10a. The gradient of a tangent to the curve y = 4x\(^3\) at points P and Q is 108. Find the coordinates of P and Q

bi. Given \(\hat{A}\) = 45º, \(\hat{B}\) = 30º, sin(A + B) = sinA sinB + sinB sinA and cos(A + B) = cosA cosB - sinA sinB. Show that sin 15º = \(\frac{\sqrt{6} - \sqrt{2}}{4}\)

and cos15º = \(\frac{\sqrt{6} + \sqrt{2}}{4}\)

ii. Hence, find tan 15º

11a. Using the substitution U = 5 - x\(^2\)

evaluate \(\int _1^2 \frac{\text{x}}{\sqrt{5 - x^2}}\) dx

b. If y = px\(^2\) + qx, \(\frac{\text{dy}}{\text{dx}}\) = 7 and \(\frac{d^2y}{dx^2}\) = 6. Find the values of p and q.

12a. A man P has 5 red, 3 blue, and 2 white buses. Another man, Q, has 3 red, 2 blue, and 4 white buses. A bus owned by P is involved in an accident with a bus belonging to Q. Calculate the probability that the two buses are not the same. Leave your answer in this format ' a/b.'

b. A man travels from Nigeria to Ghana by air and from Ghana to Liberia by ship. He returns by the same means. He has 6 airlines and 4 shipping lines to choose from. In how many ways can he make his journey without using the same airline or shipping line?

13a. The table below shows the distribution of hours spent at work by the employees of a factory in a week

Draw an Ogive for the distribution

b. Using your graph,  estimate

i. the median.

ii. estimate the lower quartile

iii. 40th percentile

iv. number of employees that spent at least 50 hours 30 mins.

14a.  Two pupils are chosen at random from a group of 4 boys and 5 girls. Find the probability that the two pupils chosen would be boys. Leave your answer in fraction 'a/b.'

b. Twenty percent of the total production of transistors produced by a machine are below standard. If a random sample of six transistors produced by the machine is taken, what is the probability of getting 

i. Exactly 2

ii. Exactly 1. Leave your answer in six decimal places " a.bcdefg."

iii. At least 2. Leave your answer in five decimal places " a. bcdeg."

iv. At most 2 standard transistors