neco 2017 further_mathematics | Essay

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Question 1 View Details
Let \(A=\{1,2,3,4,5\}\). A binary operation \(*\) is defined on \(A\) by \[a * b = (a + 2b) \bmod 5,\] where the result of the modulo operation is taken to be an element of \(A\) (the value \(0\) is replaced by \(5\)).
Question Parts
(a)
Show that the operation \(*\) is closed on \(A\).
(b)
Determine whether an identity element \(e\) exists for \(*\) on \(A\). If it exists, find its value.
(c)
Find the inverse of the element \(3\) under \(*\) (if it exists).
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Question 2 View Details
A geometric progression has first term \(a\) and common ratio \(r\) (with \(r>0\) and \(r\neq1\)). The sum of the first three terms is 26 and the sum of the squares of the first three terms is 350.
Question Parts
(a)
Find the exact values of \(a\) and \(r\).
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Question 3 View Details
A cubic polynomial is given by \(P(x)=2x^{3}+ax^{2}+bx-12\). It is known that \(x-2\) is a factor of \(P(x)\).
Question Parts
(a)
Determine the values of the coefficients \(a\) and \(b\).
(b)
Find the remainder when \(P(x)\) is divided by \(x+3\).
(c)
After factoring out \((x-2)\) and \((x+3)\) from \(P(x)\), obtain the remaining quadratic factor and state the sum of its roots.
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Question 4 View Details
Consider the rational function \(R(x)=\dfrac{3x^{2}+5x-2}{(x-1)(x^{2}+4)}\).
Question Parts
(a)
Express \(R(x)\) in the form \(\dfrac{A}{x-1}+\dfrac{Bx+C}{x^{2}+4}\) and determine the constants \(A, B, C\).
(b)
Evaluate the definite integral \(\displaystyle \int_{0}^{2} R(x)\,dx\) using the decomposition obtained in part (a).
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Question 5 View Details
Let \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) be a \(2\times2\) matrix with integer entries. It is given that \(\det A = 6\) and that \(A\) satisfies the matrix equation \(A^{2} - 5A + 6I = 0\), where \(I\) is the \(2\times2\) identity matrix. In addition, the trace of \(A\) is \(a+d = 5\).
Question Parts
(a)
Show that the characteristic polynomial of \(A\) is \(\lambda^{2} - 5\lambda + 6 = 0\) and deduce the eigenvalues of \(A\).
(b)
Using the equation \(A^{2} - 5A + 6I = 0\), express \(A^{-1}\) in terms of \(A\) and \(I\).
(c)
Write \(A^{-1}\) explicitly in terms of the entries \(a, b, c, d\).
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Question 6 View Details
Eight students (A, B, C, D, E, F, G, H) and four teachers (T1, T2, T3, T4) are to form a committee of five members. The committee must contain exactly two teachers and three students. In addition, student A and teacher T1 are not allowed to serve on the committee together.
Question Parts
(a)
How many different committees satisfy the above conditions?
(b)
If the committee formed is to elect a chairperson and the chairperson must be a teacher, how many possible ways can the committee and the chairperson be chosen?
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Question 7 View Details
A manufacturing firm models its profit (in thousand Naira) as \(P = (x+2)^{5} - (x-1)^{5}\), where \(x\) represents the number of thousand units produced beyond a base level. \n\nUsing the Binomial Theorem, answer the following:
Question Parts
(a)
Expand \((x+2)^{5}\) and \((x-1)^{5}\) up to and including the term in \(x^{3}\).
(b)
Hence determine the coefficient of \(x^{3}\) in the simplified expression for \(P\).
(c)
Find the constant term (term independent of \(x\)) in \(P\).
(d)
If the profit must be at least \(500\) (thousand Naira), find the smallest positive integer value of \(x\) that satisfies \(P \ge 500\).
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Question 8 View Details
In a circle of radius \(r\) cm, a chord \(AB\) subtends a central angle \(\theta\) (in degrees). The length of \(AB\) is \(12\) cm and it is given that \(\sin\frac{\theta}{2}=\frac{3}{5}\). Using trigonometric relationships, answer the following:
Question Parts
(a)
Find the radius \(r\) of the circle.
(b)
Determine the measure of \(\theta\) in degrees (correct to two decimal places).
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