WAEC Further Mathematics 2013 Past Questions

(a) If the coefficient of \(x^{2}\) and \(x^{3}\) in the expansion of \((p + qx)^{7}\) are equal, express q in terms of p.

(b) A man makes a weekly contribution into a fund. In the first week, he paid N180.00, second week N260.00, third week N340.00 and so on. How much would he have contributed in 16 weeks?

Calculate the gradient of the curve \(x^{3} + y^{3} - 2xy = 11\) at (2, -1).

A side of a rectangle is three times the other. If the perimeter increases by 2%, find the percentage increase in the area of the rectangle.

The line \(2y = x + 3\) meets the circle \(x^{2} + y^{2} - 2x + 6y - 15 = 0\) at points M and N, where N is in the first quadrant. Find the coordinates of M and N.

Three school prefects are to be chosen from four girls and five boys. What is the probability that :

(a) only boys will be chosen ;

(b) more girls than boys will be chosen ?

The table shows the distribution of ages of 22 students in a school.

Using an assumed mean of 19, calculate, correct to three significant figures, the :

(a) mean age ; (b) standard deviation ; of the distribution.

The initial velocity of a particle of mass 0.1kg is 40 m/s in the direction of the unit vector j. The velocity of the particle changed to 30 m/s in the direction of the unit vector i. Find the change in momentum. 

A stone is dropped vertically downwards from the top of a tower of height 45m with a speed of 20 ms\(^{-1}\). Find the :

(a) time it takes to reach the ground ;

(b) speed with which it hits the ground. [Take \(g = 10 ms^{-2}\)].

(a) Differentiate \(\frac{x^{2} + 1}{(x + 1)^{2}}\) with respect to x.

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

(ii) Using the answer in (b)(i), solve the system of equations.

\(x + 2y - z = 4\)

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

\(-x + y + 3z = -1\).

(a) Using a scale of 2 cm to 30° on the x- axis, 2 cm to 0.2 units on the y- axis, on the same graph sheet, draw the graphs of \(y = \sin 2x\) and \(y = \cos x\) for \(0° \leq x \leq 210°\) at intervals of 30°.

(b) Using the graphs in (a), find the truth set of :

(i) \(\sin 2x = 0\) ; (ii) \(\sin 2x - \cos x = 0\).

(a) The sum of the first three terms of a decreasing exponential sequence (G.P) is equal to 7 and the product of these three is equal to 8. Find the :

(i) common ratio ; (ii) first three terms of the sequence.

(b) Using the trapezium rule with the ordinates at x = 1, 2, 3, 4 and 5, calculate, correct to two decimal places, the value of \(\int_{1} ^{5} (x + \frac{2}{x^{2}}) \mathrm {d} x\).

(a) Find the maximum and minimum points of the curve \(y = 2x^{3} - 3x^{2} - 12x + 4\).

(b) Sketch the curve in (a) above.

The table gives the relationship between the height, in metres, of a plant and the number of days it is left to grow.

(x) (a) Using a scale of 2 cm to represent 0.5 units on the y- axis and 2cm to 10 units on the x- axis, draw a scatter diagram for the information.

(b) Find \(\bar{x}\), the mean of x, and \(\bar{y}\), the mean of y, and plot \((\bar{x}, \bar{y})\) on the diagram.

(c) Draw the line of best fit to pass through \((\bar{x}, \bar{y})\) and \((10, 1)\).

(d) From graph, find the :

(i) equation of the line of best fit ; (ii) height of plant in 75 days.

(a) A bag contains 5 blue, 4 green and 3 yellow balls. All the balls are identical except for colour. Three balls are drawn at random without replacement. Find the probability that : (i) all three balls have the same colour ; (ii) two balls have the same colour.

(b) The table shows the ranks of the marks scored by 7 candidates in Physics and Chemistry tests.

Calculate the Spearman's rank correlation coefficient.

(a) The probability that a man wins a race is 0.8. In four different races, what is the probability that he wins : (i) all races ; (ii) no race ; (iii) at most 3 races ?

(b) A class consists of 5 girls and 10 boys. If a committee of 5 is chosen at random from the class, find the probability that :

(i) 3 boys are selected ; (ii) at least one girl is selected.

(a) Three vectors a, b and c are \(\begin{pmatrix} 8 \\ 3 \end{pmatrix}, \begin{pmatrix} 6 \\ -5 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\) respectively. Find the vector d such that \(|d| = \sqrt{41}\) and d is in the direction of \(a + b - 2c\).

(b) The coordinates of A and B are (3, 4) and (3, n) respectively. If AOB = 30°, find, correct to 2 decimal places, the values of n.

 A particle is under the action of forces \(P = (4N, 030°)\) and \(R = (10N, 300°)\). Find the force that will keep the particle in equilibrium.

The displacement S metres of a particle from a fixed point O at time t seconds is given by \(S = t^{2} - 6t + 5\).

(a) On a graph sheet, draw a displacement- time graph for the interval \(0 \leq x \leq 6\).

(b) From the graph, find the : (i) time at which the velocity is zero ; (ii) average velocity over the interval \(0 \leq x \leq 4\) ; (iii) total distance covered in the interval \(0 \leq x \leq 5\).