CBSE Class 12 Mathematics — Real Previous-Year Papers

The order and degree of the differential equation d dx(ey) = 0 respectively are 1 (A) 0, 1 (B) 1, 1 (C) 2, 1 (D) 1, not defined

The value of p for which vectors hati + 2hatj + 3hatk and 2hati - phatj + hatk are perpendicular to each other is (A) 0 (B) 1 (C) (5)/(2) (D) -(5)/(2)

The value of m for which the points with position vectors -hati - hatj + 2hatk, 2hati + mhatj + 5hatk and 3hati + 11hatj + 6hatk are collinear, is (A) 8 (B) -8 (C) 2 (D) (5)/(2)

The feasible region of a linear programming problem with objective function Z = 5x + 7y is shown below : 1 The maximum value of Z – minimum value of Z is (A) 8 (B) 29 (C) 35 (D) 43

If Δ₁ = 1 0 0 \0 2 0 \0 0 3 and Δ₂ = 0 2 0 \1 0 0 \0 0 6 , then (A) Δ₁ = 2Δ₂ (B) Δ₂ = -2Δ₁ (C) Δ₁ = Δ₂ (D) Δ₂ = -Δ₁

If ∫ (3ax)/(b² + c²x²) dx = A log |b² + c²x²| + K, then the value of A is: (A) 3a (B) (3a)/(2b²) (C) (3a)/(b²c²) (D) (3a)/(2c²)

If 2cos⁻¹x = y, then (A) 0 ≤ y ≤ π (B) -π ≤ y ≤ π (C) 0 ≤ y ≤ 2π (D) -π ≤ y ≤ 0

The least value of f(x) = x³ - 12x, x ∈ [0, 3] is (A) -16 (B) -9 (C) 0 (D) 16

The value of ∫-1¹ (x³)/(x² + 2|x| + 1)dx is (A) 0 (B) log 2 (C) 2log 2 (D) (1)/(2)log 2

If |veca| = 8, |vecb| = 3 and |veca × vecb| = 12, then the value of |veca · vecb| is (A) 6√(3) (B) 8√(3) (C) 12√(3) (D) None of these

The length of the perpendicular from the point (2, 5, 7) on the line (x)/(1) = (y)/(0) = (z)/(0) is (A) 2 (B) 5 (C) √(74) (D) √(78)

The degree of an objective function of a linear programming problem is (A) 0 (B) 1 (C) 2 (D) Any natural number

Assertion (A): In an experiment of throwing an unbiased die, the probability of getting a prime number given that the number appearing on the die is odd is (2)/(3). Reason (R): For any two events A and B, P(A|B) = (P(A ∪ B))/(P(B)). (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): The lines x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular to each other when pp' + rr' = 1. Reason (R): Two lines vecr = veca₁ + λvecb₁ and vecr = veca₂ + μvecb₂ are perpendicular to each other if vecb₁ · vecb₂ = 0. (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.

A room freshener bottle in the shape of an inverted cone sprays at regular intervals, due to which the volume of perfume in the bottle decreases at the rate of 1 mm³/min. If the semi-vertical angle of the conical bottle is (π)/(6), then find the rate at which the level of perfume in the bottle is decreasing, when the level of perfume in the bottle is 10 mm.

Find a vector of magnitude 14 in the direction of vecQP for the points P(1, 3, 2) and Q(-1, 0, 8).

Vectors veca = 3hati - 2hatj + 2hatk and vecb = hati + 2hatk represent the adjacent sides of a parallelogram. Find the vectors representing the diagonals of the parallelogram. Also, find their lengths.

(a) Check whether the function f(x) = (|x-3|)/(2(x-3)), x < 3 ; [2mm] (x-6)/(6), x ≥ 3 is continuous at x = 3 or not. OR (b) If √(3)(x² + y²) = 4xy, then find (dy)/(dx) at ((1)/(2), dfrac√(3)2).

(a) Simplify: tan⁻¹((cos 2x - sin 2x)/(cos 2x + sin 2x)), where 0 < x < (π)/(4). OR (b) Evaluate: tan(sin⁻¹ 1 - cos⁻¹(-(1)/(2))).

If I₁ = ∫-π/4π/4 (dx)/(1 + cos 2x) and I₂ = ∫-1/21/2 |x| dx, then show that I₁ - 4I₂ = 0.

