WAEC Further Mathematics 2010 Past Questions

(a) Express \(\frac{2\sqrt{2}}{\sqrt{48} - \sqrt{8} - \sqrt{27}}\) in the form \(p + q\sqrt{r}\), where p, q and r are rational numbers. 

(b) If \(V = A\log_{10} (M + N)\), express N in terms of M, V and A.

If the quadratic equation \((2x - 1) - p(x^{2} + 2) = 0\), where p is a constant, has real roots :

(a) show that \(2p^{2} + p - 1 < 0\);

(b) find the values of p.

The equation of a curve is \(y = x(3 - x^{2})\). Find the equation of its normal of the point where x = 2.

If \(3x^{2} + 2y^{2} + xy + x - 7 = 0\), find \(\frac{\mathrm d y}{\mathrm d x}\) at the point (-2, 1).

Five students are to be selected from a large population. If 60% of them are boys and the rest are girls, find the probability that :

(a) exactly 3 of them are boys;

(b) at least 3 of them are girls.

The table shows the marks obtained by a group of students in a class test.

(a) Draw a histogram for the distribution ;

(b) Use your histogram to estimate the median of the distribution.

The position vector of a particle of mass 3 kg moving along a space curve is given by \(r = (4t^{3} - t^{2})i - (2t^{2} - t)j\) at any time t seconds. Find the force acting on it at t = 2 seconds.

A uniform plank PQ of length 8m and mass 10kg is supported horizontally at the end P and at point R, 3 metres from Q. A boy of mass 20 kg walks along the plank starting from P. If the plank is in equilibrium, calculate the 

(a) reactions at P and R when he walked 1.5 metres;

(b) distance he had walked when the two reactions are equal;

(c) distance he walked before the plank tips over.

(a) The nth term of a sequence is given by \(T_{n} = 4T_{n - 1} - 3\). If twice the third term is five times the second term, find the first three terms of the sequence.

(b) Given that \(\begin{pmatrix} 2 & 0 & 1 \\ 5 & -3 & 1 \\ 0 & 4 & 6 \end{pmatrix} \begin{pmatrix} 1 \\ m \\ r \end{pmatrix} = \begin{pmatrix} k \\ 2 \\ 26 \end{pmatrix}\), find the values of the constants k, m and r.

(a) Copy and complete the table for the relation: \(y = 2\cos x + 3\sin x\) for \(0° \leq x \leq 360°\).

(b) Using a scale of 2 cm to 60° on the x- axis and 2 cm to one unit on the y- axis, draw the graph of \(y = 2\cos x + 3\sin x\) for \(0° \leq x \leq 360°\).

(c) From the graph, find the : (i) maximum value of y, correct to two decimal places ; (ii) solution of the equation \(\frac{2}{3} \cos x + \sin x = \frac{5}{6}\).

(a) If \(y = (2x + 3)^{7} + \frac{x + 1}{2x - 1}\), find the value of \(\frac{\mathrm d y}{\mathrm d x}\) at x = -1.

(b) Using the substitution, \(u = x + 2\), evaluate \(\int_{1} ^{2} \frac{x - 1}{(x + 2)^{4}} \mathrm d x\).

The images of (3, 2) and (-1, 4) under a linear transformation T are (-1, 4) and (7, 11) respectively. P is another transformation where \(P : (x, y) \to (x + y, x + 2y)\).

(a) Find the matrices T and P of the linear transformations T and P;

(b) Calculate TP.

(c) Find the image of the point X(4, 3) under TP.

The table gives the distribution of marks of 60 candidates in a test.

(a) Draw a cumulative frequency curve of the distribution.

(b) From your curve, estimate the : (i) 80th percentile ; (ii) median ; (iii) semi-interquartile range.

(a) Eight coins are tossed at once. Find, correct to three decimal places, the probability of obtaining :

(i) exactly 8 heads ; (ii) at least 5 heads ; (iii) at most 1 head.

(b) In how many ways can four letters from the word SHEEP be arranged (i) without any restriction ; (ii) with only one E.

(a) A manufacturer produces light bulbs which are tested in the following way. A batch is accepted in either of the following cases:

(i) a first sample of 5 bulbs contains no faulty bulbs ; (ii) a first sample of 5 bulbs contains at least one faulty bulb but a second sample of size 5 has no faulty bulb. If 10% of the bulbs are faulty, what is the probability that the batch is accepted?

(b) A bag contains 15 identical marbles of which 3 are black, Keshi picks a marble at random from the bag and replaces it. If this is repeated 10 times; what is the probability that he :

(i) did not pick a black ball? (ii) picked a black ball at most three times?

(a) Two ships M and N, moving with constant velocities, have position vectors (3i + 7j) and (4i + 5j) respectively. If the velocities of M and N are (5i + 6j) and (2i + 3j) and the distance covered by the ships after t seconds are in metres, find (i) MN ; (ii) |MN|, when t = 3 seconds.

(b) A particle is acted upon by forces \(F_{1} = 5i + pj ; F_{2} = qi + j ; F_{3} = -2pi + 3j\) and \(F_{4} = -4i + qj\), where p and q are constants. If the particle remains in equilibrium under the action of these forces, find the values of p and q.

A particle moves from point O along a straight line such that its acceleration at any time, t seconds is \(a = (4 - 2t) ms^{-2}\). At t = 0, its distance from O is 18 metres while its velocity is \(5 ms^{-1}\).

(a) At what time will the velocity be greatest?

(b) Calculate the : (i) time ; (ii) distance of the particle from O when the particle is momentarily at rest.

(a) The position vectors of the points X and Y are \(x = (-2i + 5j)\) and \(y = (i - 7j)\) respectively. Find :

(i) (3x + 2y) ; (ii) \(|(y - 2x)|\) ; (iii) the angle between x and y ; (iv) the unit vector in the direction of \((x + y)\).

(b) A bullet of mass 0.084kg is fired horizontally into a 20 kg block of wood at rest on a smooth floor. If they both move at a velocity of \(0.24 ms^{-1}\) after impact; Calculate, correct to two decimal places, the initial velocity of the bullet.