Further Mathematics JAMB, WAEC, NECO AND NABTEB Official Past Questions

A uniform beam, XY, 4m long and weighing 350N rests on two pivots P and Q. It is kept in equilibrium by weights of 80N attached at X and 1000N attached at a point between P and Q such that it is 0.6m from Q. If XP = 0.8m and PQ = 2.2m.

(a) calculate the reactions at P and Q ;

(b) if the 1000N weight is replaced with a 1200N weight, at what point from Q should it be placed in order to maintain the equilibrium.

The position vectors of points A, B and C with respect to the origin are (8i – 2j), (2i + 6j) and (-10i + 4j) respectively. If ABCN is a parallelogram, find :

(a) the position vector of N;

(b) AN and AB ;

(c) correct to two decimal place, the acute angle between AN and AB.

The probabilities that Ali, Baba and Katty will gain admission to college are \(\frac{2}{3}, \frac{3}{4}\) and \(\frac{4}{5}\) respectively. Find the probability that:

(a) only Katty and Baba will gain admission ;

(b) none of them will gain admission ;

(c) at most two of them will gain admission.

Ten coins were tossed together a number of times. The distribution of the number of heads obtained is given in the following table :

Calculate, correct to three decimal places, the :

(a) mean number of heads ;

(b) probability of getting an even head ;

(c) probability of getting an odd number.

(a)(i) Write down the binomial expansion of \((2 – \frac{1}{2}x)^{5}\) in ascending powers of x.

(ii) Using the expansion in (a)(i), find, correct to two decimal places, the value of \((1.99)^{5}\).

(b) The polynomial \(x^{3} + qx^{2} + rx + 9\), where q and r are constants, has (x + 1) as a factor and has a remainder -17 when divided by (x + 2). Find the values of q and r.

(a) Solve : \(2^{3y + 2} – 7(2^{2y + 2}) – 31(2^{y}) – 8 = 0, y \in R\).

(b) Find \(\int (\sqrt{x^{2} + 1}) xdx\).

A circle is drawn through the points (3, 2), (-1, -2) and (5, -4). Find the :

(a) coordinates of the centre of the circle ;

(b) radius of the circle ;

(c) equation of the circle.

A body of mass 20kg moving with a velocity of 80ms\(^{-1}\) collides with another body of mass 30kg moving with a velocity of 50ms\(^{-1}\). If they both moved in the same direction after collision, find their common velocity if they moved in the :

(a) same direction before collision ; (b) opposite direction before collision.

Given that \(m = 3i – 2j ; n = 2i – 3j\) and \(p = -i + 6j\), find \(4m + 2n – 3p\).

(a) The probability that Kunle solves a particular question is \(\frac{1}{3}\) while that of Tayo is \(\frac{1}{5}\). If both of them attempt the question, find the probability that only one of them will solve the question.

(b) A committee of 8 is to be chosen from 10 persons. In how many ways can this be done if there is no restriction?

Two panel of judges, X and Y, rank 8 brands of cooking oil as follows :

Calculate the Spearmann’s rank correlation coefficient.

The sum of the first twelve terms of an Arithmetic Progression is 168. If the third term is 7, find the values of the common difference and the first term.

(a) Using the substitution \(u = x – 2\), write \(\frac{x^{3} + 5}{(x – 2)^{4}}\) as an expression in terms of u.

(b) Using the answer in (a), express \(\frac{x^{3} + 5}{(x – 2)^{4}}\) in partial fractions.

Given that \(\log_{3} x – 3\log_{x} 3 + 2 = 0\), find the values of x.

If \(\begin{vmatrix} x – 3 & -4 & 3 \\ 5 & 2 & 2 \\ 2 & -4 & 6 – x \end{vmatrix} = -24 \), find the values of x.

Two functions f and g are defined on the set of real numbers by \(f : x \to x^{2} + 1\) and \(g : x \to x – 2\). Find f o g.

A car is moving at 120\(kmh^{-1}\). Find its speed in \(ms^{-1}\).

A particle starts from rest and moves through a distance \(S = 12t^{2} – 2t^{3}\) metres in time t seconds. Find its acceleration in 1 second.

Find the constant term in the binomial expansion \((2x^{2} + \frac{1}{x})^{9}\)

Find the angle between forces of magnitude 7N and 4N if their resultant has a magnitude of 9N.

A body of mass 28g, initially at rest is acted upon by a force, F Newtons. If it attains a velocity of \(5.4ms^{-1}\) in 18 seconds, find the value of F.