CBSE 12 MATHS 2025 SET5

Let both AB and BA be defined for matrices A and B. If order of A is n × m, then the order of B is:

If A = [[1, 0, 0], [0, -3, 0], [0, 0, 5]], then A is a/an:

The following graph is a combination of:

Sum of two skew-symmetric matrices of same order is always a/an:

sec⁻¹(−2) − tan⁻¹(1/√3) is equal to:

If f(x) = [log(1 + ax) − log(1 − bx)] / x, for x ≠ 0, and f(x) = k, for x = 0, is continuous at x = 0, then the value of k is:

If tan⁻¹(x² − y²) = a, where 'a' is a constant, then dy/dx is:

If y = a cos(log x) + b sin(log x), then x²y₂ + xy₁ is:

Let f(x) = |x|, x ∈ R. Then, which of the following statements is incorrect?

Let f′(x) = (2/x³)(x⁴ + 2x) − 5, f(1) = 0. Then, f(x) is:

∫ (xeˣ + 5 − 6) / (x + 6)² dx is equal to:

The order and degree of the following differential equation are, respectively: d⁴y/dx⁴ − 2e^(dy/dx) + y = 0

The solution for the differential equation dy/dx = log(3x + 4y) is:

For a Linear Programming Problem (LPP), the given objective function is Z = x + 2y. The feasible region PQRS determined by the set of constraints is shown as a shaded region in the graph. P = (3/13, 24/13), Q = (3/2, 15/4), R = (7/2, 3/4), S = (18/7, 2/7). Which of the following statements is correct?

In a Linear Programming Problem (LPP), the objective function Z = 2x + 5y is to be maximised under the following constraints: x + y ≤ 4, 3x + 3y ≤ 18, x, y ≥ 0. Study the graph and select the correct option. The solution of the given LPP:

Let |a⃗| = 5 and −2 ≤ λ ≤ 1. Then, the range of |λa⃗| is:

The area of the region bounded by the curve y² = x between x = 0 and x = 1 is:

A box has 4 green, 8 blue and 3 red pens. A student picks up a pen at random, checks its colour and replaces it in the box. He repeats this process 3 times. The probability that at least one pen picked was red is:

Assertion: If |a⃗ × b⃗|² + |a⃗ · b⃗|² = 256 and |b⃗| = 8, then |a⃗| = 2. Reason: sin²θ + cos²θ = 1 and |a⃗ × b⃗| = |a⃗||b⃗|sinθ and a⃗ · b⃗ = |a⃗||b⃗|cosθ.

Assertion: Let f(x) = eˣ and g(x) = log x. Then (f + g)(x) = eˣ + log x where domain of (f + g) is R. Reason: Dom(f + g) = Dom(f) ∩ Dom(g).

Find the domain of f(x) = sin⁻¹(−x²).

(a) Differentiate 2^(eˣ) with respect to e^(2x) for x > 0. OR (b) If (x)^y = (y)^x, then find dy/dx.

Determine the values of x for which f(x) = (x − 4)/(x + 1), x ≠ −1, is an increasing or a decreasing function.

(a) If a⃗ and b⃗ are position vectors of point A and point B respectively, find the position vector of point C on BA produced such that BC = 3BA. OR (b) Vector r⃗ is inclined at equal angles to the three axes x, y and z. If magnitude of r⃗ is 5√3 units, then find r⃗.

Determine if the line r = (î + ĵ − k̂) + λ(3î − ĵ) and r = (4î − k̂) + μ(2î + 3k̂) intersect with each other.

Let A = [[1, 4], [2, −1]] and C = [[3, 4, 2], [12, 16, 8], [−6, −8, −4]] be two matrices. Then, find the matrix B if AB = C.

(a) Differentiate y = sin⁻¹(3x − 4x³) w.r.t. x, if x ∈ (−1/2, 1/2). OR (b) Differentiate y = cos⁻¹((1 − x²)/(1 + x²)) with respect to x, when x ∈ (0, 1).

