Question

From the differential equation of equation y = a cos 2x + b sin 2x, where a and b are constant.

If P(A) = 0.6, P(B) = 0.5 and P(A|B) = 0.3, then find P(A ∪ B).

One bag contains 3 red and 5 black balls. Another bag contains 6 red and 4 black balls. A ball is transferred from first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is red.

If E and F are independent events, then show that (i) E and F' are independent events. (ii) E' and F are also independent events.

The radius r of a right circular cylinder is increasing at the rate of 5 cm/min and its height h, is decreasing at the rate of 4 cm/min. When r= 8 cm and h = 6 cm, find the rate of change of the volume of cylinder.

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$