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A particle $P$ of mass $m$ is free to move on the smooth inner surface of a fixed hollow sphere of radius $a$. The centre of the sphere is $O$. The points $A$ and $A ^ { \prime }$ are on the inner surface of the sphere, on opposite sides of the vertical through $O$; the radius $O A$ makes an angle $\alpha$ with the downward vertical and the radius $O A ^ { \prime }$ makes an angle $\beta$ with the upward vertical. The point $B$ is on the inner surface of the sphere, vertically below $O$. The point $B ^ { \prime }$ is on the inner surface of the sphere and such that $O B ^ { \prime }$ makes an angle $2 \beta$ with the upward vertical through $O$ (see diagram). It is given that $\cos \alpha = \frac { 1 } { 16 }$.\\
(i) $P$ is projected from $A$ with speed $u$ along the surface of the sphere downwards towards $B$. Subsequently it loses contact with the sphere at $A ^ { \prime }$. Show that $u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )$.\\

(ii) $P$ is now projected from $B$ with speed $u$ along the surface of the sphere towards $B ^ { \prime }$. Subsequently it loses contact with the sphere at $B ^ { \prime }$. Find $\cos \beta$.\\

(iii) In part (i), the reaction of the sphere on $P$ when it is initially projected at $A$ is $R$. Find $R$ in terms of $m$ and $g$.\\

\hfill \mbox{\textit{CAIE FP2 2017 Q11 EITHER}}