Answer only one of the following two alternatives.

**EITHER**

\includegraphics{figure_11a}

A uniform plane object consists of three identical circular rings, $X$, $Y$ and $Z$, enclosed in a larger circular ring $W$. Each of the inner rings has mass $m$ and radius $r$. The outer ring has mass $3m$ and radius $R$. The centres of the inner rings lie at the vertices of an equilateral triangle of side $2r$. The outer ring touches each of the inner rings and the rings are rigidly joined together. The fixed axis $AB$ is the diameter of $W$ that passes through the centre of $X$ and the point of contact of $Y$ and $Z$ (see diagram). It is given that $R = \left(1 + \frac{2}{3}\sqrt{3}\right)r$.

\begin{enumerate}[label=(\roman*)]
\item Show that the moment of inertia of the object about $AB$ is $\left(7 + 2\sqrt{3}\right)mr^2$. [8]
\end{enumerate}

The line $CD$ is the diameter of $W$ that is perpendicular to $AB$. A particle of mass $9m$ is attached to $D$. The object is now held with its plane horizontal. It is released from rest and rotates freely about the fixed horizontal axis $AB$.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find, in terms of $g$ and $r$, the angular speed of the object when it has rotated through $60°$. [6]
\end{enumerate}

**OR**

Fish of a certain species live in two separate lakes, $A$ and $B$. A zoologist claims that the mean length of fish in $A$ is greater than the mean length of fish in $B$. To test his claim, he catches a random sample of 8 fish from $A$ and a random sample of 6 fish from $B$. The lengths of the 8 fish from $A$, in appropriate units, are as follows.

$$15.3 \quad 12.0 \quad 15.1 \quad 11.2 \quad 14.4 \quad 13.8 \quad 12.4 \quad 11.8$$

Assuming a normal distribution, find a 95% confidence interval for the mean length of fish in $A$. [5]

The lengths of the 6 fish from $B$, in the same units, are as follows.

$$15.0 \quad 10.7 \quad 13.6 \quad 12.4 \quad 11.6 \quad 12.6$$

Stating any assumptions that you make, test at the 5% significance level whether the mean length of fish in $A$ is greater than the mean length of fish in $B$. [7]

Calculate a 95% confidence interval for the difference in the mean lengths of fish from $A$ and from $B$. [2]

\hfill \mbox{\textit{CAIE FP2 2014 Q11 [28]}}