Answer only one of the following two alternatives.

EITHER

A light elastic string, of natural length $l$ and modulus of elasticity $4mg$, is attached at one end to a fixed point and has a particle $P$ of mass $m$ attached to the other end. When $P$ is hanging in equilibrium under gravity it is given a velocity $\sqrt{(gl)}$ vertically downwards. At time $t$ the downward displacement of $P$ from its equilibrium position is $x$. Show that, while the string is taut,

$$\ddot{x} = -\frac{4g}{l}x.$$ [4]

Find the speed of $P$ when the length of the string is $l$. [4]

Show that the time taken for $P$ to move from the lowest point to the highest point of its motion is

$$\left(\frac{\pi}{3} + \frac{\sqrt{3}}{2}\right)\sqrt{\left(\frac{l}{g}\right)}.$$ [6]

OR

\includegraphics{figure_11}

The scatter diagram shows a sample of size 5 of bivariate data, together with the regression line of $y$ on $x$. State what is minimised in obtaining this regression line, illustrating your answer on a copy of this diagram. [2]

State, giving a reason, whether, for the data shown, the regression line of $y$ on $x$ is the same as the regression line of $x$ on $y$. [1]

A car is travelling along a stretch of road with speed $v$ km h$^{-1}$ when the brakes are applied. The car comes to rest after travelling a further distance of $z$ m. The values of $z$ (and $\sqrt{z}$) for 8 different values of $v$ are given in the table, correct to 2 decimal places.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$v$ & 25 & 30 & 35 & 40 & 45 & 50 & 55 & 60 \\
\hline
$z$ & 2.83 & 4.63 & 4.84 & 5.29 & 9.73 & 10.30 & 14.82 & 15.21 \\
\hline
$\sqrt{z}$ & 1.68 & 2.15 & 2.20 & 2.30 & 3.12 & 3.21 & 3.85 & 3.90 \\
\hline
\end{tabular}

[$\sum v = 340$, $\sum v^2 = 15500$, $\sum \sqrt{z} = 22.41$, $\sum z = 67.65$, $\sum v\sqrt{z} = 1022.15$.]

\begin{enumerate}[label=(\roman*)]
\item Calculate the product moment correlation coefficient between $v$ and $\sqrt{z}$. What does this indicate about the scatter diagram of the points $(v, \sqrt{z})$? [4]

\item Given that the product moment correlation coefficient between $v$ and $z$ is 0.965, correct to 3 decimal places, state why the regression line of $\sqrt{z}$ on $v$ is more suitable than the regression line of $z$ on $v$, and find the equation of the regression line of $\sqrt{z}$ on $v$. [5]

\item Comment, in the context of the question, on the value of the constant term in the equation of the regression line of $\sqrt{z}$ on $v$. [2]
\end{enumerate}

\hfill \mbox{\textit{CAIE FP2 2009 Q11 [28]}}