neco 2018 mathematics | Essay

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Question 1 View Details
Find the smallest positive integer \(N\) that satisfies the three conditions below and show all reasoning steps: - When \(N\) is divided by 7 the remainder is 3. - When \(N\) is divided by 9 the remainder is 5. - When \(N\) is divided by 11 the remainder is 7.
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Question 2 View Details
A shopkeeper mixes two acid solutions to obtain a 25‑litre solution that is 50 % acid. - Solution A contains 30 % acid and costs ₦120 per litre. - Solution B contains 80 % acid and costs ₦200 per litre. (a) Determine how many litres of each solution are required. (b) Calculate the total cost of the mixture. (c) If the final 25‑litre solution is sold at ₦250 per litre, find the profit made by the shopkeeper.
Question Parts
(a)
Determine the number of litres of Solution A and Solution B needed.
(b)
Calculate the total cost of the mixture.
(c)
Find the profit if the final solution is sold at ₦250 per litre.
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Question 3 View Details
Consider the expression \(E = \frac{(2^{3} \cdot 5^{2})^{1/2}}{10^{\log_{10} 8}}\).
Question Parts
(a)
Simplify the denominator using the definition of logarithm.
(b)
Simplify the numerator as a surd in its simplest form.
(c)
Hence find the value of \(E\) in the form \(\frac{p\sqrt{q}}{r}\) where \(p, q, r\) are integers, \(q\) is not divisible by any perfect square greater than 1. State the values of \(p, q, r\).
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Question 4 View Details
A survey of 75 senior secondary students was carried out to find out their participation in three extracurricular clubs: Drama (A), Science (B) and Music (C). The results are represented in the Venn diagram shown.
Question Parts
(a)
Find the probability that a randomly selected student is a member of exactly one club.
(b)
Find the probability that a student belongs to at least two of the clubs.
(c)
Given that a student is a member of the Science club, find the probability that the student also belongs to the Drama club.
(d)
Let X be the random variable representing the number of clubs to which a randomly selected student belongs.
() Find the expected value E(X).
() Find the variance Var(X).
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Question 5 View Details
Let a and b be two integers such that a + b = 10 and a \times b = 21.
Question Parts
(a)
Determine the values of a and b.
(b)
Using the values obtained, evaluate \(\frac{a^{3}+b^{3}}{a+b}\).
(c)
Let c = a - b. Find the value of c^{2}.
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Question 6 View Details
A contractor pays daily wages to skilled workers (₦x per worker) and unskilled workers (₦y per worker). On Monday, 5 skilled and 8 unskilled workers were employed and the total wage paid was ₦690. On Tuesday, 7 skilled and 4 unskilled workers were employed and the total wage paid was ₦750.
Question Parts
(a)
Determine the daily wages x and y.
(b)
Using the wages found, calculate the total wage that would be paid if on Wednesday 6 skilled and 6 unskilled workers were employed.
(c)
Find the percentage difference between the daily wages of skilled and unskilled workers, expressed as \(\frac{|x-y|}{y}\times100\%\).
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Question 7 View Details
A rectangular plot has length \(L\) cm and width \(W\) cm. It is known that \(L = 2W + 5\) and the area of the plot is \(200\) cm\(^2\).
Question Parts
(a)
Form the quadratic equation that \(W\) satisfies and solve for the possible value(s) of \(W\).
(b)
Using the value of \(W\) obtained, determine the corresponding length \(L\).
(c)
If fencing costs \(\text{₦}120\) per metre, calculate the total cost to fence the plot. (Give your answer in naira, rounded to the nearest naira.)
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Question 8 View Details
The profit (in thousand naira) from producing \(x\) thousand units of a widget is given by \(P = -2x^{2} + 12x - 5\). The company requires the profit to be at least \(7\) thousand naira.
Question Parts
(a)
Solve the inequality \(-2x^{2} + 12x - 5 \ge 7\) for \(x\).
(b)
Express the range of \(x\) (in thousand units) that satisfies the profit requirement as an inequality.
(c)
If the company can only produce whole thousand‑unit batches, list all integer values of \(x\) that meet the requirement.
(d)
The production cost is \(\text{₦}1500\) per thousand units. Determine the minimum total cost (in naira) corresponding to the feasible production levels found in part (c).
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Question 9 View Details
A sequence \(\{a_n\}\) is defined by \(a_1 = 2\) and for every integer \(n \ge 1\) the recurrence relation \(a_{n+1}=3a_n+2\) holds.
Question Parts
(a)
Find the value of \(a_4\).
(b)
Prove, by mathematical induction, that for all positive integers \(n\), \(a_n = 3^{,n}-1\).
(c)
Using the formula obtained in part (b), find the sum of the first five terms \(S_5 = a_1+a_2+\dots +a_5\).
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Question 10 View Details
The time \(T\) (in seconds) required for a particle to travel a distance \(s\) metres under a force that varies directly as the square of its speed \(v\) (in m s⁻¹) and inversely as its mass \(m\) (in kg) is described by a joint variation. In addition, the speed \(v\) varies inversely as the square root of the distance travelled. When the mass is \(2\) kg and the distance is \(8\) m, the time taken is \(4\) s.
Question Parts
(a)
Express the relationship between \(T\), \(v\) and \(m\) using a constant of variation \(k\).
(b)
Write the relationship between \(v\) and \(s\) using a constant \(c\).
(c)
Using the information given (\(m=2\) kg, \(s=8\) m, \(T=4\) s), determine the combined constant \(K\) in the expression \(T = \frac{K}{sm}\).
(d)
Find the time taken when the mass is \(5\) kg and the distance travelled is \(18\) m. Give your answer as a simplified fraction.
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Question 11 View Details
The graph shown consists of a quadratic curve and a straight line.
Question Parts
(a)
Determine the coordinates of the points where the quadratic curve and the straight line intersect.
(b)
Calculate the exact area enclosed between the quadratic curve and the straight line from the leftmost to the rightmost point of intersection.
(c)
Find the x‑coordinate at which the vertical separation between the two curves is greatest and state that maximum separation.
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Question 12 View Details
In the coordinate plane, triangle ABC is positioned such that point A is at the origin O(0,0) and side AB lies along the positive x‑axis. The length of AB is 8 cm, \(\angle A = 40^{\circ}\) and \(\angle B = 70^{\circ}\).
Question Parts
(a)
Determine the coordinates of point C (give your answer to two decimal places).
(b)
Find the equation of line AC.
(c)
Calculate the length of the altitude from C to side AB.
(d)
Find the area of triangle ABC (express your answer to two decimal places).
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Question 13 View Details
In triangle \(ABC\) the side \(AB = 12\) cm, \(\angle A = 40^{\circ}\) and \(\angle B = 70^{\circ}\). Point \(D\) lies on side \(BC\) such that \(BD : DC = 1 : 2\). The labelled diagram of the triangle is provided.
Question Parts
(a)
Find the measure of \(\angle C\).
(b)
Calculate the length of side \(BC\) (give your answer to two decimal places).
(c)
Find the area of \(\triangle ABC\) (give your answer to one decimal place).
(d)
Point \(D\) divides \(BC\) in the ratio \(1:2\). Determine the length of \(AD\) (give your answer to two decimal places).
(e)
Find the radius \(R\) of the circumcircle of \(\triangle ABC\) (give your answer to two decimal places).
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