Cameroon GCE advanced level June 2025 further mathematics 3

Here are the extracted questions from the provided images:

Page 1

1. The force F=(4i−7j−4k)N acts at the point with position vector (3i+4j−6k)m.

   i) Write down a vector equation of the line of action of F.

   ii) Find the work done when a particle moves under the influence of F from the point A(2,3,4) to the point B(6,−1,10).

   iii) Find the vector moment of F about the origin.

   iv) Find the vector moment of F about the point with position vector (i+2j−3k)m.

   v) Find the distance from the origin to the line of action of F.

2. The function y=f(x) satisfies the differential equation

   dxdy​=f(x,y)where f(x,y)=2−x2y​

   and the initial condition y(1)=1.

   a) Use the formula

   yr+1​≈yr​+hf(xr​,yr​)with h=0.05

   to obtain an approximation for y(1.1), giving your answer correct to three decimal places.

   c) Hence, use Simpson’s rule to estimate the value of ∫11.1​y(x)dx.

3. A particle, P, executes simple harmonic motion along a straight line with centre, O. The period of motion is 2π seconds and the amplitude is 5 metres. Initially, P passes through the point, A, while moving with velocity −3m s−1. Find,

   (i) the distance OA,

   (ii) the time, in seconds, to four significant figures, taken by P to move directly from A to O.

4. A car of mass 800kg moves along a straight level road against a resistance of magnitude (4+kv2)N where vm s−1 is the speed and k is a positive constant. The constant tractive force exerted by the engine is 404N. The maximum speed of the car is U.

   i) Show that

   vdxdv​=2U2(U2−v2)​

   (ii) Find the distance covered as the speed increases from 0 to 2U​.

5. A sphere A of mass 2m moving with velocity 3ui collides obliquely with another sphere, B, of same radius but of mass m moving with velocity u(−2i+4j). Just before the spheres collide, their line of centres is parallel to the unit vector i. The coefficient of restitution between the two spheres is 51​.

   (i) Find the velocities of A and B immediately after impact.

   (ii) Find the total kinetic energy of the spheres before impact.

Page 2

5. Show that

   (iii) the total kinetic energy of the spheres after impact is 11mu2,

   (iv) the kinetic energy loss as a result of the impact is 8mu2.

6. A particle, P, moves on the curve with polar equation

   r=1+cosθ1​.

   Given that at any time t during the motion, r2dtdθ​=2.

   (i) write an expression for rdtdθ​ in terms of θ.

   (ii) Show that dtdr​=2sinθ.

   When θ=2π​, find

   (iii) the speed of P,

   (iv) the radial component of the acceleration of P.

7. A uniform circular disc of mass, m, and radius, a, performs small oscillations about a smooth horizontal axis in the plane of the disc, which is of distance x from the diameter of the disc, where 0<x<a.

   (i) Show that the least period of oscillation is 2πga​​.

   (ii) Find the length of the equivalent simple pendulum in this case.

   (You may assume that the moment of inertia of the disc about its diameter is 41​ma2)

8. a) A discrete random variable Y follows a binomial distribution with mean 1 and variance 0.8. Find, correct to 4 decimal places,

   (i) P(Y=2)

   (ii) P(Y<2)

   (iii) P(Y≥1)

   b) The marks X in an examination are normally distributed with mean μ and standard deviation 8. Find the value of μ to the nearest whole number, given that the probability that a candidate scores a mark more than 30 is 0.1038.