WAEC Further Mathematics 2008 Past Questions

A function F is defined on the set R, of real numbers by \(f : x \to px^{2} + qx + 2\), where p and q are constants. If \(f(-2) = 0\) and \(f(1) = 3\), find \(f(-4)\).

If \(\sin A = \frac{3}{5}\) and \(\cos B = \frac{15}{17}\), where A is an obtuse angle and B is acute, find the value of \(\cos (A + B)\).

Differentiate, with respect to x, \(x^{3} + 2x\) from the first principle.

A straight line passes through the point P(-1, 3). Another line which passes through Q(-4, 4) intersects the first line at the point R(k, 5), where k is a constant. If \(<PRQ = 90°\), find the values of k.

In a hotel, the breakfast is a choice between yam (Y) or plantain (P) or both. The Venn diagram shows the choices made by 25 guests of the hotel.

(a) Find the value of x;

(b) What is the probability that a guest chosen at random chose only one of the two?

The table shows the distribution of marks obtained by some candidates in a test.

Draw a histogram for the distribution.

A car travelling at a velocity of 50kmh\(^{-1}\), covers a distance of 20km. If it was accelerating at 6kmh\(^{-1}\), calculate, correct to one decimal place, the time the car took to cover the distance.

The magnitude of a force \(xi + 15j\) is 17N. Find the :

(a) possible values of x ; 

(b) directions of the forces, correct to the nearest degree.

(a) The sum of the first n  terms of a sequence is given by \(S_{n} = \frac{5n^{2}}{2} + \frac{5n}{2}\). Write down the first four terms of the sequence and an expression for the nth term.

(b) The equation of a circle is given by \(x^{2} + y^{2} - 10x - 8y + 25 = 0\).

(i) Show that the circle touches the x- axis ; (ii) Find the coordinates of the point of contact.

(a) If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} + 5x - 6 = 0\), find the equation whose roots are \((\alpha - 2)\) and \((\beta - 2)\).

(b) Given that \(\int_{0} ^{k} (x^{2} - 2x) \mathrm {d} x = 4\), find the values of k.

(a) Express \(\frac{2x^{2} - 5x + 1}{x^{3} - 4x^{2} + 3x}\) in partial fractions.

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

(a) The point P(3, -5) is rotated through an angle 60° anticlockwise about the origin. (i) Obtain the matrix for the rotation ; (ii) Find the image P' of the point P under the rotation.

(b) A linear transformation is given by \(N : (x, y) \to (2x + 3y, 3x - y)\).

(i) Write down the matrix N of the transformation ; (ii) If \(N^{2} + aN + bI = 0\), where a, b \(\in\) R, \(I\) is the \(2 \times 2\) matrix and \(0\) is the \(2 \times 2\) null matrix, find the values of a and b.

A survey indicated that 65% of the families in an area have cars. Find, correct to three decimal places, the probability that among 7 families selected at random in the area

(a) exactly 5 ;

(b) 3 or 4 ;

(c) at most 2 of them have cars.

The following table shows the distribution of marks obtained by some students in an examination.

(a) Construct a cumulative frequency table for the distribution

(b) Draw an ogive for the distribution

(c) Use your graph in (b) to determine : (i) semi- interquartile range ; (ii) number of students who failed, if the pass mark for the examination is 37 ; (iii) probability that a student selected at random scored between 20% and 60%.

(a) Five female and seven male teachers applied for 4 vacancies in a Junior High School. The teachers are equally qualified. Find the number of ways of employing the 4 teachers, if : (i) there is no restriction ; (ii) at least 2 of them are females.

(b) The table shows the positions awarded to 7 contestants by Judges X and Y in a competition.

(i) Calculate, correct to one decimal place, the Spearman's rank correlation coefficient.

(ii) Interpret your answer in b(i) above.

The position vectors of points P, Q and R with respect to the origin are \((4i - 5j), (i + 3j)\) and \((-5i + 2j)\) respectively. If PQRM is a parallelogram, find:

(a) the position vector of M ;

(b) \(|\overrightarrow{PM}|\) and \(|\overrightarrow{PQ}|\) ;

(c) the acute angle between \(\overrightarrow{PM}\) and \(\overrightarrow{PQ}\), correct to 1 decimal place ;

(d) the area of PQRM.

(a) Forces \(F_{1} = \begin{pmatrix} -5 & 4 \end{pmatrix} N; F_{2} = \begin{pmatrix} 2 \\ 5 \end{pmatrix} N; F_{3} = \begin{pmatrix} 2 & -1 \end{pmatrix} N\) and \(F_{4} = \begin{pmatrix} 3 & -5 \end{pmatrix} N\) act on a body. Find the : 

(i) resultant of these forces ; (ii) fifth force that will keep the body in equilibrium.

(b) A body moving at 20 ms\(^{-1}\) accelerates uniformly at 2.5 ms\(^{-2}\) for 4 seconds. It continues the journey at the speed for 8 seconds, before coming to rest in t seconds with a uniform retardation. If the ratio of the acceleration to the retardation is 3 : 4, 

(i) sketch the velocity- time graph of the journey ; (ii) find t ; (iii) find the total distance of the journey.

(a) An object is thrown up a smooth plane inclined at an angle of 30° to the horizontal. If the plane is 15m long and the object comes to rest at the top, find the :

(i) initial speed of the object ; (ii) time taken to reach the top.

(b) Force of magnitudes \(5 N, 5\sqrt{3} N, 10 N, 5\sqrt{3} N\) and \(5 N\) act on a body P, of mass 5 kg as shown in the diagram. Find the :

(i) magnitude of the resultant force ; (ii) acceleration of the body.