Mathematics JAMB, WAEC, NECO AND NABTEB Official Past Questions

A box contains 5 blue balls, 3 black balls and 2 red balls of the same size. A ball is selected at random from the box and then replaced. A second ball is then selected. Find the probability of obtaining

(a) two red balls ;

(b) two blue balls or two black balls ;

(c) one black and one red ball in any order.

The table shows the scores of 2000 candidates in an entrance examination into a private secondary school.

(a) Prepare a cumulative frequency table and draw the cumulative frequency curve for the distribution.

(b) Use your curve to estimate the : (i) cut off mark, if 300 candidates are to be offered admission ; (ii) probability that a candidate picked at random, scored at least 45%.

(a) A man travels from a village X on a bearing of 060° to a village Y which is 20km away. From Y, he travels to a village Z, on a bearing of 195°. If Z is directly east of X, calculate, correct to three significant figures, the distance of :

(i) Y from Z ; (ii) Z from X .

(b) An aircraft flies due South from an airfield on latitude 36°N, longitude 138°E to an airfield on latitude 36°S, longitude 138°E. 

(i) Calculate the distance travelled, correct to three significant figures ; (ii) if the speed of the aircraft is 800km per hour, calculate the time taken, correct to the nearest hour.

[Take \(\pi = \frac{22}{7}\), R = 6400km].

(a) Using a ruler and a pair of compasses only, construct triangle ABC with /AB/ = 7.5cm, /BC/ = 8.1cm and < ABC = 105°.

(b) Locate a point D on BC such that /BD/ : /DC/ is 3 : 2.

(c) Through D, construct a line I perpendicular to BC.

(d) If the line I meets AC at P, measure /BP/.

(a) Copy and complete the table for the relation \(y = 2 \cos 2x – 1\).

(b) Using a scale of 2cm = 30° on the x- axis and 2cm = 1 unit on the y- axis, draw the graph of \(y = 2 \cos 2x – 1\) for \(0° \leq x \leq 180°\).

(c) On the same axis, draw the graph of \(y = \frac{1}{180} (x – 360)\)

(d) Use your graphs to find the : (i) values of x for which \(2 \cos 2x + \frac{1}{2} = 0\); (ii) roots of the equation \(2 \cos 2x – \frac{x}{180} + 1 = 0\).

(a) Given that \(p = x + ym^{3}\), find m in terms of p, x and y.

(b) Using the method of completing the square, find the roots of the equation \(x^{2} – 6x + 7 = 0\), correct to 1 decimal place.

(c) The product of two consecutive positive odd numbers is 195. By constructing a quadratic equation and solving it, find the two numbers.

(a) Using mathematical tables, find ; (i) \(2 \sin 63.35°\) ; (ii) \(\log \cos 44.74°\);

(b) Find the value of K given that \(\log K – \log (K – 2) = \log 5\);

(c) Use logarithm tables to evaluate \(\frac{(3.68)^{2} \times 6.705}{\sqrt{0.3581}}\)

The frequency table shows the marks scored by 32 students in a test.

Find the :

(a)(i) mean ; (ii) median ; (iii) mode of the marks;

(b) percentage of the students who scored at least 8 marks.

(a) A pack of 52 playing cards is shuffled and a card is drawn at random. Calculate the probability that it is either a five or a red nine.

[Hint : There are 4 fives and 2 red nines in a pack of 52 cards]

(b) P, Q and R are points in the same horizontal plane. The bearing of Q from P is 150° and the bearing of R from Q is 060°. If /PQ/ = 5m and /QR/ = 3m, find the bearing of R from P, correct to the nearest degree.

(a) In the diagram, PQSR and SRYZ are parallelograms and PQYZ is a straight line. If /QY/ = 2cm and /RS/ = 3cm, find /PZ/.

(b) P and Q are two towns on the earth’s surface on latitude 56°N. Thei longitudes are 25°E and 95°E respectively. Find the distance PQ along their parallel of latitude, correct to the nearest km. [Take radius of the earth as 6400km and \(\pi = \frac{22}{7}\)]

The quantity y is partly constant and partly varies inversely as the square of x.

(a) Write down the relationship between x and y.

(b) When x = 1, y = 11 and when x = 2, y = 5, find the value of y when x = 4.

(a) Factorise : \(px – 2px – 4qy + 2py\)

(b) Given that the universal set U = {1, 2, 3, 4,5, 6, 7, 8, 9, 10}, P = {1, 2, 4, 6, 10} and Q = {2, 3, 6, 9}; show that \((P \cup Q)’ = P’ \cap Q’\)

From a box containing 2 red, 6 white and 5 black balls, a ball is randomly selected. What is the probability that the selected ball is black?

Two fair dice are tossed together once. Find the probability that the sum of the outcome is at least 10.

Find the median of the following numbers 2.64, 2.50, 2.72, 2.91 and 2.35.

The mean of 20 observations in an experiment is 4, lf the observed largest value is 23, find the mean of the remaining observations.

What is the mode of the numbers 8, 10, 9, 9, 10, 8, 11, 8, 10, 9, 8 and 14?

The bearing of two points Q and R from a point P are 030° and 120° respectively, lf /PQ/ = 12 m and /PR/ = 5 m, find the distance QR.

In the diagram above, ∠PRQ = 90°, ∠QPR = 30° and /PQ/ = 10 cm. Find y.

Without using tables, find the value of \(\frac{\sin 20°}{\cos 70°} + \frac{\cos 25°}{\sin 65°}\)

In ΔABC above, BC is produced to D, /AB/ = /AC/ and ∠BAC = 50^o. Find ∠ACD