Answer only one of the following two alternatives.

**EITHER**

A particle $P$, of mass $m$, is able to move in a vertical circle on the smooth inner surface of a sphere with centre $O$ and radius $a$. Points $A$ and $B$ are on the inner surface of the sphere and $AOB$ is a horizontal diameter. Initially, $P$ is projected vertically downwards with speed $\sqrt{\left(\frac{21}{2}ag\right)}$ from $A$ and begins to move in a vertical circle. At the lowest point of its path, vertically below $O$, the particle $P$ collides with a stationary particle $Q$, of mass $4m$, and rebounds. The speed acquired by $Q$, as a result of the collision, is just sufficient for it to reach the point $B$.

(i) Find the speed of $P$ and the speed of $Q$ immediately after their collision. [7]

In its subsequent motion, $P$ loses contact with the inner surface of the sphere at the point $D$, where the angle between $OD$ and the upward vertical through $O$ is $\theta$.

(ii) Find $\cos \theta$. [5]

**OR**

A farmer grows two different types of cherries, Type A and Type B. He assumes that the masses of each type are normally distributed. He chooses a random sample of 8 cherries of Type A. He finds that the sample mean mass is 15.1 g and that a 95% confidence interval for the population mean mass, $\mu$ g, is $13.5 \leqslant \mu \leqslant 16.7$.

(i) Find an unbiased estimate for the population variance of the masses of cherries of Type A. [3]

The farmer now chooses a random sample of 6 cherries of Type B and records their masses as follows.
$$12.2 \quad 13.3 \quad 16.4 \quad 14.0 \quad 13.9 \quad 15.4$$

(ii) Test at the 5% significance level whether the mean mass of cherries of Type B is less than the mean mass of cherries of Type A. You should assume that the population variances for the two types of cherry are equal. [9]

\hfill \mbox{\textit{CAIE FP2 2019 Q11 [24]}}