Commerce (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

If A `= [("cos x", - "sin x"),("sin x", "cos x")]`, find AAT.

If A `= [(0,-1,2),(1,0,3),(-2,-3,0)],` then A + 2A^T equals

If the matrix A `= [(5,2,"x"),("y",2,-3),(4, "t",-7)]` is a symmetric matrix, then find the value of x, y and t respectively.

If a matrix A is both symmetric and skew-symmetric, then ____________.

The matrix `[(0,5,-7),(-5,0,11),(7,-11,0)]` is ____________.

The matrix A `=[(0,1),(1,0)]` is a ____________.

The matrix `[(0,-5,8),(5,0,12),(-8,-12,0)]`  is a ____________.

If `[(1,2),(3,4)],` then A² - 5A is equal to ____________.

Find the angle between the lines whose direction cosines are given by the equations: 3l + m + 5n = 0 and 6mn – 2nl + 5lm = 0.

Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l² + m² – n² = 0.

The differential coefficient of `"tan"^-1 ((sqrt(1 + "x") - sqrt (1 - "x"))/(sqrt (1+ "x") + sqrt (1 - "x")))` is ____________.

Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.

If l₁, m₁, n₁; l₂, m₂, n₂; l₃, m₃, n₃ are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to l₁ + l₂ + l₃, m₁ + m₂ + m₃, n₁ + n₂ + n₃ makes equal angles with them.

Maximise and Minimise Z = 3x – 4y subject to x – 2y ≤ 0, – 3x + y ≤ 4, x – y ≤ 6, x, y ≥ 0

Corner points of the feasible region determined by the system of linear constraints are (0, 3), (1, 1) and (3, 0). Let Z = px + qy, where p, q > 0. Condition on p and q so that the minimum of Z occurs at (3, 0) and (1, 1) is ______.

The probability of simultaneous occurrence of at least one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then prove that P(A′) + P(B′) = 2 – 2p + q.

10% of the bulbs produced in a factory are of red colour and 2% are red and defective. If one bulb is picked up at random, determine the probability of its being defective if it is red.

Two dice are thrown together. Let A be the event ‘getting 6 on the first die’ and B be the event ‘getting 2 on the second die’. Are the events A and B independent?

If A and B are independent events such that 0 < P(A) < 1 and 0 < P(B) < 1, then which of the following is not correct?

If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of A, B) = `5/9`, then p = ______.