neco 2023 further_mathematics | Essay

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Question 1 View Details
Consider the set A = {1, 2, 3, 4}. A relation R on A is defined by the set of ordered pairs R = {(1,2), (2,4), (3,1), (4,3)}. A binary operation * on A is defined by a * b = (a + b) \bmod 5, where the result 0 is written as 5. Answer the following questions:
Question Parts
(a)
State whether R is a function from A to A. Justify your answer.
(b)
Find the composition R \circ R and list its ordered pairs.
(c)
For the operation * on A: (i) Determine whether * is associative. (ii) Find the identity element, if any, in A.
() Determine whether * is associative.
() Find the identity element in A, if it exists.
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Question 2 View Details
An investment grows according to the formula A = 2000 \times (3)^{t/2} \times (\sqrt{5})^{t} \text{ Naira}, where t is the number of years the money has been invested. Answer the following:
Question Parts
(a)
Show that the expression for A can be written in the form A = 2000 (\sqrt{15})^{t}.
(b)
Find the smallest integer value of t for which the amount A exceeds ₦10,000.
(c)
If the investment is left for 5 years, give the exact value of A in surd form and also its approximate value to the nearest Naira.
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Question 3 View Details
Let \(f(x)=2x^{3}+ax^{2}+bx+12\). It is known that when \(f(x)\) is divided by \((x-3)\) the remainder is \(6\), and when divided by \((x+2)\) the remainder is \(-4\). Also, the sum of the roots of the equation \(f(x)=0\) is \(5\).
Question Parts
(a)
Determine the values of the coefficients \(a\) and \(b\).
(b)
Using the values of \(a\) and \(b\) found in part (a), evaluate \(f(1)\).
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Question 4 View Details
Express the rational function \(\displaystyle \frac{3x^{2}+5x-2}{(x-1)(x^{2}+4)}\) as a sum of partial fractions of the form \(\displaystyle \frac{A}{x-1}+\frac{Bx+C}{x^{2}+4}\). Then, using the obtained decomposition, find the value of the original expression at \(x=2\).
Question Parts
(a)
Determine the constants \(A,\;B\) and \(C\).
(b)
Using the constants found in part (a), evaluate the original rational function at \(x=2\).
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Question 5 View Details
A 2\times2 matrix \(A\) is defined by \(A = \begin{pmatrix} a & 2 \\ 3 & b \end{pmatrix}\) where \(a\) and \(b\) are integers. It is given that \(A^{2}=kA\) for some constant \(k\).
Question Parts
(a)
Find the relationship between \(a\) and \(b\) and determine the value of \(k\).
(b)
Given that \(a\) and \(b\) are positive integers, list all ordered pairs \((a,b)\) that satisfy the relationship found in part (a).
(c)
For each ordered pair found in part (b), state whether \(A\) is invertible. If it is invertible, write down its inverse matrix.
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Question 6 View Details
A school science club has 12 members: 5 boys and 7 girls. A committee of 4 members is to be formed subject to the following conditions: (i) at least two girls must be on the committee, (ii) the chairperson must be a boy, and (iii) the secretary must be a girl.
Question Parts
(a)
How many different committees can be formed satisfying all the above conditions?
(b)
If the committee is formed randomly, what is the probability that the chairperson is the oldest boy in the club?
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Question 7 View Details
Consider the expansion of (2x - 3)^5 using the Binomial Theorem.
Question Parts
(a)
Write the general term T_{k+1} in the expansion of (2x - 3)^5.
(b)
Find the coefficient of x^3 in the expansion.
(c)
Find the sum of the coefficients of all terms in which the power of x is an even number.
(d)
Evaluate the expansion at x = \frac{1}{2} and verify that the result equals the sum of all terms obtained by adding the constant term, the coefficient of x^3 multiplied by (\frac{1}{2})^3, and the sum found in part (c).
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Question 8 View Details
A surveyor measures the angle of elevation of the top of a tower from two points A and B on level ground. The points A and B are 50 m apart along a straight line, with B nearer to the tower. The angle of elevation from A is 30°, and from B is 45°.
Question Parts
(a)
Let the horizontal distance from B to the base of the tower be d metres. Using the given angles, set up two equations involving d and the height h of the tower, and solve for h and d.
(b)
From the top of the tower, a point C on the ground lies 120 m horizontally from point A on the opposite side of the tower. Find the angle of depression from the top of the tower to point C.
(c)
If a flag is placed at the top of the tower and the sun’s altitude angle (angle above the horizon) is 60°, determine the length of the shadow cast by the tower on the ground.
(d)
Verify that tan(30°) = h / (d + 50) using the numerical values obtained for h and d.
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