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neco 2017 further_mathematics | Objective

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Question 1 View Details

If \( \log_{2}(8x) = 5 \) and \( x > 0 \), find \( x \).

Correct A. 4

B. 5/8

C. 2

D. 32

Correct Answer: A

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Question 2 View Details

A simply supported beam AB of length 5 m has a pin support at A and a roller at B. A point load of 12 kN acts 1.5 m from A. The beam also carries a uniformly distributed load of \( w \) kN/m over its entire length. If the reaction at A is twice the reaction at B, determine the value of \( w \) (in kN/m).

A. 0.64

B. 0.56

C. 0.42

Correct D. 0.48

Correct Answer: D

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Question 3 View Details

Find the greater possible radius of a circle that passes through the point \( P(2,3) \), is tangent to the line \( x + y = 4 \), and has its centre on the line \( y = 2x - 1 \). Give your answer correct to two decimal places.

Correct A. 13.78

B. 14.02

C. 13.05

D. 12.45

Correct Answer: A

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Question 4 View Details

The word "MATHEMATICS" consists of 11 letters with repetitions: M(2), A(2), T(2), H, E, I, C, S. How many distinct arrangements of all the letters have a vowel as the first letter and the two T's not adjacent?

A. 14515200

B. 1451522

Correct C. 1451520

D. 1451525

Correct Answer: C

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Question 5 View Details

Let \( z \) be a complex number such that \( |z - 2i| = 3 \) and \( \arg(z) = \frac{\pi}{6} \). Find the exact value of \( z^{2} \).

A. (\\frac{7}{2} - \\sqrt{6}) + i(\\frac{7\\sqrt{3}}{2} + 3\\sqrt{2})

Correct B. (\frac{7}{2} + \sqrt{6}) + i(\frac{7\sqrt{3}}{2} + 3\sqrt{2})

C. (\\frac{7}{2} + \\sqrt{6}) + i(\\frac{7\\sqrt{3}}{2} - 3\\sqrt{2})

D. (\\frac{5}{2} + \\sqrt{6}) + i(\\frac{7\\sqrt{3}}{2} + 3\\sqrt{2})

Correct Answer: B

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Question 6 View Details

The rational function \(\frac{2x^{2}+5x+3}{(x-1)(x+2)^{2}}\) can be expressed as \(\frac{A}{x-1}+\frac{B}{x+2}+\frac{C}{(x+2)^{2}}\). Find the value of \(A+B+C\).

A. 7/3

B. 4/3

C. 5/2

Correct D. 5/3

Correct Answer: D

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Question 7 View Details

A quality inspector examines 20 items from a production line and finds exactly 4 defective items. Assuming each item is defective independently with probability \(p\), (a) find the maximum‑likelihood estimate of \(p\); (b) using this estimate, compute the probability that in the next batch of 10 items at most one is defective. Give your answer to three decimal places.

A. 0.452

Correct B. 0.376

C. 0.107

D. 0.244

Correct Answer: B

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Question 8 View Details

Given that \(\sin\theta = \frac{3}{5}\) where \(\theta\) is an acute angle, find the exact value of \(\tan\left(2\theta + \frac{\pi}{6}\right)\).

A. (24\\sqrt{3}+7)\/(7\\sqrt{3}+24)

B. (24\\sqrt{3}-7)\/(7\\sqrt{3}+24)

C. (7\\sqrt{3}+24)\/(24\\sqrt{3}-7)

Correct D. (24\sqrt{3}+7)/(7\sqrt{3}-24)

Correct Answer: D

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Question 9 View Details

Points \(A(2,3)\) and \(B(8,7)\) lie in the plane. Determine the equation of the circle that passes through \(A\) and \(B\) and whose centre lies on the line \(x - y = 1\). State the centre coordinates and the radius.

A. (23\/5, 26\/5) and radius 9\\sqrt{2}\/5

B. (27\/5, 22\/5) and radius 12\\sqrt{2}\/5

Correct C. (27/5, 22/5) and radius 13\sqrt{2}/5

D. (31\/5, 18\/5) and radius 11\\sqrt{2}\/5

Correct Answer: C

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Question 10 View Details

Let \(A = \begin{bmatrix}2 & 1 & 0\\ -1 & 3 & 1\\ 4 & 0 & 5\end{bmatrix}\). (a) Find \(\det A\) and state whether \(A\) is invertible. (b) If \(A\) is invertible, give the element in the second row, third column of \(A^{-1}\).

A. -4/39

Correct B. -2/39

C. -2/13

D. 2/39

Correct Answer: B

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Question 11 View Details

The polynomial \(P(x)=x^{3}-9x^{2}+ax+b\) satisfies that the remainder when divided by \((x-3)\) is \(4\) and that \((x^{2}-4x+k)\) is a factor of \(P(x)\). Find the values of \(a\) and \(k\) and then evaluate \(P(1)\).

A. 12

B. 10

Correct C. 8

D. 6

Correct Answer: C

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Question 12 View Details

Let the function \(f\) be defined on the set \{1,2,3,4\} by \(f(1)=a,\; f(2)=b,\; f(3)=a,\; f(4)=c\). Determine the image \(f(\{2,3,4\})\).

Correct A. {a,b,c}

B. {a,b,c,d}

C. {b,c}

D. {a,b}

Correct Answer: A

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Question 13 View Details

In triangle \(ABC\) we have \(\angle B = 60^{\circ}\), side \(AB = 8\) cm and side \(BC = 6\) cm. (i) Find the length of side \(AC\). (ii) Determine \(\sin A\).

