A particle $$\(P\)$$ is moving along the $$\(x\)$$-axis. At time $$\(t\)$$ seconds the displacement of $$\(P\)$$ from the origin $$\(O\)$$ is $$\(x\)$$ metres, where $$\(x=4 \cos \left(\frac{1}{5} \pi t\right)\)$$ (a) Prove that $$\(P\)$$ is moving with simple harmonic motion. (3) (b) Find the period of the motion. (2) (c) State the amplitude of the motion. (1) (d) Find, in terms of $$\(\pi\)$$, the maximum speed of $$\(P\)$$ (2) The points $$\(A\)$$ and $$\(B\)$$ lie on the $$\(x\)$$-axis, on opposite sides of $$\(O\)$$, with $$\(O A=1.5 \mathrm{~m}\)$$ and $$\(O B=2.5 \mathrm{~m}\)$$. (e) Find the time taken by $$\(P\)$$ to move directly from $$\(A\)$$ to $$\(B\)$$. (4)

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