Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is :

If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of these boxes contain exactly 2 and 3 balls is :

A random variable X has the following probability distribution : .tg {border-collapse:collapse;border-spacing:0;width:100%} .tg td{font-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;} .tg th{font-family:Arial, sans-serif;font-size:14px;font-weight:normal;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;} .tg .tg-baqh{text-align:center;vertical-align:top} .tg .tg-wa1i{font-weight:bold;text-align:center;vertical-align:middle} .tg .tg-nrix{text-align:center;vertical-align:middle} X: 1 2 3 4 5 P(X): K2 2K K 2K 5K2 Then P(X > 2) is equal to :

In a box, there are 20 cards, out of which 10 are lebelled as A and the remaining 10 are labelled as B. Cards are drawn at random, one after the other and with replacement, till a second A-card is obtained. The probability that the second A-card appears before the third B-card is :

Let A and B be two events such that the probability that exactly one of them occurs is {2 \over 5} and the probability that A or B occurs is {1 \over 2} , then the probability of both of them occur together is :

Let A and B be two independent events such that P(A) = {1 \over 3} and P(B) = {1 \over 6}. Then, which of the following is TRUE?

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is {{1 \over 4}} . If the probability that at most two machines will be out of service on the same day is {\left( {{3 \over 4}} \right)^3}k, then k is equal to :

An unbiased coin is tossed 5 times. Suppose that a variable X is assigned the value of k when k consecutive heads are obtained for k = 3, 4, 5, otherwise X takes the value -1. Then the expected value of X, is :

Let ƒ and g be differentiable functions on R such that fog is the identity function. If for some a, b $$\in$$ R, g'(a) = 5 and g(a) = b, then ƒ'(b) is equal to :

The derivative of {\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right) with respect to {\tan ^{ - 1}}\left( {{{2x\sqrt {1 - {x^2}} } \over {1 - 2{x^2}}}} \right) at x = {1 \over 2} is :

If $$\left(a+\sqrt{2}b\cos x\right)\left(a-\sqrt{2}b\cos y\right)=a^2-b^2$$ where a > b > 0, then {{dx} \over {dy}}\,\,at\left( {{\pi \over 4},{\pi \over 4}} \right) is :

If y2 + loge (cos2x) = y, x \in \left( { - {\pi \over 2},{\pi \over 2}} \right), then :

If $$x=2\sin \theta -\sin 2\theta$$ and $$y=2\cos \theta -\cos 2\theta$$, $$\theta \in \left[0,2\pi \right]$$, then {{{d^2}y} \over {d{x^2}}} at $$\theta$$ = $$\pi$$ is :

Let ƒ(x) = (sin(tan–1x) + sin(cot–1x))2 – 1, |x| > 1. If {{dy} \over {dx}} = {1 \over 2}{d \over {dx}}\left( {{{\sin }^{ - 1}}\left( {f\left( x \right)} \right)} \right) and y\left( {\sqrt 3 } \right) = {\pi \over 6}, then y($$-\sqrt{3}$$) is equal to :

Let y = y(x) be a function of x satisfying $$y\sqrt{1-x^2}=k-x\sqrt{1-y^2}$$ where k is a constant and y\left( {{1 \over 2}} \right) = - {1 \over 4}. Then {{dy} \over {dx}} at x = {1 \over 2}, is equal to :

If y\left( \alpha \right) = \sqrt {2\left( {{{\tan \alpha + \cot \alpha } \over {1 + {{\tan }^2}\alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }}} ,\alpha \in \left( {{{3\pi } \over 4},\pi } \right) {{dy} \over {d\alpha }}\,\,at\,\alpha = {{5\pi } \over 6}is :

Let xk + yk = ak, (a, k > 0 ) and {{dy} \over {dx}} + {\left( {{y \over x}} \right)^{{1 \over 3}}} = 0, then k is:

If y = \left( {{2 \over \pi }x - 1} \right) cosec\,x is the solution of the differential equation, {{dy} \over {dx}} + p\left( x \right)y = {2 \over \pi } cosec\,x, 0 < x < {\pi \over 2}, then the function p(x) is equal to :

The general solution of the differential equation $$\sqrt{1+x^2+y^2+x^2{y}^2}$$ + xy{{dy} \over {dx}} = 0 is : (where C is a constant of integration)

Let y = y(x) be the solution of the differential equation cosx{{dy} \over {dx}} + 2ysinx = sin2x, x $$\in$$ \left( {0,{\pi \over 2}} \right). If y\left( {{\pi \over 3}} \right) = 0, then y\left( {{\pi \over 4}} \right) is equal to :