WAEC Further Mathematics 2006 Past Questions

The sum of the 2nd and 5th terms of an arithmetic progression (AP) is 42. If the difference between the 6th and 3rd term is 12, find the 

(i) Common difference

(ii) first term

(iii) 20th term.

If \(2^{2x - 3y} = 32\) and \(\log_{y} x = 2\), find the values of x and y.

If (x + 2) and (x - 1) are factors of \(f(x) = 6x^{4} + mx^{3} - 13x^{2} + nx + 14\), find the 

(a) values of m and n.

(b) remainder when f(x) is divided be (x + 1).

Find the equation of the tangent to the curve \(y = \frac{x - 1}{2x + 1}, x \neq -\frac{1}{2}\) at the point (1, 0).

(a) There are 6 points in a plane. How many triangles can be formed with the points?

(b) A family of 6 is to be seated in a row . In how many ways can this be done if the father and mother are not to be seated together?

The table shows the distribution of the ages of a group of people in a village.

Using an assumed mean of 24.5, calculate the mean of the distribution.

A body of mass 5 kg resting on a smooth horizontal plane, is acted upon by force 6i + 2j, 5i + 4j and 4i - j. Calculate the:

(a) velocity of the body

(b) Magnitude of its velocity after 4s.

An object is projected vertically upwards with a velocity of 80 m/s. Find the :

(a) Maximum height reached

(b) Time taken to return to the point of projection. [Take g = \(10 ms^{-2}\)].

(a) Simplify \(\frac{\sqrt{75} - 3}{\sqrt{3} + 1}\), leaving your answer in the form \(a + b\sqrt{c}\); where a, b and c are rational numbers.

(b) The points (7, 3), (2, 8) and (-3, 3) lie on a circle. Find the (i) equation and (ii) radius of the circle.

(a) The gradient of the tangent to the curve \(y = 4x^{3}\) at points P and Q is 108. Find the coordinates of P and Q.

(b) Given that \(A = 45°, B = 30°, \sin (A + B) = \sin A \cos B + \sin B \cos A\) and \(\cos (A + B)  = \cos A \cos B - \sin A \sin B\)

(i) Show that \(\sin 15° = \frac{\sqrt{6} - \sqrt{2}}{4}\) and \(\cos 15° = \frac{\sqrt{6} + \sqrt{2}}{4}\)

(ii) hence find \(\tan 15°\).

(a) Using the substitution \(u = 5 - x^{2}\), evaluate \(\int_{1}^{2} \frac{x}{\sqrt{5 - x^{2}}} \mathrm {d} x\).

(b) If \(y = px^{2} + qx; \frac{\mathrm d y}{\mathrm d x} = 6x + 7\) and \(\frac{\mathrm d^{2} y}{\mathrm d x^{2}} = 6\), find the values of p and q. 

(a) A man P has 5 red, 3 blue and 2 white buses. Another man Q has 3 red, 2 blue and 4 white buses. A bus owned by P is involved in an accident with a bus belonging to Q. Calculate the probability that the two buses are not of the same color.

(b) A man travels from Nigeria to Ghana by air and from Ghana to Liberia by ship. He returns by the same means. He has 6 airlines and 4 shipping lines to choose from. In how many ways can he make his journey without using the same airline or shipping line twice?

The table shows the distribution of hours spent at work by the employees of a factory in a week.

(a) Draw an ogive for the distribution.

(b) Using your graph, estimate the (i) lower quartile   (ii) median  (iii) 40th percentile  (iv) number of employees that spent at least 50 hours 30 minutes at work.

(a) Two pupils are chosen at random from a group of 4 boys and 5 girls. Find the probability that the two pupils chosen would be boys.

(b) Twenty percent of the total production of transistors produced by a machine are below standard. If a random sample of 6 transistors produced by the machine is taken, what is the probability of getting (i) exactly 2 standard transistors  (ii) exactly 1 standard transistor  (iii) at least 2 standard transistors  (iv) at most 2 standard transistors?

(a) A body of mass 15 kg is suspended at a point P by two light inextensible strings \(\overrightarrow{XP}\) and \(\overrightarrow{YP}\). The strings are inclined at 60° and 40° respectively to the downward vertical. Find, correct to two decimal places, the tensionsin the strings. [Take g = \(10 ms^{-2}\)].

(b) The height h metres, of a ball thrown into the air is \(2 + 20t + kt^{2}\), after t seconds. If it takes 2 seconds for the ball to reach its highest point, find

(i) the value of k  (ii) its highest point from the point of throw.

(a) A ball P moving with velocity \(2u ms^{-1}\), collides with a similar ball Q, of different mass, which is at rest. After collision, Q moves with \(u ms^{-1}\) and P with velocity \(\frac{1}{2} u ms^{-2}\), in the opposite direction. Find the ratio of the masses of P and Q.

(b) Two forces of magnitudes 3 N and 7 N have a resultant of magnitude 5 N. Calculate, correct to one decimal place, the angle between the two forces.

(c) \(AB = \begin{pmatrix} -4 \\ 6 \end{pmatrix}\) and \(CB = \begin{pmatrix} 2 \\ -3 \end{pmatrix}\) are two vectors in the XY- plane. If V is the midpoint of AB, find CV.

(a) A body of mass 5 kg is placed on a smooth plane inclined at an angle 30° to the horizontal. Find the magnitude of the force: (i) acting parallel to the plane                               (ii) required to keep the body in equilibrium. [Take g = \(10 ms^{-2}\)].

(b) A uniform plank PQ of length 10m and mass m kg rests on two supports A and B, where |PA| = |BQ| = 1m. A load of mass 8 kg is placed on the plank at point C such that |AC| = 3.5 m. If the reaction at B is 100 N, calculate the (i) value of m  (ii) reaction at A.  [Take g = \(10 ms^{-2}\)].