WAEC Further Mathematics 2009 Past Questions

Solve \(x^{2} - 2x - 8 > 0\).

A.x < -4 or x > 2
B.x < -2 or x > 4
C.-2 < x < 4
D.-4 < x < 2

(a) Solve : \(2x^{2} + x - 6 < 0\)

(b) Express \(\frac{5 - 2\sqrt{10}}{3\sqrt{5} + \sqrt{2}}\) in the form \(m\sqrt{2} + n\sqrt{5}\) where m and n are rational numbers.

Theory

Question 2

If (x + 3) is a factor of the polynomial \(x^{3} + 3x^{2} + nx - 12\), where n is a constant, find the value of n.

A.-1
B.-2
C.-3
D.-4

Objective

Question 2

Solve the simultaneous equations : \(\log_{2} x - \log_{2} y = 2 ; \log_{2} (x - 2y) = 3\)

Theory

Question 3

The line \(y = mx - 3\) is a tangent to the curve \(y = 1 - 3x + 2x^{3}\) at (1, 0). Find the value of the constant m.

A.-4
B.-1
C.3
D.4

Objective

Question 3

If the quadratic equation \((x + 1)(x + 2) = k(3x + 7)\) has equal roots, find the possible values of the constant k.

Theory

Question 4

The coordinates of the centre of a circle is (-2, 3). If its area is \(25\pi cm^{2}\), find its equation.

A.\(x^{2} + y^{2} - 4x - 6y - 12 = 0\)
B.\(x^{2} + y^{2} - 4x + 6y - 12 = 0\)
C.\(x^{2} + y^{2} + 4x + 6y - 12 = 0\)
D.\(x^{2} + y^{2} + 4x - 6y - 12 = 0\)

Objective

Question 4

Given that \(\tan 2A = \frac{2 \tan A}{1 - \tan^{2} A}\), evaluate \(\tan 15°\), leaving your answer in surd form.

Theory

Question 5

Given \(\sin \theta = \frac{\sqrt{3}}{2}, 0° \leq \theta \leq 90°\), find \(\tan 2\theta\) in surd form.

A.\(- \sqrt{3}\)
B.\(-\frac{\sqrt{3}}{2}\)
C.\(\frac{\sqrt{3}}{2}\)
D.\(\sqrt{3}\)

Objective

Question 5

The probabilities of Rotey obtaining the highest mark in Mathematics, Physics and Biology tests are 0.9, 0.75 and 0.8 respectively. Calculate the probability of getting the highest marks in at least two of the subjects.

Theory

Question 6

Find the coefficient of \(x^{4}\) in the binomial expansion of \((2 + x)^{6}\).

A.120
B.80
C.60
D.15

Objective

Question 6

A student representative council consists of 8 girls and 6 boys. If an editorial board consisting of 5 persons is to be formed, what is the probability that the board consists of

(a) 3 girls and 2 boys ;

(b) either all girls or all boys.

Theory

Question 7

Which of the following binary operations is not commutative?

A.\(a * b = \frac{1}{a} + \frac{1}{b}\)
B.\(a * b = a + b - ab\)
C.\(a * b = 2a + 2b + ab\)
D.\(a * b = a - b + ab\)

Objective

Question 7

The coordinates of the vertices of triangle ABC are A(-2, 1), B(4, -2) and C(1, 8) respectively. If D(x, y) is the foot perpendicular from A to BC, find

(a) an equation connecting x and y ;

(b) the unit vector in the direction of BC.

Theory

Question 8

Express \(\frac{2}{3 - \sqrt{7}} \text{ in the form} a + \sqrt{b}\), where a and b are integers.

A.\(6 + \sqrt{7}\)
B.\(3 + \sqrt{7}\)
C.\(3 - \sqrt{7}\)
D.\(6 - \sqrt{7}\)

Objective

Question 8

The position vector of a body, with respect to the origin, is given by \(r = 4ti + (12 - 3t)j\) at any time t seconds.

(a) Find the velocity of the body ;

(b) Calculate the magnitude of the displacement between t = 0 and t = 5.

Theory

Question 9

The roots of the quadratic equation \(2x^{2} - 5x + m = 0\) are \(\alpha\) and \(\beta\), where m is a constant. Find \(\alpha^{2} + \beta^{2}\) in terms of m.

A.\(\frac{25}{4} - m\)
B.\(\frac{25}{4} - 2m\)
C.\(\frac{25}{4} + m\)
D.\(\frac{25}{4} + 2m\)

Objective

Question 9

(a) The 3rd and 6th terms of a Geometric Progression (G.P) are 2 and 54 respectively. Find the : (i) common ratio ; (ii) first term ; (iii) sum of the first 10 terms, correct to the nearest whole number.

