WAEC Further Mathematics 2011 Past Questions

If \(f(x) = 6x^{3} + 13x^{2} + 2x - 5\) and \(f(-1) = 0\), find the factors of f(x).

If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} - 7x + 4 = 0\), find the equation whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\).

(a) Write down the binomial expansion of \((2 - x)^{5}\) in ascending powers of x.

(b) Use your expansion in (a) to evaluate \((1.98)^{5}\) correct to four decimal places.

Solve \(2^{(2y + 2)} - 9(2^{y}) = -2\).

(a) Write the following as column vectors: \(r = (10N, 090°) ; q = (8N, 135°)\).

(b) Use your answer in (a) to find \((r + q)\).

(a) Evaluate \(\frac{^{9}P_{3}}{^{15}C_{3}} + \frac{^{5}C_{3}}{^{3}P_{2}}\) correct to two decimal places.

(b) A committee of 2 tutors and 5 pupils is to be formed among 6 tutors and 10 pupils. In how many ways can this be done if one particular tutor must be on the committee and two particular pupils must not be on the committee?

A tyre manufacturing company researched into the life span of one type of their motorcycle tyres. The results were as follows :

(a) Draw a histogram for the distribution.

(b) Use the histogram to estimate the mode.

A particle of mass 400g is moving under the action of two forces \(F_{1} = (35N, 210°), F_{2} = (35\sqrt{3} N, 300°)\) and a resistance of 40N. Find the magnitude of the 

(a) resultant of \(F_{1}\) and \(F_{2}\).

(b) resultant force acting on the particle.

(a) Find, from first principles, the derivative of \(f(x) = (2x + 3)^{2}\).

(b) Evaluate : \(\int_{1} ^{2} \frac{(x + 1)(x^{2} - 2x + 2)}{x^{2}} \mathrm {d} x\)

(a) If \(A = \begin{pmatrix} -2 & 5 \\ 4 & 3 \end{pmatrix}\) and \(B = \begin{pmatrix} 3 & 1 \\ 2 & 3 \end{pmatrix}\), find the values of x and y such that \(BA = 2\begin{pmatrix} 3 & 7 \\ -2 & x \end{pmatrix} + \begin{pmatrix} y & 4 \\ 12 & -3 \end{pmatrix}\).

(b) Two functions, f and g are defined by \(f : x \to \frac{1}{2}x + 1\) and \(g : x \to \frac{5x - 1}{3}\). Find :

(i) \(g^{-1}\) ; (ii) \(g^{-1} \circ f\).

The images of points (2, -3) and (4, 5) under a linear transformation A are (3, 4) and (5, 6) respectively. Find the :

(a) matrix A ; (b) inverse of A ; (c) point whose image is (-1, 1).

(a) Find the equation of the tangent to curve \(\frac{x^{2}}{4} + y^{2} = 1\) at point \(1, \frac{\sqrt{3}}{2}\).

(b) Express \(\frac{3x + 2}{x^{2} + x - 2}\) in partial fractions.

A survey conducted revealed that four out of every twenty taxi drivers do not have a valid driving license. If 6 drivers are selected at random, calculate, correct to three decimal places, the probability that

(a) exactly 2 ;

(b) more than 3 ;

(c) at least 5; have valid driving license.

The table shows the frequency distribution of marks scored by some candidates in an examination.

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

(b) Use your graph to estimate the :

(i) semi-interquartile range of the distribution; (ii) percentage of candidates who passed with distinction if the least mark for distinction was 72.

A bag contains 4 red, 6 blue and 8 green identical marbles.

(a) If three marbles are drawn at random, without replacement, calculate the probability that :

(i) all will be green ; (ii) all will have the same colour.

(b) If each marble is replaced before another is drawn, calculate the probability that all will have the same colour.

(a) An object P of mass 6.5kg is suspended by two light inextensible strings, AP and BP. The strings make angles 50° and 60° respectively with the downward vertical.

(i) Express the forces acting on P in component form; (ii) If P is at rest, write down the vector equation connecting all the forces; (iii) Calculate, correct to one decimal place, the tensions in the strings.

(b) A particle of mass 5 kg moves with initial velocity \(\frac{1}{2} m/s\) and final velocity \(\frac{3}{4} m/s \). Find the magnitude of its change in momentum.

(a) \(m \begin{pmatrix} 2 \\ 1 \end{pmatrix} + n \begin{pmatrix} -1 \\ 2 \end{pmatrix} = \begin{pmatrix} 5 \\ -4 \end{pmatrix}\) where m and n are scalars. Find the value of (m + n).

(b) A(-1, 3), B(2, -1) and C(5, 3) are the vertices of \(\Delta\) ABC.

(i) Express in column notation, the unit vectors parallel to AB and AC.

(ii) Use a dot product to calculate \(\stackrel \frown{BAC}\), correct to the nearest degree. 

(a) Forces \(F_{1} = (3N, 210°)\) and \(F_{2} = (4N, 120°)\) act on a particle of mass 7kg which is at rest. Calculate the  :

(i) acceleration of the particle ; (ii) velocity of the particle after 3 seconds.

(b) \(F_{1} = (2i + 3j)N, F_{2} = (-5j)N \) and \(F_{3} = (6i - 4j)N\) act on a body. Find the magnitude and direction of the fourth force that will keep the body in equilibrium.