Correct A. AC = 2\sqrt{13} cm; \sin A = \frac{3\sqrt{39}}{26}

B. AC = 2\sqrt{7} cm; \sin A = \frac{3\sqrt{39}}{26}

C. AC = 4\sqrt{5} cm; \sin A = \frac{3\sqrt{13}}{26}

D. AC = 2\sqrt{13} cm; \sin A = \frac{\sqrt{39}}{26}

Correct Answer: A

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Question 14 View Details

Evaluate the definite integral \(\displaystyle\int_{0}^{1} x^{2}e^{x}\,dx\).

A. e - 1

B. 2 - e

Correct C. e - 2

D. e - 3

Correct Answer: C

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Question 15 View Details

A uniform beam \(AB\) is 8 m long and weighs 200 N, acting at its centre. The beam is supported by a pin at \(A\) and by a cable attached at \(B\) that makes an angle of \(45^{\circ}\) with the beam and is in tension. Determine (i) the tension \(T\) in the cable, and (ii) the magnitude and direction of the reaction at \(A\).

Correct A. T = 100\sqrt{2}\text{ N}; Reaction at A = 100\text{ N upward and }100\text{ N horizontal (resultant }100\sqrt{2}\text{ N at }45^{\circ})

B. T = 100\\text{ N}; Reaction at A = 100\\text{ N upward and }100\\text{ N horizontal (resultant }150\\text{ N at }45^{\\circ})

C. T = 200\\sqrt{2}\\text{ N}; Reaction at A = 200\\text{ N upward and }0\\text{ N horizontal (resultant }200\\text{ N at }90^{\\circ})

D. T = 100\\sqrt{2}\\text{ N}; Reaction at A = 50\\text{ N upward and }50\\text{ N horizontal (resultant }70.7\\text{ N at }45^{\\circ})

Correct Answer: A

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Question 16 View Details

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

A. 1+\ln\frac{3}{2}

Correct B. 1-\ln\frac{3}{2}

C. 1-\ln\frac{5}{2}

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

Correct Answer: B

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Question 17 View Details

A committee of five members is to be formed from 8 men and 6 women. The chairperson must be a man and the committee must contain at least two women. How many different committees can be formed?

Correct A. 3760

B. 3640

C. 5720

D. 470

Correct Answer: A

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Question 18 View Details

Find the centre and radius of the circle that passes through the points \(A(1,2)\) and \(B(4,6)\) and is tangent to the line \(3x - 4y + 5 = 0\).

A. Centre = (-60\/14, 64\/7); Radius = 125\/14

B. Centre = (-61\/14, 64\/7); Radius = 126\/14

C. Centre = (-61\/14, 63\/7); Radius = 125\/14

Correct D. Centre = (-61/14, 64/7); Radius = 125/14

Correct Answer: D

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Question 19 View Details

In the expansion of \((2x - 3)^{8}\), find the sum of the coefficients of all terms that contain an even power of \(x\).

Correct A. 195313

B. 195314

C. 390626

D. 195312

Correct Answer: A

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Question 20 View Details

Find the coefficient of \(x^{5}\) in the expansion of \((3x^{2} - 2x^{-1})^{7}\).

A. -68040

Correct B. -22680

C. 22680

D. -13608

Correct Answer: B

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Question 21 View Details

If \\( \\sin \\alpha = \\frac{3}{5} \\) where \\( \\alpha \\) is an acute angle, and \\( \\cos \\beta = -\\frac{4}{5} \\) where \\( \\beta \\) is an obtuse angle, find the exact value of \\( \\sin (\\alpha + \\beta) \\).

A. 12/25

Correct B. 0

C. 24/25

D. -12/25

Correct Answer: B

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Question 22 View Details

Let \\(z\\) be a complex number such that \(|z- (2 + i)| = |z- (2 - i)|\) and \\(\\arg(z) = \\frac{\\pi}{6}\\). Find \\(z\\) and then compute \\(z^{2}\\).

Correct A. \\frac{8}{3} + \\frac{8}{\\sqrt{3}} i

B. \\frac{8}{3} - \\frac{8}{\\sqrt{3}} i

C. \\frac{8}{3} + \\frac{4}{\\sqrt{3}} i

D. \\frac{16}{3} + \\frac{8}{\\sqrt{3}} i

Correct Answer: A

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Question 23 View Details

Simplify \\(\\sqrt{12} + \\sqrt{27} - \\sqrt{48}\\) and express your answer in simplest surd form.

Correct A. \\sqrt{3}

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

C. 2\sqrt{3}

D. \sqrt{6}

Correct Answer: A

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Question 24 View Details

Vectors \\(\\mathbf{a}\\) and \\(\\mathbf{b}\\) in three dimensions satisfy \\(|\\mathbf{a}| = 5\\), \\(|\\mathbf{b}| = 7\\) and \\(\\mathbf{a}\\cdot\\mathbf{b} = 20\\). Let \\(\\mathbf{c} = 2\\mathbf{a} - 3\\mathbf{b}\\). Find the magnitude \\(|\\mathbf{c}|\\).

A. \\sqrt{361}

Correct B. \\sqrt{301}

C. \\sqrt{289}

D. \\sqrt{325}

Correct Answer: B

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Question 25 View Details

Let the function \\(f:\\mathbb{R}\\to\\mathbb{R}\\) be defined by \\(f(x) = 2x - 3\\). Find the image set \\(f(\\{1,4,7\\})\\).

A. {-1,4,11}

B. {0,5,11}

C. {-1,5,9}

Correct D. {-1,5,11}

Correct Answer: D

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