CBSE Class 12 Mathematics 2025 — Set 65/2/1

If E and F are two events such that P(E) > 0 and P(F) ≠ 1, then P(E'|F') is: (A) (P(E'))/(P(F')) (B) 1 - P(E'|F) (C) 1 - P(E|F) (D) (1 - P(E ∪ F))/(P(F'))

Which of the following can be both a symmetric and skew-symmetric matrix? (A) Unit Matrix (B) Diagonal Matrix (C) Null Matrix (D) Row Matrix

The equation of a line parallel to the vector 3hati+hatj+2hatk and passing through the point (4,-3,7) is: (A) x = 4t+3,y = -3t+1,z = 7t+2 (B) x = 3t+4,y = t+3,z = 2t+7 (C) x = 3t+4,y = t-3,z = 2t+7 (D) x = 3t+4,y = -t+3,z = 2t+7

Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify 4AB + 3(AB + BA) - 4BA, where A and B are both matrices of order 2 × 2. It is known that A ≠ B ≠ I and A⁻¹ ≠ B. Their answers are given as. Who answered it correctly? (A) Abhay: 6AB (B) Bina: 7AB - BA (C) Chhaya: 8AB (D) Devesh: 7BA - AB

A cylindrical tank of radius 10 cm is being filled with sugar at the rate of 100π cm³/s. The rate at which the height of the sugar inside the tank is increasing is: (A) 0.1 cm/s (B) 0.5 cm/s (C) 1 cm/s (D) 1.1 cm/s

Let vecp and vecq be two unit vectors and α the angle between them. Then (vecp+vecq) will be a unit vector if the value of α is: (A) (π)/(4) (B) (π)/(3) (C) (π)/(2) (D) (2π)/(3)

The line x = 1 + 5μ,y = -5 + μ,z = -6 - 3μ passes through which of the following point? (A) (1, -5, 6) (B) (1, 5, 6) (C) (1, -5, -6) (D) (-1, -5, 6)

The area of the shaded region (figure) represented by the curves y = x²,0 ≤ x ≤ 2, and the y-axis is given by: (A) ∫₀² x²dx (B) ∫₀² √(y)dy (C) ∫₀⁴ x²dx (D) ∫₀⁴ √(y)dy

A factory produces two products X and Y. The profit earned by selling X and Y is represented by the objective function Z = 5x + 7y, where x and y are the number of units of X and Y respectively sold. Which of the following statement is correct? (A) The objective function maximizes the difference of the profit earned from products X and Y. (B) The objective function measures the total production of products X and Y. (C) The objective function maximizes the combined profit earned from selling X and Y. (D) The objective function ensures the company produces more of product X than product Y.

If A and B are square matrices of order m such that A² - B² = (A - B)(A + B), then which of the following is always correct? (A) A = B (B) AB = BA (C) A = 0 or B = 0 (D) A = I or B = I

If p and q are respectively the order and degree of the differential equation (d)/(dx)(((dy)/(dx))³) = 0, then (p - q) is: (A) 0 (B) 1 (C) 2 (D) 3

Assertion (A): A = operatornamediag[352] is a scalar matrix of order 3 × 3. Reason (R): If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix. (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.

Assertion (A): Every point of the feasible region of a Linear Programming Problem is an optimal solution. Reason (R): The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region. (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.

(a) A vector veca makes equal angles with all the three axes. If the magnitude of the vector is 5√(3) units, find veca. OR (b) If vecα and vecβ are position vectors of two points P and Q respectively, then find the position vector of a point R in QP produced such that vecQR = (3)/(2)vecQP.

Evaluate: ∫₀π/4 √(1 + sin 2x) dx

Find the values of a for which f(x) = sin x - ax + b is increasing on mathbbR.

If veca and vecb are two non-collinear vectors, then find x such that vecα = (x - 2)veca + vecb and vecβ = (3 + 2x)veca - 2vecb are collinear.

(a) If x = ex/y, then prove that (dy)/(dx) = (x - y)/(x log x). OR (b) If f(x) = 2x - 3, -3 ≤ x ≤ -2 x + 1, -2 < x ≤ 0 , check the differentiability of f(x) at x = -2.

