Q1. (10 marks) Using the Laplace transform table, obtain the Laplace transform of the following function: sinh(4t), where ”sinh” stands for hyperbolic sine and sinh(x) = e x-e-x 2 . Q2. (10 marks) Using the Laplace transform table, obtain the Laplace transforms of the following function: cosh(4t), where ”cosh” stands for hyperbolic cosine and cosh(x) = e x+e-x 2 . Q3. (10 marks) Using the Laplace transform table, obtain the Laplace transforms of the following function: t 2 e -2t - 5 cos(5t) Q4. (10 marks) Find the inverse Laplace transform of the following function. G(s) = 2s + 1 (s - 2)(s + 2)2 Q5. (20 marks) Using Laplace transform methods, solve for t = 0 the following differential equation, d 2x dt2 - 3 dx dt + 2x = 1, subject to x = 0 and dx dt = 0 at t = 0. Q6. (20 marks) Using Laplace transform methods, solve for t = 0 the following differential equation, 9 d 2 y dt2 + 12 dy dt + 4y = e -2t , subject to y = 1 and dy dt = 1 at t = 0. Q7. (10 marks) The following f(x) is a periodic function of period p = 2L = 4 defined over the period -L < x < L. (In other words L = 2 in this case is half a period). f(x) =  -16 when - L < x < 0 16 when 0 < x < L , p = 4 Express f(x) as a Fourier series expansion