bece model questions vol1 2022 mathematics | Essay

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Question 1 View Details
A positive integer N satisfies the following conditions: when divided by 7 it leaves a remainder of 3, and when divided by 9 it leaves a remainder of 5.
Question Parts
(a)
Find the smallest positive integer N that satisfies both conditions.
(b)
If 30 is added to this smallest N, what remainder is obtained when the resulting number is divided by 7?
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Question 2 View Details
A bag contains only red and blue marbles. The probability of drawing a red marble at random is \(\frac{2}{5}\). The total number of marbles in the bag is 45.
Question Parts
(a)
Determine the number of red marbles and the number of blue marbles in the bag.
(b)
Six blue marbles are removed from the bag. Find the new probability of drawing a red marble, expressed as a fraction in lowest terms.
(c)
If instead three red marbles are added to the original bag (before any removal), what would be the probability of drawing a red marble then? Give your answer as a fraction in lowest terms.
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Question 3 View Details
A school canteen prepares a special fruit punch for a celebration. The recipe requires mixing three juices in the ratio 2:3:5 by volume.
Question Parts
(a)
If the total volume of the punch to be prepared is 48 litres, calculate the volume of each juice required.
(b)
The canteen decides to use only orange juice and mango juice, keeping their original ratio, and wants to make 30 litres of punch using only these two juices.
(c)
After preparation, 5.6 litres of the punch are discarded due to spillage. Express the remaining volume as a mixed number of litres and fractions of a litre.
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Question 4 View Details
A farmer sells yams at a market. The price per kilogram is ₦1,250. He sells 3.75 kg in the morning and 2.48 kg in the afternoon.
Question Parts
(a)
Calculate the total revenue for the day.
(b)
The farmer wants to save 20 % of his revenue for future planting. How much money will he set aside and what amount remains after the saving?
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Question 5 View Details
A shopkeeper bought 120 kg of rice for a total cost of ₦48,000. He intends to sell part of the rice at a profit of 10 % on the cost price per kilogram and the remaining part at a loss of 5 % on the cost price per kilogram. After selling all the rice his total revenue is ₦51,600.
Question Parts
(a)
Determine how many kilograms of rice are sold at a profit and how many at a loss.
(b)
State the selling price per kilogram for the rice sold at profit and for the rice sold at loss.
(c)
Calculate the overall profit percentage earned by the shopkeeper.
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Question 6 View Details
Container A holds a mixture of acid and water in the ratio 3 : 5. Container B holds a mixture of acid and water in the ratio 2 : 7. A chemist needs 500 ml of a new mixture in which the acid and water are in the ratio 1 : 2.
Question Parts
(a)
Let x ml be taken from container A and y ml from container B. Write the two equations that relate x and y to the required total volume and the required acid proportion.
(b)
Solve the equations to find the exact volumes taken from each container (express your answers as fractions and to two decimal places).
(c)
Verify that the mixture obtained from the volumes found in part (b) indeed has an acid‑to‑water ratio of 1 : 2.
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Question 7 View Details
A water tank can be filled or emptied by three pipes. Pipe A can fill the tank in 10 hours when operating alone. Pipe B can fill the tank in an unknown time. Pipe C is an outlet that empties the tank. The following observations are made: - When all three pipes are opened together, the tank is filled in 6 hours. - When only pipe A and the outlet C are opened, the tank empties in 4 hours. Using this information answer the following.
Question Parts
(a)
Determine the rate of pipe B in tanks per hour and the time it would take pipe B alone to fill the tank.
(b)
Find the time required to fill the tank if only pipe B is opened.
(c)
The three pipes are opened together for 2 hours, after which pipe A is closed while pipes B and C remain open. How much additional time is required to fill the tank completely?
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Question 8 View Details
The function f(x)=√x is to be approximated near x = 49.
Question Parts
(a)
Using the linear approximation at a = 49, estimate √50. Show the derivative calculation.
(b)
Using the second‑order Taylor expansion about a = 49, obtain a refined estimate of √50.
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Question 9 View Details
In a certain experiment the quantity of a substance after t minutes is given by \(N = 2^{t}\,\times\,5^{k-t}\), where \(k\) is a constant. It is observed that when \(t = 2\) minutes the quantity is \(N = 2500\).
Question Parts
(a)
Determine the value of the constant \(k\).
(b)
Using the value of \(k\) found in part (a), find the smallest positive integer value of \(t\) greater than 2 for which \(N\) is a perfect square.
(c)
For the value of \(t\) obtained in part (b), evaluate \(\log_{10} N\) (give your answer correct to two decimal places).
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Question 10 View Details
A rectangular garden has length \(L = 3x + 4\) metres and width \(W = x - 2\) metres. Its area can be written in the form \(3x^{2} + a x + b\) square metres.
Question Parts
(a)
Expand \((3x + 4)(x - 2)\) and determine the values of \(a\) and \(b\).
(b)
If the perimeter of the garden is 70 metres, find the value of \(x\).
(c)
Using the value of \(x\) from part (b), calculate: i. The cost of fencing the garden at a rate of ₦250 per metre. ii. The cost of planting grass over the whole garden at a rate of ₦150 per square metre. iii. State the total expense.
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