A die is rolled. If E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5}, find (a) P(E U F)/G] (b)P [(E ∩ F)/G].

A particle moves along the curve 6y = x³ + 2. Find the points on the curve at which y-coordinate is changing 2 times as fast as x-coordinate.

Find the differential equation of the family of curves y²= 4ax.

For the curve y = 5x – 2x³, if x increases at the rate of 2 units/sec, then find the rate of change of the slope of the curve when x = 3.

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.

If P(E) = 6/11, P(F) = 5/11 and P(E ∪ F) = 7/11 then find (i) P(E/F) (ii) P(F/E)

If P(E) = 7/13, P(F) = 9/3 and P(E' / F') = 4/3, then evaluate : (i) P(E / F) (ii) P(E / F)

The radius r of a right circular cone is decreasing at the rate of 3 cm/minute and the height h is increasing at the rate of 2 cm/minute. When r = 9 cm and h = 6 cm, find the rate of change of its volume.

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.

If A and B are two independent events, prove that A’ and B are also independent.

If E and F are two events such that P(E) = 1/4, P(F) = 1/2 and P(E ∩ F) = 1/8, find (i) P(E or F) (ii) P(not E and not F).

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.

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 volume of a cube is increasing at the rate of 9 cm³/sec. How fast is the surface area increasing when the length of an edge is 10 cm.