Question

Find : $\textstyle\int$$\left({\frac{1-x}{1+x^{2}}}\right)^{2}e^{x}d x~.$

Show that the function f(x) = x³ – 3x² + 6x – 100 is increasing on R.

The radius r of the base of a right circular cone is decreasing at the rate of 2 cm/min. and height h is increasing at the rate of 3 cm/min. When r = 3.5 cm and h = 6 cm, find the rate of change of the volume of the cone. $[\mathrm{Use}\:\pi=\:\frac{22}{7}\:]$

Find the sum of the order and the degree of the following differential equations : ${\frac{d^{2}y}{d x^{2}}}+{3{\sqrt\frac{{d y}}{d x}}}+(1+x)=0$

Find : $\textstyle\int$$\frac{x}{\sqrt{32-x^{2}}}\,d x$

A couple has 2 children. Find the probability that both are boys, if it is known that (i) one of them is a boy (ii) the older child is a boy.

Three cards are drawn without replacement from a pack of 52 cards. Find the probability that (i) the cards drawn are king, queen and jack respectively. (ii) the cards drawn are king, queen and jack.