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neco 2018 further_mathematics | Objective

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Question 1 View Details

A committee of five members is to be chosen from \(8\) boys and \(7\) girls such that the committee contains at least two girls and at most three boys. After the committee is formed, a chairperson is elected from among the girls on the committee. In how many different ways can the committee and the chairperson be selected?

A. 6300

Correct B. 6517

C. 6527

D. 2457

Correct Answer: B

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Question 2 View Details

A linear transformation \(T\) on \(\mathbb{R}^{2}\) satisfies \(T\begin{pmatrix}1\\2\end{pmatrix}=\begin{pmatrix}3\\5\end{pmatrix}\) and \(T\begin{pmatrix}4\\-1\end{pmatrix}=\begin{pmatrix}2\\k\end{pmatrix}\). The image of \(\begin{pmatrix}7\\9\end{pmatrix}\) under \(T^{-1}\) has second component equal to \(2\). Find \(k\) and compute \(T^{-1}\begin{pmatrix}7\\9\end{pmatrix}\).

Correct A. (\frac{43}{7}, 2)

B. (\\frac{42}{7}, 2)

C. (\\frac{43}{7}, 3)

D. (\\frac{44}{7}, 2)

Correct Answer: A

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Question 3 View Details

Evaluate \(\displaystyle \int \frac{2x^{2}+3x+5}{(x-1)(x^{2}+4)}\,dx\).

A. 2\ln|x-1|+\frac{3}{2}\tan^{-1}\left(\frac{x}{4}\right)+C

B. 2\ln|x-1|+\frac{1}{2}\tan^{-1}\left(\frac{x}{2}\right)+C

C. \ln|x-1|+\frac{3}{2}\tan^{-1}\left(\frac{x}{2}\right)+C

Correct D. 2\ln|x-1|+\frac{3}{2}\tan^{-1}\left(\frac{x}{2}\right)+C

Correct Answer: D

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Question 4 View Details

Let \(P(x)=x^{4}+ax^{3}+bx^{2}+cx+d\). When \(P(x)\) is divided by \((x-1)(x-2)\) the remainder is \(5x-1\). Find the value of \(a+b+c+d\).

Correct A. 3

B. 4

C. 5

D. 2

Correct Answer: A

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Question 5 View Details

A particle is acted upon by three coplanar forces: \(\vec{F}_{1}\) of magnitude \(10\text{ N}\) directed \(30^{\circ}\) above the positive \(x\)-axis, \(\vec{F}_{2}\) of unknown magnitude \(P\) directed \(150^{\circ}\) from the positive \(x\)-axis, and \(\vec{F}_{3}\) of unknown magnitude directed at an angle \(\theta\) below the positive \(x\)-axis. The particle is in equilibrium and the angle between \(\vec{F}_{2}\) and \(\vec{F}_{3}\) is \(120^{\circ}\). Determine \(P\) (to two decimal places) and \(\theta\) (in degrees, to the nearest integer).

A. P = 5.00 N, \\theta = 120^{\\circ}

B. P = 12.00 N, \\theta = 60^{\\circ}

C. P = 10.00 N, \\theta = 45^{\\circ}

Correct D. P = 10.00 N, \theta = 90^{\circ}

Correct Answer: D

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Question 6 View Details

If \( \sin \theta + \sqrt{3}\cos \theta = 2 \), find \( \tan \theta \).

A. -\\frac{\\sqrt{3}}{3}

B. \\frac{2}{\\sqrt{3}}

Correct C. \\frac{\\sqrt{3}}{3}

D. \\frac{\\sqrt{3}}{2}

Correct Answer: C

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Question 7 View Details

A rectangular field is to be fenced. The two length sides use material costing \(\text{\u20A6}5\) per metre and the two width sides cost \(\text{\u20A6}3\) per metre. If the area must be \(2000\) m\(^2\), find the dimensions that give the minimum total cost.

