11 Answer only one of the following two alternatives.

\\EITHER\\
A particle $P$ of mass $m$ is suspended from a fixed point by a light elastic string of natural length $l$, and hangs in equilibrium. The particle is pulled vertically down to a position where the length of the string is $\frac { 13 } { 7 } l$. The particle is released from rest in this position and reaches its greatest height when the length of the string is $\frac { 11 } { 7 } l$.\\
(i) Show that the modulus of elasticity of the string is $\frac { 7 } { 5 } \mathrm { mg }$.\\
(ii) Show that $P$ moves in simple harmonic motion about the equilibrium position and state the period of the motion.\\
(iii) Find the time after release when the speed of $P$ is first equal to half of its maximum value.

\\OR\\
For a random sample of 12 observations of pairs of values $( x , y )$, the equation of the regression line of $y$ on $x$ and the equation of the regression line of $x$ on $y$ are

$$y = b x + 4.5 \quad \text { and } \quad x = a y + c$$

respectively, where $a , b$ and $c$ are constants. The product moment correlation coefficient for the sample is 0.6 .\\
(i) Test, at the $5 \%$ significance level, whether there is evidence of positive correlation between the variables.\\
(ii) Given that $b - a = 0.5$, find the values of $a$ and $b$.\\
(iii) Given that the sum of the $x$-values in the sample data is 66, find the value of $c$ and sketch the two regression lines on the same diagram.

For each of the 12 pairs of values of $( x , y )$ in the sample, another variable $z$ is considered, where $z = 5 y$.\\
(iv) State the coefficient of $x$ in the equation of the regression line of $z$ on $x$ and find the value of the product moment correlation coefficient between $x$ and $z$, justifying your answer.

\hfill \mbox{\textit{CAIE FP2 2014 Q11}}