neco 2020 further_mathematics | Essay

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Question 1 View Details
Consider the universal set \(U = \{1,2,3,4,5,6,7,8,9,10\}\). Let \(A = \{x \in U \mid x \text{ is a multiple of 2}\}\) and \(B = \{x \in U \mid x > 4\}\). Define a binary operation \(*\) on the set \(S = \{1,2,3,4,5\}\) by \(a * b = (a + b) \bmod 5\), where the result \(0\) is identified with \(5\).
Question Parts
(a)
List the elements of sets \(A\) and \(B\). Then find \(A \cap B\) and \(A \cup B\).
(b)
Find the cardinality of the complement of \(A \cup B\) in \(U\).
(c)
Construct the operation table for \(*\) on \(S\).
(d)
State whether \(*\) is associative on \(S\). Justify your answer with a brief verification using any three elements of \(S\).
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Question 2 View Details
Answer the following questions on indices, logarithms and surds.
Question Parts
(a)
Solve for \(x\) in the equation \(2^{x} = 3^{\sqrt{x}}\).
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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 \(P(x)\) and that when \(P(x)\) is divided by \((x+3)\) the remainder is \(15\).
Question Parts
(a)
Determine the values of the constants \(a\) and \(b\).
(b)
Using the values of \(a\) and \(b\) found in part (a), state the sum of the three zeros of \(P(x)\).
(c)
Evaluate \(P(1)\) using the polynomial with the determined coefficients.
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Question 4 View Details
Consider the rational expression \(\dfrac{2x+k}{x^{2}+5x+6}\). The denominator can be factorised as \((x+2)(x+3)\).
Question Parts
(a)
Find the integer value of \(k\) for which the partial‑fraction decomposition \(\dfrac{2x+k}{(x+2)(x+3)}=\dfrac{A}{x+2}+\dfrac{B}{x+3}\) yields integer constants \(A\) and \(B\).
(b)
For the value \(k=4\) found in part (a), determine the constants \(A\) and \(B\).
(c)
Verify the decomposition by recombining the fractions for \(k=4\).
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Question 5 View Details
A 3 \times 3 matrix \(A\) is given by \(A = \begin{bmatrix} 2 & k & 1 \\ 0 & m & 4 \\ p & 5 & 0 \end{bmatrix}. It is known that \(\det A = 20\). If the element \(k\) is increased by 2 (i.e. replaced by \(k+2\)), the matrix becomes singular. \(\)Determine the values of \(k, m\) and \(p\). Then find the element in the second row, third column of \(A^{-1}\).
Question Parts
(a)
Find the value of \(p\) using the information that the matrix becomes singular when \(k\) is replaced by \(k+2\).
(b)
Using \(\det A = 20\) and the value of \(p\) found in part (a), determine the values of \(k\) and \(m\).
(c)
Find the element in the second row, third column of \(A^{-1}\).
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Question 6 View Details
A committee of eight members is to be formed from a pool of 5 teachers and 7 students. The committee must contain at least two teachers and at most three students. In addition, the chairperson of the committee must be a teacher and the secretary must be a student.
Question Parts
(a)
How many different committees can be formed satisfying all the given conditions?
(b)
If the committee is formed randomly under the above conditions, what is the probability that a particular teacher, Mr. A, is appointed as the chairperson?
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Question 7 View Details
Consider the binomial expansion of \( (x + \frac{1}{x})^{6} \).
Question Parts
(a)
Find the term in the expansion that is independent of \(x\) and state its numerical value.
(b)
If \(x\) satisfies \(x + \frac{1}{x} = 3\), determine the value of \(x^{2} + \frac{1}{x^{2}}\) and then calculate the product of this value with the independent term found in part (a).
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Question 8 View Details
A lighthouse stands on a straight coastline. From point A on the coast the angle of elevation to the top of the lighthouse is \(30^{\circ}\). From point B, which is 500 m nearer to the lighthouse along the same line, the angle of elevation is \(45^{\circ}\). The light makes one complete rotation every 20 s.
Question Parts
(a)
Determine the height of the lighthouse (in metres), correct to the nearest metre.
(b)
Using the height found in part (a) and \(\pi = 3.14\), compute the linear speed of the light beam at the top of the lighthouse (in metres per second), correct to the nearest whole number.
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