WAEC Further Mathematics 2016 Past Questions

A binary operation \(*\) is defined on the set, R, of real numbers by \(m * n = m + n + 2\). Find the :

(a) identity element under the operation ;

(b) inverse of n under the operation .

Given that (5, 2), (-4, k) and (2, 1) lie on a straight line, find the value of k.

(a) If \(f(x + 2) = 6x^{2} + 5x - 8\), find \(f(5)\).

(b) Express \(\frac{7\sqrt{2} + 3\sqrt{3}}{4\sqrt{2} - 2\sqrt{3}}\) in the form \(p + q\sqrt{r}\), where p, q and r are rational numbers. 

When \(f(x) = 2x^{3} + mx^{2} + nx + 11\) is divided by \(x^{2} + 5x + 1\), the quotient is \(2x - 5\) and the remainder is \(30x + 16\). Find the values of m and n.

The probabilities that Ago, Sulley and Musa will gain admission to a certain university are \(\frac{4}{5}, \frac{3}{4}\) and \(\frac{2}{3}\) respectively. Find the probability that :

(a) none of them will gain admission ; 

(b) only Ago and Sulley will gain admission.

The mean of the numbers 1, 4, k, (k + 4) and 11 is (k + 1). Calculate the :

(a) value of k ;

(b) standard deviation.

(a) A body of mass 3kg moves with a velocity of 8ms\(^{-1}\). It collides with a second body moving in the same direction with a velocity of 5ms\(^{-1}\). After collision, the bodies move together with a velocity of 6ms\(^{-1}\). Find the mass of the second body.

(b) If the second body in (a) moves with a velocity of 5ms\(^{-1}\) in the opposite direction as that of the 3kg body with a velocity of 8ms\(^{-1}\), find, correct to two decimal places, the common velocity of the two bodies if they move together after collision.

Given that \(p = \begin{pmatrix} 5 \\ 3 \end{pmatrix}, q = \begin{pmatrix} -1 \\ 2 \end{pmatrix}\) and \(r = \begin{pmatrix} 17 \\ 5 \end{pmatrix}\) and \(r = \alpha r + \beta q\), where \(\alpha\) and \(\beta\) are scalars, express q in terms of r and p.

(a) Without using mathematical tables or calculator, evaluate \(\frac{\frac{3}{2}\log 27 - 3\log 5\sqrt{5}}{\log 0.6}\)

(b) Two linear transformations A and B in the \(O_{xy}\) plane, are defined by :

\(A : (x, y)   (x + 2y, -x + y)\)

\(B : (x, y)   (2x + 3y, x + 2y)\).

(i) Write down the matrices A and B; (ii) Find the image of the point P(-2, 2) under the linear transformation A followed by B.

(a)(i) Write down the expansion of \((1 + x)^{7}\) in ascending powers of x.

(ii) If the coefficients of the fifth, sixth and seventh terms in the expansion in (a)(i) above form a linear sequence(A.P), find the common difference of the A.P.

(b) Using the trapezium rule with ordinates at 1, 2, 3, 4 and 5, calculate, correct to two decimal places, 

\(\int_{1}^{5} \sqrt{(2x + 8x^{2})} \mathrm {d} x\).

(a) If \(^{k}P_{2} = 72\), find the value of k.

(b) Solve the equation : \(2\cos^{2} \theta - 5\cos \theta = 3; 0° \leq \theta \leq 360°\)

The table shows the distribution of marks scored by some students in a test.

(a)(i) Construct a cumulative frequency table for the distribution ; (ii) Draw a cumulative frequency curve for the distribution.

(b) Use the curve to estimate the :

(i) number of students who scored marks between 32 and 74 ; (ii) pass mark, if 18% of the students failed ; (iii) lowest mark for distinction, if 8% of the students passed with distinction.

(a) Two Mathematics books, 5 different Physics books and 3 different Chemistry are to be arranged on a shelf. How many arrangements are possible if ;

(i) books on the same subject must stand together?  (ii) only the Physics books must stand together?

(b) In a certain community, 13 out of every 20 persons speak English. If 8 persons are selected at random from the community, find, correct to three significant figures, the probability that at least 3 of them speak English.

Four vectors \(r = \alpha i + \beta j\), where \(\alpha \text{ and } \beta\) are constants, \(s = 2i -j, m = 3i + 2j\) and \(n = i + j\) are such that the magnitude of r is three times as s and is parallel to the vactor (m - n).

(a) Find the values of \(\alpha\) and \(\beta\).

(b) Calculate the magnitude and direction of (r - s).

A particle is projected vertically upwards from the ground with speed \(30ms^{-1}\). Calculate the :

(a) maximum height reached by the particle;

(b) time taken by the particle to return to the ground;

(c) time(s) taken for the particle to attain a height of 40m above the ground. [Take \(g = 10ms^{-2}\)]