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

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 \(0 \leq x \leq 5\).

 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.

(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) 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) 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.

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

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

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

(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) 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) 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 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 \(g = 10 ms^{-2}\)].

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. 

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.

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 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.

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.

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

(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?

A body is kept at rest by three forces \(F_{1} = (10N, 030°), F_{2} = (10N, 150°)\) and \(F_{3}\). Find \(F_{3}\).

If \(\frac{^{n}C_{3}}{^{n}P_{2}} = 1\), find the value of n.

Find the equation of the straight line that passes through (2, -3) and perpendicular to the line 3x – 2y + 4 = 0.