A. Length = 15\\sqrt{3} m, Width = \\frac{200\\sqrt{3}}{3} m

Correct B. Length = 20\\sqrt{3} m, Width = \frac{100\\sqrt{3}}{3} m

C. Length = 25\\sqrt{3} m, Width = \\frac{80\\sqrt{3}}{3} m

D. Length = 20\\sqrt{2} m, Width = \\frac{100\\sqrt{2}}{3} m

Correct Answer: B

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Question 8 View Details

In a factory, 60% of the items are produced by Machine A and 40% by Machine B. The defect rates are 2% for Machine A and 5% for Machine B. An item is selected at random and found to be defective; a second item from the same production batch is then selected and found to be non‑defective. What is the probability that the first defective item was produced by Machine B?

A. 2\/3

B. 1\/2

C. 3\/8

Correct D. 5/8

Correct Answer: D

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Question 9 View Details

Evaluate the integral \(\displaystyle \int \frac{2x+3}{(x-1)(x^{2}+4)}\,dx\).

A. \\ln|x-1| - \\frac{1}{2}\\ln(x^{2}+4) + \\arctan\\left(\\frac{x}{2}\\right) + C

B. \\ln|x-1| - \\frac{1}{2}\\ln(x^{2}+4) - \\frac{1}{2}\\arctan\\left(\\frac{x}{2}\\right) + C

Correct C. \\ln|x-1| - \\frac{1}{2}\\ln(x^{2}+4) + \\frac{1}{2}\\arctan\\left(\\frac{x}{2}\\right) + C

D. \\ln|x-1| + \\frac{1}{2}\\ln(x^{2}+4) + \\frac{1}{2}\\arctan\\left(\\frac{x}{2}\\right) + C

Correct Answer: C

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Question 10 View Details

The number of defects occurring in a day at a manufacturing plant follows a Poisson distribution. The probability that a day has no defects is 0.2. Determine the mean number of defects per day, and then find the probability that on a given day there are at least two defects. Give the probability to three decimal places.

Correct A. Mean \lambda = \ln 5 \approx 1.609; Probability \approx 0.478

B. Mean \\lambda = \\ln 3 \\approx 1.099; Probability \\approx 0.610

C. Mean \\lambda = \\ln 4 \\approx 1.386; Probability \\approx 0.523

D. Mean \\lambda = \\ln 6 \\approx 1.792; Probability \\approx 0.412

Correct Answer: A

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Question 11 View Details

Let \(z\) be a complex number satisfying \(|z-(2+3i)|=5\) and \(\arg(z)=\pi/4\). Find \(z\) and then compute \(z^{2}+\overline{z}\).

Correct A. 6+66i

B. 5+66i

C. 6-66i

D. 6+56i

Correct Answer: A

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Question 12 View Details

Let \(B=\begin{pmatrix}p&1\\k&p\end{pmatrix}\) where \(p\) and \(k\) are integers. Suppose that \(B^{-1}\) has integer entries. Find the smallest positive integer value of \(p\) for which this is possible and give the determinant of \(B^{-1}\) for that \(p\).

A. 3

B. 2

C. 4

Correct D. 1

Correct Answer: D

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Question 13 View Details

Circle \(C\) passes through the points \(A(1,2)\) and \(B(4,6)\) and its centre lies on the line \(x-y=1\). Determine the equation of \(C\) and hence find its radius.

A. 7\\sqrt{74}\/14

Correct B. 5\sqrt{74}/14

C. 5\\sqrt{73}\/14

D. 5\\sqrt{74}\/12

Correct Answer: B

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Question 14 View Details

Let the function \(f:\{1,2,3,4,5\}\to\{a,b,c,d\}\) be defined by \(f(1)=a,\;f(2)=b,\;f(3)=a,\;f(4)=c,\;f(5)=d\). State the range of \(f\) and indicate whether \(f\) is onto.

A. Range = {a,b,c}; f is onto

B. Range = {a,b,c,d}; f is not onto

C. Range = {a,b,d}; f is not onto

Correct D. Range = {a,b,c,d}; f is onto

Correct Answer: D

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Question 15 View Details

Evaluate the definite integral \(I=\int_{0}^{1}\frac{x\ln x}{1+x^{2}}\,dx\).

A. -\pi^{2}\/24

B. -\pi^{2}\/64

C. -\pi^{2}\/12

Correct D. -\pi^{2}/48

Correct Answer: D

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Question 16 View Details

The curve is given by \(f(x)=2x^{3}-9x^{2}+12x+5\). Find all \(x\)-coordinates at which the tangent to the curve is parallel to the line joining the points \((1,4)\) and \((4,10)\).

