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\includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-4_433_841_255_653} A particle \(P\) is projected from a point \(O\) with initial speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) above the horizontal. \(P\) subsequently passes through the point \(A\) which is at an angle of elevation of \(30 ^ { \circ }\) from \(O\) (see diagram). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Write down expressions for \(x\) and \(y\) in terms of \(t\), and hence obtain the equation of the trajectory of \(P\).
2. Calculate the value of \(x\) when \(P\) is at \(A\).
3. Find the angle the trajectory makes with the horizontal when \(P\) is at \(A\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
   University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }