neco 2021 mathematics | Essay

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Question 1 View Details
A positive integer N leaves a remainder of 4 when divided by 9, a remainder of 6 when divided by 11 and a remainder of 8 when divided by 13. Determine the smallest such N.
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Question 2 View Details
A trader bought an item for ₦1,200. He sold the item at a profit of 20 % and later offered a discount of 10 % on the selling price. Determine the final profit or loss percentage made by the trader.
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Question 3 View Details
A positive real number N satisfies the equation N^{\log_{10} N}=1000.
Question Parts
(a)
Find the value of N, expressing your answer in exact form.
(b)
Using the value of N obtained in part (a), evaluate \log_{N} 10 and give your answer in simplest surd form.
(c)
Hence, find the value of (\log_{N} 10)^{2}.
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Question 4 View Details
A school surveyed 120 students about their participation in three sports: football (set A), basketball (set B) and volleyball (set C). The results are shown in the Venn diagram.
Question Parts
(a)
How many students play exactly two of the three sports?
(b)
How many students do not play any of the three sports?
(c)
What is the probability that a randomly selected student plays at least one sport? Express your answer as a fraction in lowest terms.
(d)
If a student is known to play football, what is the probability that they also play basketball?
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Question 5 View Details
Let \(x\) and \(y\) be integers such that \(x + y = 7\), \(xy = 10\) and \(x > y\). Define the expression \(E = 3x^{2} - 5x + 2y - xy\).
Question Parts
(a)
Find the ordered pair \((x, y)\) that satisfies the given conditions.
() Form a quadratic equation whose roots are \(x\) and \(y\).
() Solve the quadratic, using the condition \(x > y\) to select the ordered pair.
(b)
Using the values of \(x\) and \(y\) obtained, evaluate \(E\).
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Question 6 View Details
A small business finds that its total revenue \(R\) (in naira) depends linearly on the number of units \(x\) sold. When 200 units are sold the revenue is \(₦150{,}000\) and when 350 units are sold the revenue is \(₦210{,}000\).
Question Parts
(a)
Determine the unit price (slope) and the fixed cost (intercept) of the revenue function.
() Set up the two linear equations using the given data.
() Solve the equations to find the unit price \(a\).
() Find the fixed cost \(b\).
(b)
Using the revenue function, calculate the expected revenue when 500 units are sold.
() Substitute \(x = 500\) into \(R = ax + b\) and compute \(R\).
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Question 7 View Details
The quadratic equation \(3x^{2}-kx+12=0\) is known to have two distinct real roots that differ by 2 units.
Question Parts
(a)
Find the value of \(k\).
(b)
Using the value of \(k\) found in part (a), determine the product of the roots.
(c)
Solve the quadratic equation completely.
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Question 8 View Details
A company produces two products, A and B. The profit (in thousands of naira) from producing \(x\) units of A and \(y\) units of B is given by \(P = 5x + 3y\). The following constraints apply: - Each unit of A requires 2 hours of production time and each unit of B requires 3 hours; total production time cannot exceed 40 hours. - Each unit of A uses 4 kg of raw material and each unit of B uses 5 kg; total raw material available is 90 kg. - The company wants the profit to be at least 120 thousand naira. Formulate and analyse the feasible region defined by these constraints.
Question Parts
(a)
Write the system of inequalities that represent the three constraints.
(b)
Determine the coordinates of the vertices of the feasible region defined by the inequalities in part (a). List them in order of increasing \(x\).
(c)
From the vertices found in part (b), state the possible range of values of \(x\) (the number of units of product A) within the feasible region.
(d)
If the company can only produce whole units, list all integer pairs \((x, y)\) that satisfy the constraints and give a profit of at least 120 thousand naira.
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Question 9 View Details
The sum of the first \(n\) terms of an arithmetic sequence is given by \(S_n = 3n^2 + 5n\).
Question Parts
(a)
Find the first term and the common difference of the sequence.
(b)
Determine the value of the 10th term.
(c)
Find the smallest integer \(n\) for which \(S_n\) exceeds 500.
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Question 10 View Details
The time \(t\) (in seconds) taken for a particle to travel a distance \(s\) (in metres) under a constant force \(F\) (in newtons) varies directly as the square root of the distance and inversely as the cube of the force. The particle's mass \(m\) (in kilograms) varies directly as the time taken. When \(s = 200\) m and \(F = 4\) N, the time taken is \(t = 5\) s and the mass is \(m = 10\) kg.
Question Parts
(a)
Determine the constant of variation \(k\) in the relation \(t = k\,\frac{\sqrt{s}}{F^{3}}\).
(b)
Using the constant found, calculate the time taken when \(s = 450\) m and \(F = 3\) N.
(c)
Determine the mass of the particle under these new conditions.
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Question 11 View Details
The graph shown represents the monthly revenue (in thousands of Naira) of a small business over a period of seven months, from month 0 to month 6. The revenue values at the indicated months are: (0,20), (1,45), (2,70), (3,95), (4,115), (5,130) and (6,125).
Question Parts
(a)
State the month in which the revenue was maximum and give the corresponding revenue.
(b)
Assuming the revenue \(R\) (in thousand Naira) varies with month \(t\) (in months) according to a quadratic model \(R = at^{2}+bt+c\), use the data for months 0, 3 and 6 to determine the constants \(a\), \(b\) and \(c\).
(c)
Using the model obtained in part (b), predict the revenue for month 7.
(d)
Discuss briefly whether the quadratic model is appropriate for forecasting revenue beyond month 6. Give two reasons to support your view.
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Question 12 View Details
In the diagram, triangle \(ABC\) is shown. Vertex \(A\) is at the origin \((0,0)\) and side \(AB\) lies along the positive x‑axis. The length \(AB = 12\) cm, \(\angle A = 40^{\circ}\) and \(\angle B = 70^{\circ}\). Point \(D\) lies on \(BC\) such that \(BD:DC = 1:2\).
Question Parts
(a)
Determine the coordinates of point \(C\).
(b)
Find the equation of the line \(AC\) in the form \(y = mx\).
(c)
Calculate the area of triangle \(ABC\) in square centimetres, giving your answer to one decimal place.
(d)
Point \(D\) divides \(BC\) internally in the ratio \(1:2\).
() Find the coordinates of \(D\).
() Determine the length of \(AD\).
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Question 13 View Details
A solid is formed as follows: - A right circular cylinder of radius \(5\) cm and height \(12\) cm. - A hemisphere of radius \(5\) cm is attached to the base of the cylinder (the circular base of the cylinder is completely covered by the hemisphere). - From the top of the cylinder a right circular cone of radius \(5\) cm and height \(8\) cm is removed, the base of the cone coinciding with the top circular face of the cylinder. The resulting solid has a cylindrical body, a hemispherical extension at the bottom and a conical cavity at the top. Answer the following questions:
Question Parts
(a)
Find the total surface area of the solid, including all exposed curved surfaces. Give your answer in terms of \(\pi\) and simplify where possible.
(b)
Determine the volume of the solid. Express your answer as a multiple of \(\pi\).
(c)
If the solid is painted on all its exposed surfaces and then completely immersed in water, calculate the ratio of the volume of water displaced (in \(\text{cm}^3\)) to the total painted surface area (in \(\text{cm}^2\)). Express the ratio in its simplest exact form and also give a decimal approximation to three decimal places.
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