(b) The ratio of the coefficient of \(x^{4}\) to that of \(x^{3}\) in the binomial expansion of \((1 + 2x)^{n}\) is \(3 : 1\). Find the value of n.

Theory

Question 10

Given that \(2^{x} = 0.125\), find the value of x.

A.0
B.-1
C.-2
D.-3

Objective

Question 10

(a) Using the same axes, sketch the curves \(y = 6 - x - x^{2}\) and \(y = 3x^{2} - 2x + 3\).

(b) Find the x- coordinates of the points of intersection of the two curves in (a).

(c) Calculatethe area of the finite region bounded by the two curves in (a).

Theory

Question 11

The gradient of point P on the curve \(y = 3x^{2} - x + 3\) is 5. Find the coordinates of P.

A.(1, 5)
B.(1, 7)
C.(1, 13)
D.(1, 17)

Objective

Question 11

(a) Evaluate \(\int_{1} ^{2} \frac{x}{\sqrt{5 - x^{2}}} \mathrm {d} x\)

(b)(i) Evaluate: \(\begin{vmatrix} 2 & -3 & 1 \\ 0 & 1 & -2 \\ 1 & 2 & -3 \end{vmatrix}\)

(ii) Using your answer in b(i), solve the simultaneous equations :

\(2x - 3y + z = 10\)

\(y - 2z = -7\)

\(x + 2y - 3z = -9\)

Theory

Question 12

An arc of length 10.8 cm subtends an angle of 1.2 radians at the centre of a circle. Calculate the radius of the circle.

A.12.6 cm
B.12.0 cm
C.9.6 cm
D.9.0 cm

Objective

Question 12

(a) Use the trapezium rule with five ordinates to evaluate \(\int_{0} ^{1} \frac{3}{1 + x^{2}} \mathrm {d} x\), correct to four significant figures.

(b) If \(A = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}\), find the image of the point (1, 2) under the linear transformation \(A^{2} + A + 2I\), where I is the \(2 \times 2\) unit matrix.

Theory

Question 13

The first term of a geometric progression is 350. If the sum to infinity is 250, find the common ratio.

A.\(\frac{-5}{7}\)
B.\(-\frac{2}{5}\)
C.\(\frac{2}{5}\)
D.\(\frac{5}{7}\)

Objective

Question 13

(a) Simplify \(^{n + 1}C_{3} - ^{n - 1}C_{3}\)

(b) A fair die is thrown five times. Calculate, correct to three decimal places, the probability of obtaining (i) at most two sixes ; (ii) exactly three sixes.

Theory

Question 14

p and q are statements such that \(p \implies q\). Which of the following is a valid conclusion from the implication?

A.\(q \implies p\)
B.\(\sim q \implies p\)
C.\(\sim q \implies \sim p\)
D.\(\sim p \implies \sim q\)

Objective

Question 14

(a) The distribution of the lives (in days) of 40 transitor batteries is shown in the table:

Battery life

(in days)

(a) Draw a histogram for the distribution.

(b) Use your graph in (a) to determine the mode for the distribution.

(c) Using an assumed mean of 43 days, calculate the mean of the distribution.

(d) What percentage number of batteries will live for less than 31 days or more than 45 days?

Theory

Question 15

The roots of a quadratic equation are -3 and 1. Find its equation.

A.\(x^{2} - 3x + 1 = 0\)
B.\(x^{2} - 2x + 1 = 0\)
C.\(x^{2} + 2x - 3 = 0\)
D.\(x^{2} + x - 3 = 0\)

Objective

Question 15

The table below shows the corresponding values of two variables X and Y.

X	33	31	28	25	23	22	19	17	16	14
Y	4	6	4	10	12	10	14	15	18	22

(a) Plot a scatter diagram to represent the data.

(b) Calculate \(\bar{x}\), the mean of X and \(\bar{y}\), the mean of Y.

(c) Draw the line of best fit to pass through \((\bar{x}, \bar{y})\).

(d) From your graph in (c), determine the (i) relationship between X and Y ; (ii) value of Y when X is 24.

Theory

Question 16

The derivative of a function f with respect to x is given by \(f'(x) = 3x^{2} - \frac{4}{x^{5}}\). If \(f(1) = 4\), find f(x).

A.\(f(x) = x^{3} - \frac{1}{x^{4}} + 2\)
B.\(f(x) = x^{3} + \frac{1}{x^{4}} + 2\)
C.\(f(x) = x^{3} - \frac{1}{x^{4}} - 2 \)
D.\(f(x) = x^{3} + \frac{1}{x^{4}} - 2\)

Objective

Question 16

(a) The position vectors of points L and M are (5i + 6j) and (13i + 4j) respectively. If point K lies on LM such that LK : KM is 2 : 3, find the position vector of K.

