11 The region bounded by the \(x\)-axis and the curve \(y = \frac { 1 } { 2 } k \left( 1 - x ^ { 2 } \right)\) for \(- 1 \leq x \leq 1\) is occupied by a uniform lamina, as shown in Fig. 11.1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{76875226-2e6c-4571-9318-ecce51ba8b9f-10_387_903_358_571}
\captionsetup{labelformat=empty}
\caption{Fig. 11.1}
\end{figure}
\section*{(i) In this question you must show detailed reasoning.}
Show that the centre of mass of the lamina is at \(\left( 0 , \frac { 1 } { 5 } k \right)\).
A shop sign is modelled as a uniform lamina in the form of the lamina in part (i) attached to a rectangle ABCD , where \(\mathrm { AB } = 2\) and \(\mathrm { BC } = 1\). The sign is suspended by two vertical wires attached at A and D , as shown in Fig. 11.2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{76875226-2e6c-4571-9318-ecce51ba8b9f-10_727_885_1327_475}
\captionsetup{labelformat=empty}
\caption{Fig. 11.2}
\end{figure}
(ii) Show that the centre of mass of the sign is at a distance
$$\frac { 2 k ^ { 2 } + 10 k + 15 } { 10 k + 30 }$$
from the midpoint of CD.
The tension in the wire at A is twice the tension in the wire at D .
(iii) Find the value of \(k\).
Fig. 12 shows \(x\) - and \(y\)-coordinate axes with origin O and the trajectory of a particle projected from O with speed \(28 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal. After \(t\) seconds, the particle has horizontal and vertical displacements \(x \mathrm {~m}\) and \(y \mathrm {~m}\).
Air resistance should be neglected.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{76875226-2e6c-4571-9318-ecce51ba8b9f-11_389_535_557_790}
\captionsetup{labelformat=empty}
\caption{Fig. 12}
\end{figure}
(i) Show that the equation of the trajectory is given by
$$\tan ^ { 2 } \alpha - \frac { 160 } { x } \tan \alpha + \frac { 160 y } { x ^ { 2 } } + 1 = 0 .$$
(ii) (A) Show that if (
) is treated as an equation with \(\tan \alpha\) as a variable and with \(x\) and \(y\) as constants, then () has two distinct real roots for \(\tan \alpha\) when \(y < 40 - \frac { x ^ { 2 } } { 160 }\).
(B) Show the inequality in part (ii) (A) as a locus on the graph of \(y = 40 - \frac { x ^ { 2 } } { 160 }\) in the Printed Answer Booklet and label it R.
S is the locus of points \(( x , y )\) where \(( * )\) has one real root for \(\tan \alpha\).
T is the locus of points \(( x , y )\) where \(( * )\) has no real roots for \(\tan \alpha\).
(iii) Indicate S and T on the graph in the Printed Answer Booklet.
(iv) State the significance of \(\mathrm { R } , \mathrm { S }\) and T for the possible trajectories of the particle.
A machine can fire a tennis ball from ground level with a maximum speed of \(28 \mathrm {~ms} ^ { - 1 }\).
(v) State, with a reason, whether a tennis ball fired from the machine can achieve a range of 80 m .
\section*{END OF QUESTION PAPER}
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