neco 2022 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: for any a, b in A, a * b is the remainder obtained when (a + 2b) is divided by 6.
Question Parts
(a)
Show that the operation * is closed on A.
(b)
Determine whether * is associative on A. State your conclusion with a brief justification.
(c)
Find the identity element e (if it exists) for * on A and justify why it works.
(d)
State whether each element of A has an inverse with respect to *. If an element has an inverse, give it; if not, explain why.
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Question 2 View Details
Consider the equation \(\dfrac{2^{x}\,5^{2x}}{10^{x-1}} = 20\).
Question Parts
(a)
Solve for the real value of x.
(b)
Using the value of x obtained in part (a), evaluate \(\log_{2}\sqrt{50}\; -\; \log_{5}(2^{x+1})\).
(c)
Express \(3^{\log_{3}7}+\sqrt{49}\) in its simplest exact form and explain each step.
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Question 3 View Details
A polynomial \(P(x) = 2x^{4} - 3x^{3} + kx^{2} + 5x - 12\) is such that \((x-2)\) is a factor of \(P(x)\) and the remainder when \(P(x)\) is divided by \((x+1)\) is \(4\).
Question Parts
(a)
Determine the value of the constant \(k\).
(b)
Using the value of \(k\) found in part (a), factor \(P(x)\) completely over the real numbers.
(c)
Find the sum of all real roots of the equation \(P(x)=0\).
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Question 4 View Details
Consider the rational function \(R(x)=\dfrac{3x^{2}+5x-2}{(x-1)(x+2)^{2}}\).
Question Parts
(a)
Express \(R(x)\) as a sum of partial fractions of the form \(\dfrac{A}{x-1}+\dfrac{B}{x+2}+\dfrac{C}{(x+2)^{2}}\).
(b)
Using the decomposition from part (a), evaluate the indefinite integral \(\displaystyle \int R(x)\,dx\).
(c)
Find the value of the definite integral \(\displaystyle I=\int_{0}^{1} R(x)\,dx\).
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Question 5 View Details
A 2 \times 2 matrix \(A\) is given by \(A = \begin{bmatrix} 2 & k \\ 3 & 5 \end{bmatrix}\) where \(k\) is an integer. The matrix is required to be invertible and its inverse must have only integer entries. Using this matrix, the system of linear equations \[\begin{cases} 2x + ky = 7 \\ 3x + 5y = 11 \end{cases}\] is to be solved. Answer the following: (a) Find the integer value of \(k\) that satisfies the above condition. (b) State the determinant of \(A\) and write down the inverse matrix \(A^{-1}\). (c) Using \(A^{-1}\), determine the ordered pair \((x, y)\) that satisfies the system.
Question Parts
(a)
Find the integer value of \(k\) that makes \(A\) invertible and ensures \(A^{-1}\) has only integer entries.
(b)
State the determinant of \(A\) and write the inverse matrix \(A^{-1}\).
(c)
Using \(A^{-1}\), solve the system and give the ordered pair \((x, y)\).
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Question 6 View Details
A club has eight members: four males (M₁, M₂, M₃, M₄) and four females (F₁, F₂, F₃, F₄). A sub‑committee of five members is to be formed under the following conditions: - The sub‑committee must contain at least two males and at least one female. - The chairperson must be a male and the secretary must be a female. How many distinct ways can the sub‑committee be formed together with the appointment of the chairperson and secretary?
Question Parts
(a)
Determine the total number of possible selections of the sub‑committee together with the designated chairperson and secretary that satisfy all the given conditions.
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Question 7 View Details
Consider the expansion of \((x+2)^{n}\) where \(n\) is a positive integer. It is given that the coefficient of \(x^{3}\) in the expansion equals 560.
Question Parts
(a)
Determine the value of \(n\).
(b)
Using the value of \(n\) obtained in part (a), find the sum of the coefficients of all terms in the expansion whose powers of \(x\) are even (i.e., the terms containing \(x^{0}, x^{2}, x^{4},\dots\)). State the method you use.
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Question 8 View Details
A surveyor measures the angle of elevation to the top of a vertical tower from two points \(A\) and \(B\) on level ground. Points \(A\) and \(B\) lie on a straight line with the foot of the tower, and \(B\) is nearer the tower. The distance \(AB\) is 50 m. The angle of elevation from \(A\) is \(30^{\circ}\) and from \(B\) is \(45^{\circ}\).
Question Parts
(a)
Find the height of the tower.
(b)
Find the horizontal distance from point \(B\) to the foot of the tower.
(c)
At the same instant, the sun’s altitude angle is \(60^{\circ}\). Determine the length of the shadow cast by the tower on the ground.
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