7 At time \(t \mathrm {~s}\), a particle \(P\) is projected with speed \(40 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The greatest height achieved by \(P\) during its flight is \(H \mathrm {~m}\) and the corresponding time is \(T \mathrm {~s}\).

1. Obtain expressions for \(H\) and \(T\) in terms of \(\theta\).
   During the time between \(t = T\) and \(t = 3 , P\) descends a distance \(\frac { 1 } { 4 } H\).
2. Find the value of \(\theta\).
3. Find the speed of \(P\) when \(t = 3\).
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