(a) Solve the differential equation 2(y + 3) - xy(dy)/(dx) = 0; given that y(1) = -2. OR (b) Solve the following differential equation: (1 + x²)(dy)/(dx) + 2xy = 4x².

Let R be a relation defined over mathbbN, where mathbbN is the set of natural numbers, defined as "mRn if and only if m is a multiple of n, m, n ∈ mathbbN." Find whether R is reflexive, symmetric and transitive or not.

Solve the following linear programming problem graphically: Minimise Z = x - 5y subject to the constraints: x - y ≥ 0, -x + 2y ≥ 2, x ≥ 3, y ≤ 4, y ≥ 0.

(a) If y = [log(√(x) + dfrac1√(x))]², then show that x(x + 1)² y₂ + (x + 1)² y₁ = 2. OR (b) If x√(1 + y) + y√(1 + x) = 0, -1 < x < 1, x ≠ y, then prove that (dy)/(dx) = -(1)/((1 + x)²).

(a) A die with numbers 1 to 6 is biased such that P(2) = (3)/(10) and the probability of other numbers is equal. Find the mean of the number of times the number 2 appears on the die, if the die is thrown twice. OR (b) Two dice are thrown. Defined are the following two events A and B: A = \(x, y) : x + y = 9\, B = \(x, y) : x ≠ 3\, where (x, y) denotes a point in the sample space. Check if events A and B are independent or mutually exclusive.

Find: ∫ (1)/(x)√((x + a)/(x - a)) dx

Using integration, find the area of the region bounded by the line y = 5x + 2, the x-axis and the ordinates x = -2 and x = 2.

Find: ∫ dfracx²+x+1(x+2)(x²+1)dx

(a) Find the shortest distance between the lines: (x+1)/(2) = (y-1)/(1) = (z-9)/(-3) and (x-3)/(2) = (y+15)/(-7) = (z-9)/(5). OR (b) Find the image A' of the point A(2, 1, 2) in the line l:vecr = (4hati + 2hatj + 2hatk) + λ(hati - hatj - hatk). Also find the equation of the line joining A and A', and the foot of the perpendicular from A on the line l.

(a) Given A = -4 4 4 \-7 1 3 \5 -3 -1 and B = 1 -1 1 \1 -2 -2 \2 1 3 , find AB. Hence, solve the system of equations: x - y + z = 4, x - 2y - 2z = 9, 2x + y + 3z = 1. OR (b) If A = 1 -2 0 \2 -1 -2 \0 -1 1 , find A⁻¹. Hence, solve the system of equations: x - 2y = 10, 2x - y - z = 8, -2y + z = 7.

A school is organising a debate competition with participants as speakers S = \S₁, S₂, S₃, S₄\ and these are judged by judges J = \J₁, J₂, J₃\. Each speaker can be assigned one judge. Let R be a relation from set S to J defined as R = \(x, y) : speaker x is judged by judge y,x ∈ S,y ∈ J\. (i) How many relations can be there from S to J? [1] (ii) A student identifies a function from S to J as f = \(S₁, J₁), (S₂, J₂), (S₃, J₂), (S₄, J₃)\. Check if it is bijective. [1] (iii)(a) How many one-one functions can be there from set S to set J? [2] OR (iii)(b) Another student considers a relation R₁ = \(S₁, S₂), (S₂, S₄)\ in set S. Write minimum ordered pairs to be included in R₁ so that R₁ is reflexive but not symmetric. [2]

Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their production share in the market is 60%, 30% and 10% respectively. Of their respective production capacities, 20%, 10% and 5% cars respectively are electric (or battery operated). (i)(a) What is the probability that a randomly selected car is an electric car? [2] OR (i)(b) What is the probability that a randomly selected car is a petrol car? [2] (ii) A car is selected at random and is found to be electric. What is the probability that it was manufactured by Comet? [1] (iii) A car is selected at random and is found to be electric. What is the probability that it was manufactured by Amber or Bonzi? [1]

A small town is analysing the pattern of a new street light installation. The lights are set up such that the intensity of light at any point x metres from the start of the street can be modelled by f(x) = exsin x, where x is in metres. (i) Find the intervals on which f(x) is increasing or decreasing, x ∈ [0, π]. [2] (ii) Verify whether each critical point when x ∈ [0, π] is a point of local maximum or local minimum or a point of inflexion. [2]