bece model questions vol1 2018 mathematics | Essay

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Question 1 View Details
A school library has a collection of books whose total number is a five‑digit integer. The digit in the ten‑thousands place is twice the digit in the thousands place. The digit in the hundreds place is three less than the digit in the tens place. The units digit is zero. If the sum of all five digits is 24, determine the exact number of books in the library.
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Question 2 View Details
A school intends to purchase new computers. Two configurations are available: • Type A costs ₦120 000 each. • Type B costs ₦150 000 each. The school must spend exactly ₦3 300 000 and must buy at least 10 of each type. Moreover, the number of Type A computers must exceed the number of Type B computers. Answer the following:
Question Parts
(a)
Determine how many computers of each type the school should purchase.
(b)
If each Type A computer has a processing speed of 2.5 GHz and each Type B computer 3.0 GHz, calculate the average speed of all the computers purchased (in GHz, to one decimal place).
(c)
The school will divide the 25 computers into 5 equal laboratory groups. How many computers will be in each group?
(d)
Each group must contain at least one Type B computer. In how many different ways can the ten identical Type B computers be distributed among the five groups? (Assume computers of the same type are indistinguishable.)
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Question 3 View Details
A chemist has 12 litres of a solution that is 2⁄5 acid and the remainder water. He wishes to dilute the solution so that the acid constitutes exactly 1⁄3 of the final mixture by adding only water.
Question Parts
(a)
Calculate the volume of water that must be added to achieve the required concentration.
(b)
State the total volume of the solution after the water has been added.
(c)
Express the final amounts of acid and water as fractions of the total volume.
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Question 4 View Details
A shop sells fabric at a price of ₦125.75 per metre. A customer purchases 3.6 metres. The shop offers a discount of 12.5 % on the total bill if the purchase exceeds 3 metres, and a sales tax of 5 % is then applied to the discounted amount.
Question Parts
(a)
Calculate the final amount the customer has to pay, correct to two decimal places.
(b)
If the customer pays with a ₦500 note, determine the amount of change he receives.
(c)
Explain why the discount is applied before tax rather than after tax.
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Question 5 View Details
A shop bought 120 metres of fabric at a cost of ₦450 per metre. The fabric was sold in two batches. The first batch of 45 metres was sold at a 20 % profit on its cost price. The remaining fabric was sold at a 15 % loss on its cost price. After the sales the shop had to pay a flat tax of ₦5,000 on the profit earned from the first batch.
Question Parts
(a)
Calculate the total revenue obtained from the first batch.
(b)
Determine the overall profit or loss on the whole transaction after accounting for the tax.
(c)
State the overall percentage gain or loss on the whole transaction (including tax), correct to one decimal place.
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Question 6 View Details
A rectangular garden is to be divided into three smaller rectangular plots. The lengths of the three plots are in the ratio 3 : 4 : 5 and all three plots have the same width. The whole garden is a rectangle whose length is twice its width, and its total area is 1 800 m².
Question Parts
(a)
Find the common width of the three plots (in metres).
(b)
State the dimensions (length × width) of each of the three plots.
(c)
If the length of each plot is increased by 20 % while the width remains unchanged, what is the new total area of the garden? Express the increase in area as a percentage of the original total area.
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Question 7 View Details
The diagram shows a right circular cylinder of radius 6 cm and height 12 cm. A right circular cone of the same base radius and height 8 cm is removed from the interior of the cylinder, leaving a conical cavity. A hemispherical dome of radius 4 cm is attached on the top of the cylinder, centred on the axis. All dimensions are labelled.
Question Parts
(a)
Find the volume of the resulting solid. Express your answer in terms of \(\pi\) and also give a numerical value correct to the nearest cubic centimetre.
(b)
Determine the total surface area of the solid, including all exposed curved surfaces and the exposed part of the top circular face of the cylinder. Express your answer in terms of \(\pi\) and also give a numerical value correct to the nearest square centimetre.
(c)
Explain why the entire top circular face of the cylinder is not exposed after the hemisphere is attached, and calculate the exact area of the exposed part.
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Question 8 View Details
In triangle ABC, \(\angle A = 40^{\circ}\), \(\angle B = 70^{\circ}\) and side \(AB = 12\) cm. Point D lies on side BC such that \(BD : DC = 1 : 2\).
Question Parts
(a)
Find \(\angle C\).
(b)
Determine the lengths of sides BC and AC.
(c)
Point D divides BC in the ratio 1:2. (i) Find the lengths of BD and DC. (ii) Find the length of AD.
(d)
Calculate: (i) The area of triangle ABC. (ii) The length of the altitude from A to BC.
() The area of triangle ABC.
() The length of the altitude from A to BC.
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Question 9 View Details
Two positive integers \(a\) and \(b\) satisfy the simultaneous index equations \[ a^{2}\,b^{3}=2^{5}\,3^{4} \quad\text{and}\quad a\,b^{2}=2^{3}\,3^{2}. \] Using only the laws of indices, answer the following:
Question Parts
(a)
Determine the exact values of \(a\) and \(b\).
(b)
Using the values of \(a\) and \(b\) found in part (a), evaluate \[ \frac{a^{3} \; b^{5}}{(a\,b)^{2}} \] and express the result as a single power of \(2\) multiplied by a single power of \(3\).
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Question 10 View Details
Consider the quartic polynomial \[ P(x)=x^{4}+k x^{3}-6x^{2}+m x+12. \] It is known that \((x-2)\) and \((x+3)\) are factors of \(P(x)\).
Question Parts
(a)
Find the values of the constants \(k\) and \(m\).
(b)
Write \(P(x)\) as a product of four linear factors.
(c)
Using the factorisation obtained in part (b), compute the product of the constant terms of each linear factor.
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