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1. [12 marks] Find the inverse Laplace transforms of = 2s-2 2. [16 marks] Solve the following initial value problems using Laplace transforms, where u (t) is the Heaviside step function (a) y" + 2y' + 2y = 2t with y (0) = 0 and y' (0) = 1 (b) y" - 6y' + 8y = 22 (t - 2 t with y(0) = 2 1 and y' (0) = 0 3. [8 marks] Suppose the location of a particle is given by t € [0,3]. (1) (a) Sketch the path indicating the starting point at t = 0 and the direction of movement with an arrow. (b) Find the velocity of the particle v(t) and hence the speed s(t) = |v(t)]. (c) Find the acceleration of the particle at any time. 4. [7 marks] Let F(t) = (e-t, cos t) and G (t) = (t,2t²). Find d G(t) and 5. [7 marks) If y = 3z and X = sin Z (with 0 < Z < 3), find the arclength of the curve. You may evaluate the integral in any way you choose (including numerically) but you need to explain how you did it. The answer should be given to 2 decimal places.

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