WAEC Further Mathematics 2015 Past Questions

A binary operation A is defined on the set of real numbers, R, by \(a \Delta b = a^{3} - b^{3}\). Without using calculator, find the value of \((\sqrt{3} + \sqrt{2}) \Delta (\sqrt{3} - \sqrt{2})\) leaving the answer in surd form.

Points (2, 1) and (6, 7) are opposite vertices of a square which is inscribed in a circle. Find the :

(a) centre of the circle ; (b) equation of the circle.

If \(f ' '(x) = 2\), \(f ' (1) = 0\) and \(f(0) = - 8\), find f(x).

Solve : \(\tan (2x - 15)° - 1 = 0\), for values of x such that \(0° \leq x \leq 360°\).

A car moving with an initial velocity, u, travels in a straight line with a constant acceleration of 3ms\(^{-2}\) until it attains a velocity of 33ms\(^{-1}\) after 6 seconds. Calculate the distance travelled by the car.

Five finalists in a beauty pageant were ranked by two judges X and Y as shown in the table :

Calculate the Spearman's rank correlation coefficient.

There are 8 boys and 6 girls in a class. If two students are selected at random from the class, find the probability that they are of

(a) the same sex ;

(b) different sex.

Forces of magnitude 3N, 4N and 2N act along the vectors \(j ; -i + j\) and \(i + j\) respectively. Calculate, correct to one decimal place, the magnitude of the resultant of the forces.

(a) The functions \(f : x \to x^{2} + 1\) and \(g : x \to 5 - 3x\) are defined on the set of the real numbers, R.

(i) State the domain of \(f^{-1}\), the inverse of f ; (ii) find \(g^{-1} (2)\).

(b) Evaluate : \(\int \frac{(x + 3)}{x^{2} + 6x + 9} \mathrm {d} x\)

(a) If \(\frac{\sqrt{5} + 4}{3 - 2\sqrt{5}} - \frac{2 + \sqrt{5}}{4 - 2\sqrt{5}} = a + b\sqrt{5}\), find the values of a and b.

(b)(i) Evaluate : \(\begin{vmatrix} 2 & -1 & 2 \\ 1 & 3 & 4 \\ 1 & 2 & 1 \end{vmatrix}\)

(ii) Using the result in b(i), find, correct to two decimal places, the value of x in the system of equations.

\(2x - y + 2z + 5 = 0\)

\(x + 3y + 4z - 1 = 0\)

\(x + 2y + z + 2 = 0\)

(a)(i) Write down the binomial expansion of \((1 + x)^{4}\).

(ii) Use the result in (a)(i) to evaluate, correct to three decimal places \((\frac{5}{4})^{4}\).

(b) The first, second and fifth terms of a linear sequence (A.P) are three consecutive terms of an exponential sequence (G.P). If the first term of the linear sequence is 7, find the common difference.

The table shows the distribution of the heights of a group of people.

(m) (a) Draw a histogram to illustrate the distribution.

(b) Using an assumed mean of 1.1m, find, correct to one decimal place, the mean height of the group.

(a) Edem and his wife were invited to a dinner by a family of 5. They all sat in such a way in such a way that Edem sat next to his wife. Find the number of ways of seating them in a row.

(b) A bag contains 4 red and 5 black identical balls. If 5 balls are selected at random, one after the other with replacement, find the probability that :

(i) a red ball was picked 3 times ; (ii) a black ball was picked at most 2 times.

(a) Given that \(\overrightarrow{AB} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}\) and \(\overrightarrow{BC} = \begin{pmatrix} -3 \\ 5 \end{pmatrix}\); find the :

(i) angle between the vectors AB and AC ; (ii) unit vector along \(\overrightarrow{AB} - \overrightarrow{BC}\).

(b) P, Q, R and M are points in the \(O_{XY}\) plane. If \(\overrightarrow{PQ} = 2i + 8j , \overrightarrow{PR} = 11i - 12j\) and M divides QR internally in the ratio 3 : 7, find \(\overrightarrow{PM}\).

(a) A bucket full of water with a mass of 8kg is pulled out of a well with a light inextensible rope. Find its acceleration when tha tension in the rope is 150N. [Take \(g = 10ms^{-2}\)].

(b) A mass of 12kg is acted upon by a force F, changing its speed from 15 m/s to 25 m/s after covering a distance of 50m. Find the :

(i) value of F ; (ii) distance covered when its speed is 35 m/s.