A. \\frac{9 \\pm \\sqrt{20}}{6}

Correct B. \frac{9 \pm \sqrt{21}}{6}

C. \\frac{8 \\pm \\sqrt{21}}{6}

D. \\frac{9 \\pm \\sqrt{21}}{3}

Correct Answer: B

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Question 17 View Details

Evaluate the definite integral \(\int_{0}^{1} \frac{2x+3}{(x+1)(x^{2}+1)}\,dx\).

A. \frac{1}{4}\ln 3+\frac{5\pi}{8}

Correct B. \frac{1}{4}\ln 2+\frac{5\pi}{8}

C. \frac{1}{4}\ln 2+\frac{3\pi}{8}

D. \frac{1}{2}\ln 2+\frac{5\pi}{8}

Correct Answer: B

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Question 18 View Details

Let \(z\) be a complex number satisfying \(z^{3}=8i\). Find the sum of all possible values of the expression \(z^{2}+\frac{1}{z}\).

A. -2

B. 4

Correct C. 0

D. 2

Correct Answer: C

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Question 19 View Details

Vectors \(\mathbf{a}\) and \(\mathbf{b}\) satisfy \(|\mathbf{a}|=3\), \(|\mathbf{b}|=4\) and \(\mathbf{a}\cdot\mathbf{b}=6\). Compute (i) the magnitude of \(\mathbf{a}-2\mathbf{b}\) and (ii) the cosine of the angle between \(\mathbf{a}-2\mathbf{b}\) and \(\mathbf{a}\).

A. Magnitude = 5, \\cos\\theta = -\\frac{1}{5}

B. Magnitude = 7, \\cos\\theta = -\\frac{1}{5}

Correct C. Magnitude = 7, \cos\theta = -\frac{1}{7}

D. Magnitude = 7, \\cos\\theta = \\frac{1}{7}

Correct Answer: C

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Question 20 View Details

Find the term independent of \(x\) in the expansion of \((x^{2}+\frac{3}{x})^{5}\,(2x-\frac{1}{x^{3}})^{3}\).

A. -3600

B. -324

Correct C. -3240

D. -3024

Correct Answer: C

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Question 21 View Details

Given vectors \(\mathbf{a}=2\mathbf{i}-\mathbf{j}+\mathbf{k}\) and \(\mathbf{b}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}\), the vector \(\mathbf{c}=p\mathbf{a}+q\mathbf{b}\) is perpendicular to \(\mathbf{a}\) and has magnitude \(5\). If \(p>0\), find \(p+q\).

A. \frac{3\sqrt{3}}{2}

B. \frac{3\sqrt{2}}{4}

C. \frac{5\sqrt{2}}{2}

Correct D. \frac{3\sqrt{2}}{2}

Correct Answer: D

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Question 22 View Details

A uniform ladder 5 m long and weighing 120 N rests against a smooth vertical wall. Its foot is 1.5 m from the wall on a rough horizontal ground with coefficient of static friction \(\mu = 0.2\). A person weighing 500 N climbs the ladder and stops at a distance \(x\) metres from the foot. Determine the greatest possible value of \(x\) (in metres, correct to two decimal places) for which the ladder remains in equilibrium without slipping.

A. 3.78

B. 4.10

C. 2.95

Correct D. 3.34

Correct Answer: D

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Question 23 View Details

When the polynomial \(P(x)=2x^{3}-5x^{2}+4x-7\) is divided by \(x-2\), what is the remainder?

A. -5

B. 7

C. 3

Correct D. -3

Correct Answer: D

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Question 24 View Details

Let \(A=\{1,2,3,4\}\) and \(B=\{2,4,6,8\}\). Define the function \(f:A\to B\) by \(f(x)=2x\). Find the image of the subset \(C=\{1,3\}\) under \(f\).

A. {2,4}

B. {4,6}

C. {1,6}

Correct D. {2,6}

Correct Answer: D

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Question 25 View Details

Expand \((x-3)^{4}\) and state the coefficient of \(x^{2}\).

A. 48

B. 36

C. 72

Correct D. 54

Correct Answer: D

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