neco 2019 mathematics | Essay

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Question 1 View Details
A natural number N leaves a remainder of 2 when divided by 5, a remainder of 3 when divided by 7, and a remainder of 4 when divided by 9.
Question Parts
(a)
Find the smallest such N.
(b)
Determine the greatest common divisor (GCD) of N and 126.
(c)
If N is increased by 30, what is the least number that must be subtracted to make the result divisible by 12?
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Question 2 View Details
A shopkeeper bought a batch of 240 metres of fabric at a total cost of ₦48,000. He sold part of the fabric at a 25 % profit per metre and the remaining part at a 10 % loss per metre. If his overall profit on the whole batch was ₦3,600, determine how many metres of fabric were sold at a profit.
Question Parts
(a)
Let x be the metres sold at a profit. Write an equation representing the overall profit.
(b)
Solve for x.
(c)
Express the profit per metre for the whole batch as a percentage of the original cost per metre.
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Question 3 View Details
Let \(a = 2^{3/2} \times 5^{1/2}\) and \(b = 2^{1/2} \times 5^{3/2}\).
Question Parts
(a)
Write \(a\) and \(b\) in simplest surd form.
(b)
Find the value of \(\dfrac{a^{2}-b^{2}}{ab}\).
(c)
If \(\log_{10}\left(\dfrac{a}{b}\right)=k\), determine the exact value of \(k\).
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Question 4 View Details
A survey of 100 secondary school students recorded their participation in three extracurricular clubs: Drama (set A), Science (set B) and Music (set C). The Venn diagram representing the data is described in the diagram specification.
Question Parts
(a)
How many students are members of exactly two clubs?
(b)
How many students are not members of any of the three clubs?
(c)
If a student is selected at random, what is the probability that the student belongs to at least one club? Express your answer as a fraction in lowest terms.
(d)
Given that a student is a member of the Drama club, find the probability that the same student is also a member of the Science club. Give your answer as a decimal to two places.
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Question 5 View Details
The algebraic expression \(E = 3x^{2} - 12x + k\) can be written in the form \(3(x - a)(x - b)\) where \(a\) and \(b\) are distinct positive integers. (a) Determine the value of \(k\) and the sum \(a + b\). (b) Using the values of \(a\) and \(b\) found in part (a), evaluate \(F = \dfrac{a^{3} + b^{3}}{a + b}\).
Question Parts
(a)
Determine the value of k and the sum a + b.
(b)
Using the values of a and b found in part (a), evaluate F = (a³ + b³)/(a + b).
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Question 6 View Details
A company manufactures two types of gadgets, A and B. - The profit from each unit of type A is \(\text{₦}150\). - The profit from each unit of type B is \(\text{₦}200\). - A total of 120 gadgets are produced, generating a total profit of \(\text{₦}21,500\). - The number of type B gadgets produced is twice the number of type A gadgets minus 30. (a) Form the system of linear equations that represents the above information. (b) Solve the system to find how many gadgets of each type were produced. (c) Verify that the profit condition is satisfied with your solution.
Question Parts
(a)
Form the system of linear equations that represents the above information.
(b)
Solve the system to find how many gadgets of each type were produced.
(c)
Verify that the profit condition is satisfied with your solution.
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Question 7 View Details
A quadratic equation is given by \(2x^{2} - kx + 12 = 0\). It is known that the equation has two distinct real roots whose sum is 5.
Question Parts
(a)
Determine the value of the parameter \(k\).
(b)
Find the two roots of the quadratic equation.
(c)
Compute the sum of the squares of the two roots.
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Question 8 View Details
A company charges a total cost \(C\) (in thousands of naira) for purchasing \(x\) items according to the formula \(C = 0.5x^{2} - 3x + 12\). A discount is offered only when the total cost does not exceed \(30\) thousand naira.
Question Parts
(a)
Find all integer values of \(x\) for which the discount will be offered.
(b)
If the company also requires that at least three items be purchased, state the possible values of \(x\).
(c)
For each permissible value of \(x\) in part (b), the discount amount is defined as \(D = 30 - C\) (in thousands of naira). Find the discount when a customer buys the maximum permissible number of items.
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Question 9 View Details
A sequence \(\{u_n\}\) satisfies the following conditions: the second term is 5 more than the first term, the third term is twice the second term, and from the third term onward the sequence forms an arithmetic progression.
Question Parts
(a)
Determine the first term \(u_1\) and the common difference \(d\) of the arithmetic progression.
(b)
Find the value of the 10th term \(u_{10}\).
(c)
Calculate the sum of the first 10 terms \(S_{10}\).
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Question 10 View Details
The time \(T\) (in seconds) required for a certain chemical reaction to complete varies directly as the square of the concentration \(C\) (in mol·L\(^{-1}\)) and varies inversely as the temperature difference \(\Delta t\) (in \(^\circ\)C) where \(\Delta t = \text{temperature} - 20\) and the temperature is always above 20\(^\circ\)C.
Question Parts
(a)
Write the variation formula and determine the constant of variation \(k\) using the data: when \(C = 0.5\) mol·L\(^{-1}\) and \(\Delta t = 30\) \(^\circ\)C, the reaction time is \(T = 120\) s.
(b)
Using the constant found in part (a), calculate the reaction time when \(C = 0.8\) mol·L\(^{-1}\) and \(\Delta t = 45\) \(^\circ\)C.
(c)
If the temperature difference is maintained at \(\Delta t = 40\) \(^\circ\)C, find the concentration \(C\) required for the reaction to take exactly \(90\) seconds.
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Question 11 View Details
The graph shown represents the speed (in m/s) of a car as a function of time t (in seconds) during a 20‑second interval.
Question Parts
(a)
Determine the acceleration of the car during the interval 0 \le t \le 5 \text{ s}.
(b)
Find the total distance travelled by the car over the whole 20 s interval.
(c)
At what time does the car attain its maximum speed? Justify your answer.
(d)
The area under the curve between t = 12 s and t = 20 s represents what physical quantity? State its value.
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Question 12 View Details
In the coordinate plane, triangle \(ABC\) is positioned such that point \(A\) is at the origin \((0,0)\) and side \(AB\) lies on the positive \(x\)-axis. The given data are: - \(AB = 12\) cm - \(\angle A = 40^{\circ}\) - \(\angle B = 70^{\circ}\) - Point \(D\) lies on side \(BC\) and divides it internally in the ratio \(BD : DC = 1 : 2\). Using this information, answer the following.
Question Parts
(a)
Determine the exact coordinates of point \(C\).
(b)
Find the coordinates of point \(D\) on \(BC\) that satisfy \(BD:DC = 1:2\).
(c)
Write the equation of the straight line \(AD\) in the form \(y = mx\).
(d)
Calculate the area of triangle \(ABD\) in square centimetres (to two decimal places).
(e)
Determine the perpendicular distance from point \(D\) to the line \(AC\). Give your answer to two decimal places.
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Question 13 View Details
In triangle ABC, \(\angle A = 40^{\circ}\), \(\angle B = 70^{\circ}\) and side \(AB = 10\) cm. Let \(D\) be the foot of the perpendicular from \(A\) to side \(BC\). Let \(E\) be the midpoint of side \(AB\).
Question Parts
(a)
Calculate the length of side \(BC\).
(b)
Determine the length of the altitude \(AD\).
(c)
Find the lengths of \(BD\) and \(DC\).
(d)
Show that \(DE = 5\) cm.
(e)
Find the area of triangle \(ADE\). Give your answer in exact form and also to two decimal places (cm\(^2\)).
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