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

If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N

If A = `[(α, β),(γ, -α)]` is such that A² = I, then ______.

If A is a square matrix such that A² = A, then (I + A)³ – 7A is equal to ______.

If `P(A)  = 3/5 and P(B) = 1/5` , find P (A ∩ B) if A and B are independent events.

A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not.

Let E and F be events with `P(E) = 3/5, P(F) = 3/10 and P(E ∩ F) = 1/5`.  Are E and F independent?

Given that the events A and B are such that `P(A) = 1/2, PA∪B=3/5 and P (B) = p`. Find p if they are

1. mutually exclusive
2. independent.

Let A and B be independent events with P (A) = 0.3 and P (B) = 0.4. Find 

1. P (A ∩ B)
2. P (A ∪ B)
3. P (A | B)
4. P (B | A)

If A and B are two events such that `P(A) = 1/4, P(B) = 1/2 and P(A ∩ B) = 1/8`, find P (not A and not B).

Events A and B are such that `P(A) = 1/2, P(B) = 7/12 and P("not A or not B") = 1/4` . State whether A and B are independent?

Given two independent events A and B such that P (A) = 0.3, P (B) = 0.6. Find 

1. P (A and B)
2. P(A and not B)
3. P(A or B)
4. P(neither A nor B)

Probability of solving specific problem independently by A and B are `1/2` and `1/3` respectively. If both try to solve the problem independently, find the probability that

1. the problem is solved
2. exactly one of them solves the problem.

One card is drawn at random from a well-shuffled deck of 52 cards. In which of the following case is the events E and F independent?

E : ‘the card drawn is a king or queen’

F : ‘the card drawn is a queen or jack’

Two events, A and B, will be independent if ______.

If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability `1/2`).

Find the angle between the following pair of lines:

`vecr = 2hati - 5hatj + hatk + lambda(3hati - 2hatj + 6hatk) and vecr = 7hati - 6hatk + mu(hati + 2hatj + 2hatk)`

Find the angle between the following pair of lines:

`vecr = 3hati + hatj - 2hatk + lambda(hati - hatj - 2hatk) and vecr = 2hati - hatj -56hatk + mu(3hati - 5hatj - 4hatk)`

Find the angle between the following pairs of lines: 

`(x-2)/2 = (y-1)/5 = (z+3)/(-3)` and `(x+2)/(-1) = (y-4)/8 = (z -5)/4`

Find the angle between the following pairs of lines:

`x/y = y/2 = z/1` and `(x-5)/4 = (y-2)/1 = (z - 3)/8`

Find the values of p so the line `(1-x)/3 = (7y-14)/2p = (z-3)/2` and `(7-7x)/(3p) = (y -5)/1 = (6-z)/5` are at right angles.