26   9. (i) (a) Explain why cosh x x 0 ∞∫ d is an improper integral. (1) (b) Show that cosh x x 0 ∞∫ d is divergent. (3) (ii) 4 sinh x = p cosh x        where p is a real constant Given that this equation has real solutions, determine the range of possible values for p (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 27   Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 9 is 6 marks) 28   10. θ Figure 3 The motion of a pendulum, shown in Figure 3, is modelled by the differential equation 2 2 d 1 9 cos3 d 2 θ θ t t + = where θ is the angle, in radians, that the pendulum makes with the downward vertical, t seconds after it begins to move. (a) (i) Show that a particular solution of the differential equation is 1 sin3 12 θ t t = (4) (ii) Hence, find the general solution of the differential equation. (4) Initially, the pendulum • makes an angle of 3 π radians with the downward vertical • is at rest Given that, 10 seconds after it begins to move, the pendulum makes an angle of α radians with the downward vertical, (b) determine, according to the model, the value of α to 3 significant figures. (4) Given that the true value of α is 0.62 (c) evaluate the model. (1) The differential equation 2 2 d 1 9 cos3 d 2 θ θ t t + = models the motion of the pendulum as moving with forced harmonic motion. (d) Refine the differential equation so that the motion of the pendulum is simple harmonic motion. (1) Turn over 29   Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________