(a) A student wants to pair up natural numbers in such a way that they satisfy the equation 2x + y = 41, x, y ∈ N. Find the domain and range of the relation. Check if the relation thus formed is reflexive, symmetric and transitive. Hence, state whether it is an equivalence relation or not. OR (b) Show that the function f: N → N, where N is a set of natural numbers, given by f(n) = n − 1 if n is even, and f(n) = n + 1 if n is odd, is a bijection.

Consider the Linear Programming Problem, where the objective function Z = (x + 4y) needs to be minimized subject to constraints: 2x + y ≥ 1000 x + 2y ≥ 800 x, y ≥ 0. Draw a neat graph of the feasible region and find the minimum value of Z.

(a) Find the distance of the point P(2, 4, −1) from the line (x + 5)/1 = (y + 3)/4 = (z − 6)/(−9). OR (b) Let the position vectors of the points A, B and C be 3î − ĵ − 2k̂, î + 2ĵ − k̂ and î + 5ĵ + 3k̂ respectively. Find the vector and cartesian equations of the line passing through A and parallel to line BC.

A person is Head of two independent selection committees I and II. If the probability of making a wrong selection in committee I is 0.03 and that in committee II is 0.01, then find the probability that the person makes the correct decision of selection: (i) in both committees (ii) in only one committee

(a) Find: ∫ (x² + 1) / ((x² − 1)(x + 3)) dx OR (b) Evaluate: ∫₀^(π/2) x / (sin x + cos x) dx

Draw a rough sketch for the curve y = 2 + |x + 1|. Using integration, find the area of the region bounded by the curve y = 2 + |x + 1|, x = −4, x = 3 and y = 0.

(a) Solve the differential equation: x²y dx − (x³ + y³) dy = 0. OR (b) Solve the differential equation (1 + x²) dy/dx + 2xy − 4x² = 0 subject to initial condition y(0) = 0.

Let the polished side of the mirror be along the line (x − 1)/1 = (y − 2)/2 = (z − 4)/(−6). A point P(1, 6, 3), some distance away from the mirror, has its image formed behind the mirror. Find the coordinates of the image point and the distance between the point P and its image.

Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹90. Sam pays ₹70 for 6 pens, 2 notepads and 3 erasers. Based upon the above information, answer the following questions: (i) Form the equations required to solve the problem of finding the price of each item, and express it in the matrix form AX = B. (ii) Find |A| and confirm if it is possible to find A⁻¹. (iii) (a) Find A⁻¹, if possible, and write the formula to find X. OR (iii) (b) Find A² − 8I, where I is an identity matrix.

A ladder of fixed length 'h' is to be placed along the wall such that it is free to move along the height of the wall. Based upon the above information, answer the following questions: (i) Express the distance (y) between the wall and foot of the ladder in terms of 'h' and height (x) on the wall at a certain instant. Also, write an expression in terms of h and x for the area (A) of the right triangle, as seen from the side by an observer. (ii) Find the derivative [question cut off]

A ladder of fixed length 'h' is to be placed along the wall such that it is free to move along the height of the wall. Based upon the above information, answer the following questions: (i) Express the distance (y) between the wall and foot of the ladder in terms of 'h' and height (x) on the wall at a certain instant. Also, write an expression in terms of h and x for the area (A) of the right triangle, as seen from the side by an observer. (ii) Find the derivative of the area (A) with respect to the height on the wall (x), and find its critical point. (iii) (a) Show that the area (A) of the right triangle is maximum at the critical point. OR (iii) (b) If the foot of the ladder whose length is 5 m, is being pulled towards the wall such that the rate of decrease of distance (y) is 2 m/s, then at what rate is the height on the wall (x) increasing, when the foot of the ladder is 3 m away from the wall?

A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances of their manufacturing a defective smartphone are only 5%, 4% and 2% respectively. In his inventory he has 25% smartphones from company A, 35% smartphones from company B and 40% smartphones from company C. A person buys a smartphone from this shop. (i) Find the probability that it was defective. (ii) What is the probability that this defective smartphone was manufactured by company B?