neco 2025 mathematics | Essay

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Question 1 View Details
A five‑digit integer N satisfies the following conditions: (i) when N is divided by 9 the remainder is 4; (ii) when N is divided by 11 the remainder is 7; (iii) the sum of the digits of N is 27. Determine the smallest possible value of N, or state why no such number exists.
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Question 2 View Details
A merchant bought a batch of 250 identical items for an unknown total cost C (in naira). He first increased the total price by 20 %, then gave a discount of 15 % on the increased price, and finally added a sales tax of 5 % on the discounted price. The final selling price of the whole batch was ₦13 500. Determine the original cost price per item (to two decimal places).
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Question 3 View Details
Answer the following questions on indices, logarithms and surds.
Question Parts
(a)
Simplify the expression \(\dfrac{2^{3}\times5^{2}}{10^{2}}\times\dfrac{\sqrt{50}}{5\sqrt{2}}\).
(b)
Solve for \(x\) (real values) in the equation \(\log_{2}(x^{2}-5x+6)=3\).
(c)
Positive numbers \(a\) and \(b\) satisfy \(a^{\log_{10}2}=b^{\log_{10}5}\) and \(ab=100\). Find the value of \(\dfrac{a}{b}\).
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Question 4 View Details
The Venn diagram below represents the study habits of 120 senior secondary school students in a school.
Question Parts
(a)
State the number of students who study only Mathematics.
(b)
State the number of students who study only Physics.
(c)
State the number of students who study neither Mathematics nor Physics.
(d)
If 10 of the students who study neither subject decide to enrol in a remedial Mathematics class (and do not take Physics), what percentage of the original 120 students now study Mathematics? Give your answer to one decimal place.
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Question 5 View Details
The algebraic expression E is defined as E = (3a^2 - 5ab + 2b^2) - (a^2 + ab - 2b^2).
Question Parts
(a)
Simplify E and write it in standard form.
(b)
Factor the simplified expression completely.
(c)
If the value of the expression is zero when a = k b, where k is a constant, determine the possible value(s) of k.
(d)
Find the value of E when a = 3b.
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Question 6 View Details
The following system of linear equations in the variables x and y depends on a parameter p: (1) 3x + (p - 2) y = 7 (2) (p + 1) x - 4y = p
Question Parts
(a)
Determine the value(s) of p for which the system has a unique solution.
(b)
For p = 3, solve the system for x and y.
(c)
State whether there exists any value of p for which the system has infinitely many solutions. If such a value exists, give the relationship between x and y in that case.
(d)
Using the solution obtained in part (b), compute the value of S = 2x^2 - y + 5.
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Question 7 View Details
A rectangular garden has its length (in metres) represented by \(x\), where \(x\) satisfies the quadratic equation \(2x^{2} - 12x + 7 = 0\).
Question Parts
(a)
Find the exact values of \(x\) (the possible lengths) in simplest surd form.
(b)
The area of the garden is given by \(A = x(3x - 5)\). Using the value(s) of \(x\) from part (a), determine the possible area(s) of the garden. Express each area in simplest exact form.
(c)
Only the length that yields a positive area is physically meaningful. State the feasible length and its corresponding area, and explain why the other solution is rejected.
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Question 8 View Details
A company offers a monthly subscription plan where the total cost \(C\) (in naira) for \(n\) months is given by \(C = 150n + 1200 - 30\sqrt{n}\).
Question Parts
(a)
Determine the range of integer values of \(n\) for which the monthly average cost \(\frac{C}{n}\) does not exceed \(\text{\u20A6}200\).
(b)
Solve the inequality \(\left|2n - 15\right| < n + 5\) and express the solution set in interval notation.
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Question 9 View Details
A sequence \(\{a_n\}\) is defined by \(a_1 = 3\) and for \(n \ge 1\) the recurrence relation \(a_{n+1}=2a_n+(-1)^n\).
Question Parts
(a)
Find the value of \(a_5\).
(b)
Calculate the sum \(S_5 = a_1+a_2+\dots +a_5\).
(c)
Derive a closed‑form expression for the general term \(a_n\).
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Question 10 View Details
The time \(t\) (in minutes) required to fill a cylindrical tank varies directly as the volume \(V\) (in litres) of water and inversely as the rate of flow \(r\) (in litres per minute). The flow rate \(r\) itself varies directly as the pressure \(p\) (in kPa) and inversely as the length \(L\) (in metres) of the supplying pipe.
Question Parts
(a)
Write the combined variation formula that relates \(t\) to \(V, p\) and \(L\) using a single constant \(k\).
(b)
When \(V = 500\) L, \(p = 200\) kPa, \(L = 5\) m, the measured filling time is \(t = 20\) min. Determine the numerical value of the constant \(k\).
(c)
Using the constant found in part (b), calculate the time required to fill the tank when \(V = 800\) L, \(p = 250\) kPa and \(L = 4\) m. Give your answer to one decimal place.
(d)
Briefly explain how each of the variables \(V, p\) and \(L\) influences the filling time, based on the derived formula.
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Question 11 View Details
The graph below shows the monthly revenue (in thousands of naira) of a small business over a six‑month period.
Question Parts
(a)
Using the graph, determine the average rate of increase in revenue per month between month 2 and month 5.
(b)
Assuming the revenue continues to increase at the same average rate found in part (a), estimate in which month the revenue will reach \(\displaystyle \text{₦}25{,}000\).
(c)
Comment on whether a linear model is appropriate for the whole six‑month period based on the graph.
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Question 12 View Details
In the diagram, \(\triangle ABC\) has side \(AB = 8\) cm, \(\angle A = 40^{\circ}\) and \(\angle B = 70^{\circ}\). Point \(D\) lies on \(BC\) such that \(BD:DC = 1:2\).
Question Parts
(a)
Find the coordinates of point \(C\) (give answers correct to two decimal places).
(b)
i) Determine the coordinates of point \(D\).\nii) Using the coordinates of \(A\) and \(D\), write the equation of line \(AD\) in the form \(y = mx\).
(c)
Calculate (i) the length of the altitude from \(C\) to side \(AB\) and (ii) the area of \(\triangle ABC\). Give the altitude in centimetres and the area in square centimetres, both correct to two decimal places.
(d)
i) A line through \(D\) is drawn parallel to \(AC\) and meets \(AB\) at \(E\). Find the coordinates of \(E\).\nii) Determine the area of quadrilateral \(ABED\). Give the area correct to two decimal places.
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Question 13 View Details
In triangle ABC, \(\angle A = 40^{\circ}\) and \(\angle B = 70^{\circ}\). The length of side \(AB\) is 12 cm. Point D lies on side \(BC\) such that \(BD : DC = 1 : 2\). The segment AD is drawn.
Question Parts
(a)
Find the length of side \(AC\).
(b)
Determine the length of side \(BC\) (give your answer to two decimal places).
(c)
Find the length of \(AD\) (to two decimal places).
(d)
Calculate the area of \(\triangle ABC\) (to two decimal places).
(e)
Find the length of the altitude from A to BC (to two decimal places) and comment on its relation to the length of AD.
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