Advanced level 2023 south west regional mock further mathematics 3
Advanced level 2023 south west regional mock further mathematics 3
A force F acting along the line r = 4i + 2(6i — 3j + 2k ) displaces a particle from the point A
with position vector (9i- 4j + 7k ) to the point B with position vector (13£ + 4 j + 9k )m. Given
that the work done by is 12 J, find
(a) the force F, in vector form,
(b) the magnitude of the moment of K about the origin
(c) the magnitude of the moment of F about the point with position vector (i – k )
2. A particle P moves in a resisting medium such that its displacement x from a fixed point 0 is given
by the
differential equation
(a) k < m
(b) k = m
(c) k > m
Ut 2 2k if til t ?M 2.V = 0. Discuss the motion of P when
(8 marks)
(2 marks)
(2 marks)
3. Given (a) use that theyapproximations satisfies the differential equation + 2^ -I- y = 0,
(£9- yn+l—23 /i’2n‘^yn- i and yn+12-hyn-l to show that
(1 + h )yn+1 ~ (2- l\2 )yn + (/i- 1 )yn-l * (3 marks)
(b) Using a step length of h = 0.1, find the value of y, correct to 3 decimal places, when x = 0.4 .
given that y = 1 tit x = 0 and at x = 0.1. (6 marks)
lienee.
(c) using Simpson’s rule, evaluate J° A y dx,correct to 3 decimal places. (3 marks)
4. A particle of mass m falls from rest in a medium in which the resistance to motion is of magnitude
rngkv2 , where v is the speed of the particle and k is a positive constant.
(a) Show that when the speed of the particle is , it has fallen a distance ~ In
(b) Calculate the time taken by the particle to attain this speed.
(6 marks)
(6 marks)
5. The polar coordinates of a particle P, of unit mass, moving in a plane arc such that at time t ,
—dt = 4.S-1710 and ~dt = (1— 2cos0 )z.
= 0, find the polar equation of the path traced by P. k / marks)
(6 marks)
(a) Given that r = 2 when
(b) Show that the force acli- , on P when 0 = 0 is 2N
