neco 2017 mathematics | Essay

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Question 1 View Details
A school has a total number of students that is a three‑digit integer. When the number of students is divided by 7 the remainder is 3; when divided by 9 the remainder is 5; and when divided by 11 the remainder is 7. Determine the smallest possible number of students that satisfies all three conditions.
Question Parts
(a)
State the system of congruences that represents the conditions given.
(b)
Find the smallest three‑digit integer that satisfies the system of congruences.
(c)
Verify that the number you obtained is indeed a three‑digit integer.
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Question 2 View Details
A shopkeeper bought a batch of fabric for an unknown amount. He first increased the price by 25 %, then gave a discount of 20 % on the increased price. After this, a value‑added tax (VAT) of 5 % is added to the selling price. If the final amount paid by a customer is ₦13 200, determine the original cost price of the fabric and state the overall percentage gain or loss relative to the cost price.
Question Parts
(a)
Write an algebraic expression for the final price in terms of the original cost price C.
(b)
Calculate the original cost price C.
(c)
State the overall percentage gain or loss of the shopkeeper on this transaction.
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Question 3 View Details
A company produces two types of widgets, X and Y. The profit per unit of widget X is expressed as \(2^{n}\) Naira, where \(n\) is an integer. The profit per unit of widget Y is expressed as \(\log_{2} m\) Naira, where \(m\) is a positive integer. In a particular month the following conditions are satisfied: - The total profit from selling 5 units of X and 8 units of Y is 72 Naira. - The profit per unit of X is exactly twice the profit per unit of Y. Using the information above, answer the questions that follow.
Question Parts
(a)
Determine the values of \(n\) and \(m\). State the profit per unit of each widget.
(b)
If the company sells 12 units of X and 15 units of Y, calculate the total profit for that month.
(c)
The company wishes to achieve a total profit of at least 200 Naira while keeping the same ratio of X to Y units as in the original month (5 : 8). What is the minimum whole number of X units that must be sold?
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Question 4 View Details
The Venn diagram below shows the distribution of 110 senior secondary students according to the subjects they study. - 30 students study Mathematics only. - 25 students study Physics only. - 15 students study both Mathematics and Physics. - The remaining students study neither subject. Using the information from the diagram, answer the following questions.
Question Parts
(a)
How many students study at least one of the two subjects?
(b)
What is the probability that a randomly selected student studies Mathematics but not Physics?
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Question 5 View Details
Consider the cubic polynomial P(x) = ax^3 + bx^2 + cx + d, where a, b, c and d are integers.
Question Parts
(a)
It is known that (i) (x - 1) and (x + 2) are factors of P(x); (ii) when P(x) is divided by (x - 3) the remainder is 20; (iii) the sum of the coefficients of P(x) is 10. Determine the values of a, b, c and d.
(b)
Using the values obtained, evaluate P(0).
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Question 6 View Details
A shop sells two types of pens: Type A costs ₦120 each and Type B costs ₦180 each.
Question Parts
(a)
On the previous day the shop sold a total of 30 pens and collected ₦4,560. Determine how many pens of each type were sold on that day.
(b)
On the given day the shop sold a total of 50 pens and collected ₦7,680. It is known that the number of Type B pens sold on this day was twice the number of Type A pens sold on the previous day. Find the number of Type A and Type B pens sold on the given day.
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Question 7 View Details
A solid consists of a right circular cylinder of radius \(6\) cm and height \(10\) cm. A right circular cone of the same radius \(6\) cm and height \(8\) cm is placed on top of the cylinder, sharing the circular face. The solid is shown in the labelled diagram. A horizontal plane cuts the cone at a point \(3\) cm from the apex (i.e., \(5\) cm from the base of the cone) and the part of the cone above the plane is removed.
Question Parts
(a)
Find the total surface area of the original solid (include the bottom of the cylinder).
(b)
Find the total volume of the original solid.
(c)
After the top part of the cone is removed, find the total surface area of the remaining solid.
(d)
Find the volume of the removed top part of the cone.
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Question 8 View Details
A manufacturer produces two models of a gadget, A and B. The profit per unit of model A is ₦1200 and for model B is ₦1500. Due to market constraints, the total number of units produced per day cannot exceed 200. Additionally, at least 40 units of model B must be produced each day. Let x be the number of units of model A produced daily and y the number of units of model B produced daily.
Question Parts
(a)
Write the system of inequalities that represents the production constraints.
(b)
The manufacturer wishes to achieve a daily profit of at least ₦210,000. Express this requirement as an inequality involving x and y and add it to the system.
(c)
Determine the range of possible values of x that satisfy all the constraints. Show your working.
(d)
If the manufacturer decides to produce the maximum possible number of model A units while still meeting the profit target, how many units of each model should be produced? Provide the values of x and y.
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Question 9 View Details
A sequence \(\{a_n\}\) is defined recursively by \(a_1 = 2\) and for \(n \ge 2\), \(a_n = 2\big(a_{n-1}+3\)\).
Question Parts
(a)
Write down the first five terms of the sequence.
(b)
Derive a closed‑form expression for the nth term \(a_n\).
(c)
Using the expression from part (b), find the smallest integer \(n\) for which \(a_n > 500\).
(d)
Find the sum \(S = a_1 + a_2 + \dots + a_n\) where \(n\) is the value obtained in part (c).
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Question 10 View Details
In a certain chemical process the time \(t\) (in seconds) required for the reaction to complete varies jointly as the square of the concentration \(c\) of a reactant, inversely as the absolute temperature \(T\) (in kelvin) and inversely as the pressure \(P\) (in atmospheres). That is, \(t = k\dfrac{c^{2}}{TP}\), where \(k\) is a constant of variation.
Question Parts
(a)
When \(c = 0.40\) mol·L⁻¹, \(T = 300\) K and \(P = 1.00\) atm, the reaction takes \(t = 45\) s. Determine the constant \(k\).
(b)
If the concentration is increased by 50 % and the temperature is increased by 20 % while the pressure remains unchanged, find the new reaction time (in seconds).
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Question 11 View Details
The quadratic equation \(x^2 - kx + 12 = 0\) has two distinct integer roots. Moreover, the sum of the roots is three times the smaller root. Determine the value of \(k\) and the roots.
Question Parts
(a)
Determine the pair of integer roots that satisfy the given condition.
(b)
Hence find the value of \(k\).
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Question 12 View Details
In an arithmetic progression (AP), the 5th term is 20 and the 12th term is 62. Find the first term, the common difference, and the sum of the first 20 terms.
Question Parts
(a)
Find the common difference \(d\).
(b)
Determine the first term \(a_1\).
(c)
Calculate the sum \(S_{20}\) of the first 20 terms.
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Question 13 View Details
In triangle \(ABC\), \(\angle A = 40^{\circ}\), \(\angle B = 70^{\circ}\) and side \(AB = 12\) cm. Let \(D\) be a point on \(BC\) such that \(BD:DC = 1:2\).
Question Parts
(a)
Find the length of side \(AC\).
(b)
Find the length of side \(BC\) (give your answer correct to two decimal places).
(c)
Determine the length of \(BD\) (give your answer correct to two decimal places).
(d)
Find the area of triangle \(ABD\) (give your answer correct to two decimal places).
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