CAIE FP2 (Further Pure Mathematics 2) 2010 November

1 A particle \(P\) is describing simple harmonic motion of amplitude 5 m . Its speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is 3 m from the centre of the motion. Find, in terms of \(\pi\), the period of the motion. Find also

1. the maximum speed of \(P\),
2. the magnitude of the maximum acceleration of \(P\).
   \(2 \quad\) A particle \(P\) of mass \(m\) is projected horizontally with speed \(u\) from the lowest point on the inside of a fixed hollow sphere with centre \(O\). The sphere has a smooth internal surface of radius \(a\). Assuming that the particle does not lose contact with the sphere, show that when the speed of the particle has been reduced to \(\frac { 1 } { 2 } u\) the angle \(\theta\) between \(O P\) and the downward vertical satisfies the equation $$8 g a ( 1 - \cos \theta ) = 3 u ^ { 2 }$$ Find, in terms of \(m , u , a\) and \(g\), an expression for the magnitude of the contact force acting on the particle in this position.

3 Two smooth spheres \(A\) and \(B\), of equal radius, are moving in the same direction in the same straight line on a smooth horizontal table. Sphere \(A\) has mass \(m\) and speed \(u\) and sphere \(B\) has mass \(\alpha m\) and speed \(\frac { 1 } { 4 } u\). The spheres collide and \(A\) is brought to rest by the collision. Find the coefficient of restitution in terms of \(\alpha\). Deduce that \(\alpha \geqslant 2\).

4
\includegraphics[max width=\textwidth, alt={}, center]{c7844913-5c2e-47b4-87b6-f822f4d4bf22-2_426_862_1553_644} A hemispherical bowl of radius \(r\) is fixed with its rim horizontal. A thin uniform rod rests in equilibrium on the rim of the bowl with one end resting on the inner surface of the bowl at \(A\), as shown in the diagram. The rod has length \(2 a\) and weight \(W\). The point of contact between the rod and the rim is \(B\), and the rim has centre \(C\). The rod is in a vertical plane containing \(C\). The rod is inclined at \(\theta\) to the horizontal and the line \(A C\) is inclined at \(2 \theta\) to the horizontal. The contacts at \(A\) and \(B\) are smooth. In any order, show that

1. the contact force acting on the rod at \(A\) has magnitude \(W \tan \theta\),
2. the contact force acting on the rod at \(B\) has magnitude \(\frac { W \cos 2 \theta } { \cos \theta }\),
3. \(2 r \cos 2 \theta = a \cos \theta\).

5 A uniform circular disc has diameter \(A B\), mass \(2 m\) and radius \(a\). A particle of mass \(m\) is attached to the disc at \(B\). The disc is able to rotate about a smooth fixed horizontal axis through \(A\). The axis is tangential to the disc. Show that the moment of inertia of the system about the axis is \(\frac { 13 } { 2 } m a ^ { 2 }\). The disc is held with \(A B\) horizontal and released. Find the angular speed of the system when \(B\) is directly below \(A\). The disc is slightly displaced from the position of equilibrium in which \(B\) is below \(A\). At time \(t\) the angle between \(A B\) and the vertical is \(\theta\). Write down the equation of motion, and find the approximate period of small oscillations about the equilibrium position.

6 The mean Intelligence Quotient (IQ) of a random sample of 15 pupils at \(\operatorname { School } A\) is 109 . The mean IQ of a random sample of 20 pupils at School \(B\) is 112 . You may assume that the IQs for the populations from which these samples are taken are normally distributed, and that both distributions have standard deviation 15. Find a \(90 \%\) confidence interval for \(\mu _ { B } - \mu _ { A }\), where \(\mu _ { A }\) and \(\mu _ { B }\) are the population mean IQs.

7 The discrete random variable \(X\) has a geometric distribution with mean 4.
Find

1. \(\mathrm { P } ( X = 5 )\),
2. \(\mathrm { P } ( X \geqslant 5 )\),
3. the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.9995\).

8 The owner of three driving schools, \(A , B\) and \(C\), wished to assess whether there was an association between passing the driving test and the school attended. He selected a random sample of learner drivers from each of his schools and recorded the numbers of passes and failures at each school. The results that he obtained are shown in the table below. Using a \(\chi ^ { 2 }\)-test and a \(5 \%\) level of significance, test whether there is an association between passing or failing the driving test and the driving school attended.

9 A national athletics coach suspects that, on average, 200-metre runners' indoor times exceed their outdoor times by more than 0.1 seconds. In order to test this, the coach randomly selects eight 200 -metre runners and records their indoor and outdoor times. The results, in seconds, are shown in the table. Stating suitable hypotheses and any necessary assumption that you make, test the coach's suspicion at the 2.5\% level of significance.

10 For each month of a certain year, a weather station recorded the average rainfall per day, \(x \mathrm {~mm}\), and the average amount of sunshine per day, \(y\) hours. The results are summarised below. $$n = 12 , \quad \Sigma x = 24.29 , \quad \Sigma x ^ { 2 } = 50.146 , \quad \Sigma y = 45.8 , \quad \Sigma y ^ { 2 } = 211.16 , \quad \Sigma x y = 88.415 .$$

1. Find the mean values, \(\bar { x }\) and \(\bar { y }\).
2. Calculate the gradient of the line of regression of \(y\) on \(x\).
3. Use the answers to parts (i) and (ii) to obtain the equation of the line of regression of \(y\) on \(x\).
4. Find the product moment correlation coefficient and comment, in context, on its value.
5. Stating your hypotheses, test at the \(1 \%\) level of significance whether there is negative correlation between average rainfall per day and average amount of sunshine per day.

A particle of mass 0.1 kg lies on a smooth horizontal table on the line between two points \(A\) and \(B\) on the table, which are 6 m apart. The particle is joined to \(A\) by a light elastic string of natural length 2 m and modulus of elasticity 60 N , and to \(B\) by a light elastic string of natural length 1 m and modulus of elasticity 20 N . The mid-point of \(A B\) is \(M\), and \(O\) is the point between \(M\) and \(B\) at which the particle can rest in equilibrium. Show that \(M O = 0.2 \mathrm {~m}\). The particle is held at \(M\) and then released. Show that the equation of motion is $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } = - 500 y$$ where \(y\) metres is the displacement from \(O\) in the direction \(O B\) at time \(t\) seconds, and state the period of the motion. For the instant when the particle is 0.3 m from \(M\) for the first time, find

1. the speed of the particle,
2. the time taken, after release, to reach this position.

The continuous random variable \(T\) has a negative exponential distribution with probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0
0 & \text { otherwise } \end{cases}$$ Show that for \(t \geqslant 0\) the distribution function is given by \(\mathrm { F } ( t ) = 1 - \mathrm { e } ^ { - \lambda t }\). The table below shows some values of \(\mathrm { F } ( t )\) for the case when the mean is 20 . Find the missing value. It is thought that the lifetime of a species of insect under laboratory conditions has a negative exponential distribution with mean 20 hours. When observation starts there are 100 insects, which have been randomly selected. The lifetimes of the insects, in hours, are summarised in the table below. Calculate the expected values for each interval, assuming a negative exponential model with a mean of 20 hours, giving your values correct to 2 decimal places. Perform a \(\chi ^ { 2 }\)-test of goodness of fit, at the \(5 \%\) level of significance, in order to test whether a negative exponential distribution, with a mean of 20 hours, is a suitable model for the lifetime of this species of insect under laboratory conditions.