WAEC 2025 Mathematics | Essay

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Question 1
Given that \( \mu=\{x ; 1<x<20, \text{where x is an integer} \}, P=\{x: x \text{ is a multiple of } 3\} \) and \( Q=\{x: x \text{ is a prime number } \} \) where \( P \) and \( Q \) are subsets of \( \mu \), find:
Question Parts
(a)
\( P \cap Q^{\prime} \);
(b)
\( P^{\prime} \cup Q \);
(c)
\( (P \cup Q)^{\prime} \).
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Question 2
The product of the ages of Adu and Tanko is 9 less than Akorfa's age. If Tanko is 4 years older than Adu and Akorfa's age is six times Tanko's age, find Akorfa's age.
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Question 3
A company installs solar panels in its premises to reduce its electricity cost. The monthly savings on electricity in \( \$ \), is modelled by \( S=200+50 x-2 x^{2} \), where \( x \) is the number of months after installation.
Question Parts
(a)
At what time will the savings on electricity stop increasing?
(b)
Find the maximum savings.
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Question 4
The diagram shows a tower \( T R \) and an observer at \( O \). \(|O R|=84 m\) and the angle of elevation of the top of the tower \( T \) from \( O \) is \( 57^{\circ} \).
Question Parts
(a)
Calculate, correct to three significant figures, the height of the tower.
(b)
The observer at \( O \), moved away from the tower in the same straight line until the angle of elevation of \( T \) is \( 49^{\circ} \). Find, correct to two decimal places, how far the observer moved backwards.
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Question 5
The data represent the score obtained by 9 applicants in an interview arranged in ascending order: \( (3 x+2), 22,(4 x-2), 23,25,(5 x-4), 29,29 \) and ( \( x^{2}-7 \) ).
Question Parts
(a)
Given that the range is 9, find the:
(b)
If four of the applicants who obtained the highest score were selected, determine the pass mark.
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Question 6
In a certain year, the consumption pattern of electricity charges in a town was as follows: the cost of the first 30 units was \( \$ 1.00 \) per unit; the cost of the next 30 units was \( \$ 7.00 \) per unit; the cost of each additional unit was \( \$ 5.00 \).
Question Parts
(a)
If Amaka used 420 units of electricity in January that year, calculate the amount paid.
(b)
If Amaka paid \( \$ 2,740.00 \) in the month of February, calculate the number of units of electricity consumed.
(c)
Find, correct to two decimal places, the percentage change in units of electricity consumed by Amaka in January and February.
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Question 7
Yaro drove from a town Gaja to Banga. After 2 hours in the journey, he observed that he had covered 80 km and realized that if he continued driving at the same average speed, he would end up being late for 15 minutes. If he decided to increase the average speed by \( 10 \mathrm{~km} / \mathrm{h} \), he would arrive at Banga 36 minutes earlier. Find the distance between Gaja and Banga.
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Question 8
Using a ruler and a pair of compasses only, construct; (i) a quadrilateral PQRS such that \( |PQ| = 8.5\text{cm} \), \( |QR| = 7.5\text{cm} \), \( \angle QPS = 60^{\circ} \), \( \angle PQR = 105^{\circ} \) and \( S \) is a point on the locus, \( L_{1} \), which is equidistant from \( \overline{PQ} \) and \( \overline{QR} \\ \); (ii) locus, \(L_{2}\), of points equidistant from $P$ and \( Q \\ \); (iii) locate the point \( K \), which is the point of intersection of \( L_{1} \) and \( L_{2} \). (b) Measure \( |KS| \).
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Question 9
(a) Mrs. Otoo spends \( \frac{1}{9} \) of her monthly salary on rent, \( \frac{1}{2} \) on food, \( \frac{1}{4} \) on clothes and still had \( \$195.00 \) left. How much does she earn in a month? (b) A sector of a circle of radius \( 6\text{cm}\) subtends an angle of \( 105^{\circ} \) at the centre. Calculate the; (i) perimeter; (ii) area; of the sector \( [\text{Take } \pi = 22/7] \)
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Question 10
(a) The cost, C, of feeding some students in a class is partly constant and partly varies as the number of students, \( n \), in the class. For 8 students, the cost is \$ 70.00 and for 10 students the cost is \$ 90.00. Find \( \\ \): (i) an expression for \( C \) in terms of \( n \\ \); (ii) the cost of feeding 12 students. (b) In the diagram, \( A \), \( B \), \( C \) and \( D \) are points on a circle. \( \overline{AB} \) \( \parallel \overline{DC} \), \( \overline{AC} \) and \( \overline{BD} \) intersect at \( X \) and \( \angle BDC = 40^{\circ} \\ \). Find: (i) \( \angle ACD \\ \) ; (ii) \( \angle AXD \).
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Question 11
(a) Given that \( P=\left(\begin{array}{rr}2 & 4 \\ -9 & 1\end{array}\right) \) and \( Q=\left(\begin{array}{ll}1 & -1 \\ 3 & -2\end{array}\right), \text { find } P Q+2 Q \)
Question Parts
(b)
A bag contains 8 red balls and some white balls, all of the same size. If the probability of drawing at random a white ball from the bag is half the probability of drawing a red ball, find the number of white balls in the bag.
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Question 12
[Take \( \pi=\frac{22}{7} \text{and} R=6,400 \mathrm{~km} \)]
Question Parts
(a)
The eighth term of an Arithmetic Progression (A.P.) is 46 and the sum of the first eight terms is 200. Find the:
(b)
The points \( X\left(70^{\circ} \mathrm{S}, 60^{\circ} \mathrm{E}\right) \) and \( Y\left(7^{\circ} \mathrm{S}, 60^{\circ} \mathrm{E}\right) \) lie on the surface of the earth.
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Question 13
The following are the marks scored by 20 students in a test: 15, 11, 17, 25, 13, 15, 16, 22, 24, 27, 20, 22, 15, 16, 15, 19, 22, 24, 22, 11.
Question Parts
(a)
Prepare a frequency table for the distribution using class intervals 10-12, 13-15, 16-18.
(b)
Calculate the variance of the distribution.
(c)
If the pass mark for the test was 16, find the probability that a student selected at random from the class failed.
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