(b) Three poles are situated at points A, B and C on the same horizontal plane such that \(AB = (8km, 060°)\) and \(BC = (12km, 130°)\). Calculate,

(i) |AC|, correct to three significant figures ; (ii) the bearing of C from A, correct to the nearest degree.

Theory

Question 17

Simplify \((216)^{-\frac{2}{3}} \times (0.16)^{-\frac{3}{2}}\)

A.\(\frac{125}{288}\)
B.\(\frac{2}{125}\)
C.\(\frac{4}{225}\)
D.\(\frac{2}{225}\)

Objective

Question 17

Coplanar force 4N, 8N, 6N, 4N and 5N act at a point as shown in the diagram. If the 6N force acts in the direction 090°, calculate the :

(a) magnitude of the resultant force;

(b) direction of the resultant force.

Theory

Question 18

Given that \(\log_{3}(x - y) = 1\) and \(\log_{3}(2x + y) = 2\), find the value of x.

A.1
B.2
C.3
D.4

Objective

Question 18

(a) The position vectors of points A, B and C are \(i + 5j , 3i + 9j\) and \(-i + j\) respectively. (i) Show that points A, B and C are collinear; (ii) Determine the ratio \(|AB| : |BC|\).

(b) A uniform beam XY of mass 10 kg and length 24m is hunged horizontally from a cross bar by teo vertical inextensible strings, one attached to X and the other at a point M, 4m away from Y. A mass of 50kg is suspended at a point N which is 8m from X. If the system remains in equilibrium, calculate the tensions in the strings.

Theory

Question 19

If \(\frac{^{8}P_{x}}{^{8}C_{x}} = 6\), find the value of x.

A.1
B.2
C.3
D.6

Objective

Question 20

Evaluate \(\int_{1}^{2} [\frac{x^{3} - 1}{x^{2}}] \mathrm {d} x\).

A.0.5
B.1.0
C.1.5
D.2.0

Objective

Question 21

If \(P = \begin{vmatrix} 1 & 1 \\ 2 & 1 \end{vmatrix}\), find \((P^{2} + P)\).

A.\(\begin{vmatrix} 4 & 3 \\ 6 & 1 \end{vmatrix}\)
B.\(\begin{vmatrix} 4 & 3 \\ 6 & 4 \end{vmatrix}\)
C.\(\begin{vmatrix} 2 & 2 \\ 6 & 2 \end{vmatrix}\)
D.\(\begin{vmatrix} 3 & 2 \\ 6 & 4 \end{vmatrix}\)

Objective

Question 22

Which of the following is the semi- interquartile range of a distribution?

A.\(Mode - Median\)
B.\(\text{Highest score - Lowest score}\)
C.\(\frac{1}{2}(\text{Upper quartile - Median})\)
D.\(\frac{1}{2}(\text{Upper quartile - Lower quartile})\)

Objective

Question 23

A stone is projected vertically with a speed of 10 m/s from a point 8 metres above the ground. Find the maximum height reached. \([g = 10 ms^{-2}]\).

A.13 metres
B.15 metres
C.18 metres
D.23 metres

Objective

Question 24

The velocity \(v ms^{-1}\) of a particle moving in a straight line is given by \(v = 3t^{2} - 2t + 1\) at time t secs. Find the acceleration of the particle after 3 seconds.

A.\(26 ms^{-2}\)
B.\(18 ms^{-2}\)
C.\(17 ms^{-2}\)
D.\(16 ms^{-2}\)

Objective

Question 25

Three men, P, Q and R aim at a target, the probabilities that P, Q and R hit the target are \(\frac{1}{2}\), \(\frac{1}{3}\) and \(\frac{3}{4}\) respectively. Find the probability that exactly 2 of them hit the target.

A.\(1\)
B.\(\frac{1}{2}\)
C.\(\frac{5}{12}\)
D.\(\frac{1}{12}\)

Objective

Question 26

The position vectors of A and B are (2i + j) and (-i + 4j) respectively; find |AB|.

A.\(3\sqrt{2}\)
B.\(\sqrt{34}\)
C.\(\sqrt{34}\)
D.\(9\sqrt{2}\)

Objective

Question 27

Two fair dices, each numbered 1, 2, ..., 6, are tossed together. Find the probability that they both show even numbers.

A.\(\frac{1}{3}\)
B.\(\frac{1}{4}\)
C.\(\frac{1}{6}\)
D.\(\frac{1}{12}\)

Objective

Question 28

Calculate, correct to the nearest degree, the angle between the vectors \(\begin{pmatrix} 13 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ 4 \end{pmatrix}\).