Question

The length x, of a rectangle is decreasing at the rate of 5 cm/minute and the width y, is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rate of change of the area of the rectangle.

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. $[\mathrm{Use}\:\pi=\:\frac{22}{7}\:]$

If P(A) = 2/5, P(B) = 1/3, P(A ∩ B) = 1/5, then find P(A' / B') .

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

If E and F be two events such that P(E) = 1/3, P(F) = 1/4, find P(E U F) if E and F are independent events.

Find the particular solution of the differential equation ${\frac{d y}{d x}}={\frac{1+y^{2}}{1+x^{2}}}$ given that y(0) = ${\sqrt{3}}.$

Verify that ax² + by² = 1 is a solution of differential equation $x(y y_{2}+y_{1}^{2})=y y_{1}.$