page 04 MARKS 4. A particle has displacement in metres given by 2 2 2 3 5 t t t   i j k where t is time in seconds and i, j and k are unit vectors along the x, y and z axes respectively. (a) Calculate the velocity of the particle after 3 seconds. (b) Calculate the time when the speed of the particle is 50 m s−1. 5. Use the substitution tan u x  to evaluate π tan sec 2 2 3 0 x xdx  . 6. A vertical spring has one end fixed and the other end attached to a particle. The particle is pulled vertically downwards a small distance and released. The ensuing motion is simple harmonic with period π 8  seconds. As the particle passes through the equilibrium position it has a speed of 2 m s−1. (a) Calculate the amplitude of the motion. (b) Calculate the maximum acceleration of the particle. 7. Given  2 5 3 t f t t   , find the value of k when  0 f k  , where 0 k . 2 2 4 2 1 3