To DOWNLOAD CAMEROON GCE JUNE 2006 math meach Paper 2 click on the link below
June 2006
1. The position vector of two particles A and B at time t seconds are given by rA and rB respectively where
rA = [t2i + 2t3j]m, rB = [3t2i — 4t2j]m
a. Calculate the distance between A and B when t = 2.
b. Find the velocity of A relative to B when t=2
c. Find the values of t for which the velocities of A and B are perpendicular
d. Find the values of t for which the accelerations of A and B are parallel.
Figure 1 above shows a uniform lamina in the form of a trapezium in which AB & CD are parallel and of
lengths a and b respectively.(a > b).Prove that the distance of the centre of mass from AB is
1/█(3@)h[(a+2b )/█(a+b @)] where h is the distance between AB & CD.
ii) Forces F1 and F2 act at the point with position vector r, where
F1 = (4i + 6j)N, F2 = (6i – 8j)N, r = (2i – 3j)m
Find the sum of moments of F1and F2 about the origin.
3. The forces F1 = (i + 2j + 3k)N and F2 = (2i + k)N act at points whose position vectors are
(2i + Sj + ck)m and (5£ + cj + 2k)m respectively. Given that their lines of action intersect, find the
value of c and the position vector of the point of intersection. Determine the vector equation of the line of action of these forces and show that this line passes through the point with position vector
(7i + 9j + 8 k)m.
4. A particle, P, of mass 4m kg is initially at rest on a smooth plane incline at an angle a to the horizontal. It is supported by a light inextensible string which passes over a smooth light pulley A at the top edge of the plane. The other end of the string support a particle Q, of mass 2m kg, which hangs freely. Given that the system is in equilibrium, find a and the magnitude and direction of the force exerted on the pulley by the string.
A further particle of mass M kg is now attached to Q and the system is released. Given that the
acceleration of the system is,█(@250)/█(9@) ms-1 .find in terms of m, the value M, the tension in the string and the magnitude of and direction of the force exerted on the string by the pulley.
(Take g as 10ms~2)
5. i) A car of mass 1000kg is moving on a smooth level road at a steady speed of ^ms~1 against an air resistance R. The engine works at a rate of 60KW.
Calculate the are resistance R.
When the car starts to ascend a rough plane inclined at an angle sin-1 (2/3) to the horizontal with this
speed, it turns out of fuel. Assuming that the air resistance remains the same and that the driver does not
apply the brakes, find, to two decimal places, how far up the plane the car will climb before coming::
rest, given that the coefficient of friction between the plane and the tires of the car is 0.1175.
ii) A particle moves in a horizontal circular path of radius 2m with uniform angular acceleration. It is
observed to make two revolution in the first 4 seconds of motion and 4 revolutions in the next 4 seconds
Find
the initial angular velocity of the particle giving you answer in rads -1
the angular acceleration of the particle giving your answer in rads-1
c. the total distance in meters covered in 8 seconds
6. A particle is projected with speed u at an angle of elevation θ from a point 0 on a horizontal plane. Show that the equation of trajectory referred to horizontal and vertical axes Ox and Oy respectively, is
Y=xtanθ –gx2sec2θ/2u2
Given that u = 20√5ms-1 and that the particle passes through a point which is 100m from 0
horizontally and 50m vertically above the level of O, how that there are two possible angles of projection If the angles are a and β show that tan(α+ β)= 2.
For the smaller angle, calculate
The range of the particle on the horizontal plane through the point of projection,
The time of flight of the particle.
7. Two small smooth spheres A and B of masses 3m kg and m kg respectively, are moving towards eachother on a smooth horizontal floor. A is moving with a speed of 2u ms-1 in a direction perpendicular:: smooth vertical wall and B is moving with a speed of urns’1 away from the wall.
A strikes B directly and at a distance of 6 meters from the wall. Given that the coefficient of restitution
between A and B and between B and the wall is ¼ ,
Find,
The velocity of A and B after the first impact
The kinetic energy, in terms of m and u, lost in this impact
The distance, from the wall, of the point at which the second impact of the sphere takes place,
The time, in terms of u, between the two impacts of the spheres.
8. A box contains only 4 red balls and 5 white balls. Three balls are drawn at random, successfully and
without replacement from the box. Draw a tree diagram illustrating the various possibilities.
Hence, or otherwise, find the possibility that
Exactly one white ball is drawn,
No one white ball is drawn,
At least one white ball is drawn
At least on red ball is drawn, given that at least one white ball is drawn.
