C) ± (4)/(5) (D) ± (4)/(3)

The integrating factor of the differential equation (1 - x²) (dy)/(dx) + xy = ax, -1 < x < 1 is : (A) (1)/(x² - 1) (B) frac1√(x² - 1) (C) (1)/(1 - x²) (D) frac1√(1 - x²)

If the direction cosines of a line are √(3)k, √(3)k, √(3)k, then the value of k is : (A) ± 1 (B) ± √(3) (C) ± 3 (D) ± (1)/(3)

An optimal solution of a linear programming problem is related to : (A) Logarithmic function (B) Linear function (C) Quadratic function (D) Exponential function

If P(A|B) = P(A'|B), then which of the following statements is correct ? (A) P(A) = P(A') (B) P(A) = 2 P(B) (C) P(A ∩ B) = (1)/(2) P(B) (D) P(A ∩ B) = 2 P(B)

x+1 x-1 x²+x+1 x²-x+1 is equal to : (A) 2x³ (B) 2 (C) 0 (D) 2x³ - 2

The derivative of sin(x²) with respect to x at x = √(π) is : (A) 1 (B) -1 (C) -2√(π) (D) 2√(π)

The order and degree of the differential equation [1+(fracmathrmdymathrmdx)²]³=fracmathrmd² mathrmymathrmdx² are respectively : (A) 1, 2 (B) 2, 3 (C) 2, 1 (D) 2, 6

The vector with terminal point A (2,-3,5) and initial point B (3,-4,7) is : (A) hatmathrmi-hatmathrmj+2 hatmathrmk (B) hatmathrmi+hatmathrmj+2 hatmathrmk (C) -hatmathrmi-hatmathrmj-2 hatmathrmk (D) -hatmathrmi+hatmathrmj-2 hatmathrmk

The number of corner points of the feasible region determined by the constraints x ≥ 0, y ≥ 0, x+y ≥ 4 is : (A) 0 (B) 1 (C) 2 (D) 3

∫ab f(x) dx is equal to: (A) ∫ab f(a-x) dx (B) ∫ab f(a+b-x) dx (C) ∫ab f(x-(a+b)) dx (D) ∫ab f((a-x)+(b-x)) dx

The distance of point P(a, b, c) from the y-axis is: (A) b (B) b² (C) √(a² + c²) (D) a² + c²

If A and B are two non-zero square matrices of the same order such that (A + B)² = A² + B², then: (A) AB = O (B) AB = -BA (C) BA = O (D) AB = BA

Assertion (A): For the matrix A = 1 cosθ 1 \-cosθ 1 cosθ \-1 -cosθ 1 , where θ ∈ [0, 2π], |A| ∈ [2, 4]. Reason (R): cosθ ∈ [-1, 1],∀ θ ∈ [0, 2π]. (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.

Assertion (A): A line in space cannot be drawn perpendicular to x, y and z axes simultaneously. Reason (R): For any line making angles α, β, γ with the positive directions of x, y and z axes respectively, cos²α + cos²β + cos²γ = 1. (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.

Show that the function f(x) = 4x³ - 18x² + 27x - 7 has neither maximum nor minimum value.

If two non-zero vectors veca and vecb are such that (veca + vecb) perp veca and (2veca + vecb) perp vecb, then prove that |vecb| = √(2) |veca|.

In the given figure, ABCD is a parallelogram. If vecAB = 2hati - 4hatj + 5hatk and vecDB = 3hati - 6hatj + 2hatk, then find vecAD and using it, find the area of the parallelogram ABCD. SECTION G This section contains Short Answer (SA) type questions, each carrying 3 marks.

Check whether the function f(x) = x²|x| is differentiable at x = 0 or not.

Find: ∫ x√(1 + 2x) dx

Find the particular solution of the differential equation x² (dy)/(dx) - xy = x² cos² ((y)/(2x)), given that y = (π)/(2), when x = 1.

Solve the following Linear Programming Problem graphically : Subject to the constraints x + 2y ≤ 12 2x + y ≤ 12 4x + 5y ≥ 20 x ≥ 0, y ≥ 0 Maximize z = 500x + 300y.

E and F are two independent events such that P(barE) = 0.6 and P(E ∪ F) = 0.6. Find P(F) and P(barE ∪ barF).

A relation R on set A = \1, 2, 3, 4, 5\ is defined as R = \(x, y) : |x² - y²| < 8\. Check whether the relation R is reflexive, symmetric and transitive.

If √(1 - x²) + √(1 - y²) = a(x - y), prove that (dy)/(dx) = √((1 - y²)/(1 - x²)).

Find: ∫ (x²)/((x² + 4)(x² + 9)) dx

Using integration, find the area of the region bounded by the ellipse (x²)/(16) + (y²)/(4) = 1 and the lines x = -2 and x = 2.

If the image of the point P(x, y, z) in the line (x)/(1) = (y-1)/(2) = (z-2)/(3) is P'(1, 0, 7), then find the coordinates of point P.

If A = 1 -2 0 \2 -1 -1 \0 -2 1 , find A⁻¹ and use it to solve the following system of equations: x - 2y = 10, 2x - y - z = 8, -2y + z = 7.

Evaluate: ∫₀π/4 (sin x + cos x)/(9 + 16sin 2x) dx

According to recent research, air turbulence is increasing in various regions across the world due to climate change. Air turbulence makes flying difficult and often delays flights. Assume that an aeroplane experiences severe turbulence, moderate turbulence or light turbulence with equal probability. Further, the probabilities of the aeroplane arriving late at its destination due to severe turbulence, moderate turbulence and light turbulence are 55\%, 37\% and 17\% respectively. Based on the above information, answer the following questions: (i) Find the probability that the aeroplane arrives late at its destination. 2 (ii) If the aeroplane arrives late at its destination, find the probability that it was due to moderate turbulence. 2 Case Study – 2 **37.** According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that, an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are 55\%, 37\% and 17\% due to severe, moderate and light turbulence respectively. On the basis of the above information, answer the following questions : (i) Find the probability that an airplane reached its destination late. (ii) If the airplane reached its destination late, find the probability that it was due to moderate turbulence. CASE STUDY - 3

The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is θ. On the basis of the above information, answer the following questions: (i) Express θ in terms of the height of the camera installed on the pole and x. (ii) Find (dθ)/(dx). (iii)(a) Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole.

If a function f : X → Y defined as f(x) = y is one-one and onto, then we can define a unique function g : Y → X such that g(y) = x, where x ∈ X and y = f(x), y ∈ Y. Function g is called the inverse of function f. The domain of sine function is mathbbR and function sin : mathbbR → mathbbR is neither one-one nor onto. Let sine function be defined from set A to [-1, 1] such that inverse of sine function exists, i.e., sin⁻¹x is defined from [-1, 1] to A. On the basis of the above information, answer the following questions: (i) If A is the interval other than principal value branch, give an example of one such interval. (ii) If sin⁻¹(x) is defined from [-1, 1] to its principal value branch, find the value of sin⁻¹(-(1)/(2)) - sin⁻¹(1). (iii)(a) Draw the graph of sin⁻¹x from [-1, 1] to its principal value branch.