Let `-1lt=p<1.` Show that the equation `4x^3-3x-p=0` has a unique root in the interval `[1/2,1]` and identify it. (2001, 4M)

Let 0lt alpha lt pi //2 be a fixed angle. If P=(cos theta, sin theta) and Q=(cos (alpha-theta),sin (alpha-theta)) . then Q is obtained from P by

The probability distribution of random variable X is given by : Let p=P(1 lt X lt 4"|"X lt 3) . If 5p=lambda K , then lambda is equal to _________.

Let 1 le m lt n le p . The number of subsets o the set A={1,2,3, . . .P} having m, n as the least and the greatest elements respectively, is