Solve the following Linear Programming Problem (LPP) graphically: Constraints: x+y ≤ 7 2x-3y+6 ≥ 0 x ≥ 0, y ≥ 0 Minimize Z = 13x - 15y under the given constraints. 3

Evaluate: ∫₀¹ xtan⁻¹xdx.

(a) Find: ∫ √((x+2)/(x-2))dx. OR (b) Find: ∫ (x²)/((x² + 9)(x² + 16))dx.

(a) Find the general solution of the differential equation x²(dy)/(dx) = x² + xy + y². OR (b) Find the particular solution of the differential equation xy(dy)/(dx) = (x+2)(y+2), given that y = -1 when x = 1.

(a) Out of two bags, Bag I contains 3 red and 4 white balls and Bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3, then a ball is drawn at random from Bag I, otherwise a ball is drawn at random from Bag II. Find the probability that the ball drawn is a red ball. OR (b) The probability of simultaneous occurrence of at least one of two events X and Y is a. If the probability that exactly one of X, Y occurs is b, then prove that P(X') + P(Y') = 2 - 2a + b.

If x = cos t, y = cos mt, then prove that (1 - x²) (d²y)/(dx²) - x (dy)/(dx) + m²y = 0.

Check whether the lines (x - 1)/(2) = (y - 2)/(3) = (z - 3)/(4) and (x - 4)/(5) = (y - 1)/(2) = z are parallel or not. If they are parallel, find the distance between them, otherwise if the lines are intersecting, find their point of intersection.

(a) A relation R is defined on mathbbZ, the set of integers, as R = \(x, y) : |x - y| is divisible by a prime number p,x, y ∈ mathbbZ\. Check whether R is an equivalence relation or not. OR (b) A function f : mathbbR - \(3)/(5)\ → mathbbR - \(3)/(5)\ is defined as f(x) = (3x + 2)/(5x - 3). Prove that f is one-one and onto.

(a) If A = 0 2 1 \-2 -1 -2 \1 -1 0 , find A⁻¹ and use it to solve the following system of equations: -2y + z = 7, 2x - y - z = 8, x - 2y = 10. OR (b) If 3 -1 sin 3x \-7 4 cos 2x \-11 7 2 is a singular matrix, then find all values of x where x ∈ [0, (π)/(2)].

To reduce traffic and avoid red lights, roundabouts are often built on busy roads. Such a roundabout is built such that its boundary C₁ : x² + y² = 64 is given. In the middle of this roundabout, there is a circular pond with a fountain, whose equation is C₂ : x² + y² = 4. Based on the above information, answer the following questions: (i) Represent equations C₁ and C₂ graphically. 1 (ii) For both C₁ and C₂, express y as a function of x. (y = f(x)) 1 (iii) (A) Find the area enclosed by the entire roundabout using integration. 2 OR (iii) (B) Find the area enclosed by the circular pond using integration. 2

An online delivery company in a city has 5,000 subscribers and collects an annual subscription fee of ₹300 per subscriber for unlimited free deliveries. The company wishes to increase the annual subscription fee. It is predicted that for every increase of ₹1, ten subscribers will discontinue. Based on the above information, answer the following questions: (i) How many subscribers will discontinue after an increase of ₹x in the annual subscription fee? (ii) If R(x) denotes the total revenue collected after the increase of ₹x in the subscription fee, express R(x) as a function of x. (iii)(a) Find the value of x for which R(x) is maximum. OR (iii)(b) Find the sub-intervals of (0, 5000) in which R(x) is increasing and decreasing.

In an online jackpot, there is one first prize of ₹3,00,000, two second prizes of ₹2,00,000 each and three third prizes of ₹50,000 each. A total of 1,00,000 jackpot tickets, each costing ₹100, were sold. Rohan bought one ticket. Based on the above information, answer the following questions: (i) What are the possible amounts that the person can win? (ii)(a) What is the probability that the person wins at least ₹2,00,000? OR (ii)(b) What is the probability that the person does not win any prize? (iii) Rohan also bought another ticket having a prize money of ₹5,00,000, where the chances of winning are 1 in 1,00,000. Find the probability that on exactly one of the tickets he wins.