waec model questions vol1 2019 mathematics | Essay

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Question 1 View Details
A positive integer N satisfies the following conditions: when divided by 7 it leaves a remainder of 3; when divided by 9 it leaves a remainder of 5; and when divided by 11 it leaves a remainder of 7.
Question Parts
(a)
Find the smallest such N that is less than 2000.
(b)
Using the value of N obtained in part (a), determine the greatest common divisor (GCD) of N and 693.
(c)
If N is expressed in the form 1000a + b where 0 ≤ b < 1000, find the value of a + b.
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Question 2 View Details
A merchant bought a batch of fabric for ₦12,500. He first increased the price by 20% to cover overheads, then offered a discount of 15% on the increased price to attract customers, and finally added a sales tax of 5% on the discounted price.
Question Parts
(a)
Determine the final selling price of the fabric (to the nearest naira).
(b)
Calculate the overall percentage gain or loss relative to the original cost price.
(c)
If the merchant wishes to achieve an overall profit of exactly 10% on the original cost, what single percentage discount should he apply after the 20% overhead increase, assuming the 5% sales tax is still added after the discount? (Give the discount to the nearest tenth of a percent.)
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Question 3 View Details
A certain algebraic problem involves indices, logarithms and surds.
Question Parts
(a)
Given that \(\dfrac{2^{x}\,5^{2x}}{10^{\,x-1}} = 50\), find the value of \(x\).
(b)
Using the value of \(x\) obtained in part (a), evaluate \(\log_{2}\bigl(x^{2}+3x\bigr)\).
(c)
Simplify the surd expression \(\sqrt{50}\; -\; \sqrt{18}\) and write it in the form \(a\sqrt{b}\) where \(a\) is an integer and \(b\) is a positive integer not divisible by any perfect square greater than 1.
(d)
Find the product of the answers obtained in parts (b) and (c).
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Question 4 View Details
The Venn diagram below shows the numbers of students in a senior secondary school who offered the three science subjects Mathematics (M), Physics (P) and Chemistry (C). The total number of students surveyed is 120.
Question Parts
(a)
State the number of students who offered exactly one of the three subjects.
(b)
State the number of students who offered exactly two of the three subjects.
(c)
What is the probability that a randomly selected student offered at least one of the three subjects? Express your answer as a fraction in lowest terms.
(d)
Given that a student offered Mathematics, find the probability that the same student also offered Chemistry. Give your answer as a fraction in lowest terms.
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Question 5 View Details
A quadratic expression can be written in the form \((x - a)(x - b)\) where \(a\) and \(b\) are positive integers. Expanding this product gives \(x^{2} - 9x + 20\).
Question Parts
(a)
Determine the integers \(a\) and \(b\).
(b)
Let \(x = a + b\). Evaluate the expression \((x - a)(x - b)\) and state the remainder when this value is divided by 7.
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Question 6 View Details
A stationery shop sells pens and notebooks. The total cost for 5 pens and 3 notebooks is ₦405. Each pen costs ₦15 less than a notebook.
Question Parts
(a)
Find the cost of one pen and one notebook (in naira).
(b)
A student buys twice as many pens as notebooks and spends ₦1,200. Determine how many pens and how many notebooks the student buys.
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Question 7 View Details
A solid is formed by a right circular cylinder of radius r cm and height h cm. A right circular cone is cut from the cylinder such that the cone’s base coincides with the top base of the cylinder and its vertex is at the centre of the bottom base of the cylinder. The solid after the cone is removed is shown in the diagram.
Question Parts
(a)
Find the ratio of the volume of the cone to the volume of the cylinder. Express your answer in its simplest form.
(b)
If the total surface area of the solid (including the interior surface of the cone) is 1500 cm² and the height h is 20 cm, determine the radius r of the cylinder (to the nearest centimetre).
(c)
Using the value of r found in part (b), calculate the external surface area of the solid (the area that would be exposed if the cone were removed, i.e., curved surface of the cylinder plus the bottom base only). Give your answer in cm² to the nearest whole number.
(d)
If the external surface of the solid is to be painted and one litre of paint covers 12 m², how many litres of paint are required? Give your answer correct to two decimal places.
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Question 8 View Details
Solve the inequality \(\dfrac{x-2}{x+3} \le 1\) and express the solution set in interval notation.
Question Parts
(a)
State the domain of the rational expression.
(b)
Bring all terms to one side and simplify to a single rational expression.
(c)
Solve the resulting inequality by analysing the sign of the numerator and denominator.
(d)
Write the final solution set in interval notation.
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Question 9 View Details
A sequence \(\{u_n\}\) is defined by \(u_1 = 2\) and for \(n \ge 1\) the recurrence relation \(u_{n+1}=3u_n+4\).
Question Parts
(a)
Obtain an explicit formula for \(u_n\) in terms of \(n\).
(b)
Using the formula obtained in part (a), find the sum of the first five terms \(S_5 = u_1+u_2+u_3+u_4+u_5\).
(c)
Determine the smallest integer \(n\) for which \(u_n > 500\).
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Question 10 View Details
The time \(T\) (in seconds) taken by a certain chemical reaction to reach completion varies directly as the square of the concentration \(C\) (in mol·L⁻¹) of reactant A and inversely as the absolute temperature \(\theta\) (in K). When \(C = 0.2\) mol·L⁻¹ and \(\theta = 300\) K, the reaction takes 45 seconds.
Question Parts
(a)
Find the constant of variation \(k\).
(b)
If the concentration is increased to \(0.5\) mol·L⁻¹ while the temperature is lowered to \(280\) K, calculate the new reaction time.
(c)
For a fixed temperature of \(310\) K, determine the concentration required for the reaction to complete in exactly \(30\) seconds.
(d)
Explain qualitatively how the reaction time would change if both the concentration and the temperature were doubled, using the variation law.
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Question 11 View Details
The graph below shows the monthly sales (in thousands of Naira) of a retail store over a period of six months.
Question Parts
(a)
Calculate the average rate of change of sales between month 2 and month 5.
(b)
Using linear interpolation between month 4 and month 5, estimate to two decimal places the month when sales reached 20 000 Naira.
(c)
Approximate the total sales over the six‑month period by calculating the area under the graph using the trapezoidal rule (Δx = 1 month). Give your answer in thousands of Naira.
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Question 12 View Details
The diagram shows the parabola y = x² − 4x + 3 and the straight line 2x + y = 6 drawn on the same coordinate plane.
Question Parts
(a)
Determine the coordinates of the points where the line intersects the parabola.
(b)
Calculate the exact area enclosed between the line and the parabola between the intersection points.
(c)
Find the equation of the tangent to the parabola at x = 3 and state whether this tangent is parallel to the given line 2x + y = 6.
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Question 13 View Details
In right‑angled triangle ABC, ∠C = 90°. The altitude from C meets the hypotenuse AB at D. It is given that CD = 6 cm and AD = 8 cm.
Question Parts
(a)
Find the length of DB (the other segment of the hypotenuse).
(b)
Determine the length of the whole hypotenuse AB.
(c)
Calculate the lengths of the other two sides AC and BC.
(d)
Find the area of triangle ABC.
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