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 \(\frac { 11 } { 7 } l\).
- Show that the modulus of elasticity of the string is \(\frac { 7 } { 5 } \mathrm { mg }\).
- Show that \(P\) moves in simple harmonic motion about the equilibrium position and state the period of the motion.
- 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 . - Test, at the \(5 \%\) significance level, whether there is evidence of positive correlation between the variables.
- Given that \(b - a = 0.5\), find the values of \(a\) and \(b\).
- 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 \(z = 5 y\).
- 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.