Practice neco further_mathematics 2021 Past Questions

Question 1 View Details

Let \(A = \begin{pmatrix} 2 & k \\ 5 & 3 \end{pmatrix}\). It is known that the inverse of \(A\) exists and that the sum of the elements of \(A^{-1}\) equals 1. Find the value of \(k\) and then compute the determinant of the matrix \(B = A^{2} - 4A\).

A. k = \\frac{5}{2}, \\; \\det(B)=\\frac{45}{4}

Correct B. k = \frac{3}{2}, \; \det(B)=\frac{33}{4}

C. k = \\frac{3}{2}, \\; \\det(B)=\\frac{27}{4}

D. k = 2, \\; \\det(B)=\\frac{33}{4}

Correct Answer: B

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Question 2 View Details

A circle is tangent to the x‑axis and also tangent to the line \(x + y - 6 = 0\). The circle passes through the point \((4,0)\). Find the centre of the circle and its radius.

A. Centre \\((4,\\,2\\sqrt{2}+2)\\), radius \\(2\\sqrt{2}+2\\)

B. Centre \\((4,\\,2-2\\sqrt{2})\\), radius \\(2-2\\sqrt{2}\\)

C. Centre \\((4,\\,\\sqrt{2})\\), radius \\(\\sqrt{2}\\)

Correct D. Centre \((4,\,2\sqrt{2}-2)\), radius \(2\sqrt{2}-2\)

Correct Answer: D

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Question 3 View Details

Evaluate \( \log_{2}\bigl( \sqrt{8}\times 4^{\frac{3}{2}} \bigr) \).

Correct A. \frac{9}{2}

B. \frac{10}{2}

C. \frac{8}{2}

D. \frac{9}{4}

Correct Answer: A

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Question 4 View Details

A block of mass \(5\,\text{kg}\) slides down a rough incline that makes an angle of \(30^{\circ}\) with the horizontal. The length of the incline is \(2\,\text{m}\) and the coefficient of kinetic friction is \(0.20\). At the foot of the incline the block compresses a spring of spring constant \(800\,\text{N\,m}^{-1}\) and comes to rest. Assuming \(g=9.8\,\text{m\,s}^{-2}\), find the maximum compression of the spring (in metres, correct to three significant figures).

A. 0.286

Correct B. 0.283

C. 0.350

D. 0.313

Correct Answer: B

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Question 5 View Details

A committee of five members is to be chosen from eight teachers and six engineers. The committee must contain at least two teachers and at most three engineers, and the chairperson of the committee must be a teacher. How many different committees can be formed?

Correct A. 5600

B. 5320

C. 1876

D. 700

Correct Answer: A

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Question 6 View Details

Given that \(\sin \theta + \sqrt{3}\cos \theta = 2\), find \(\tan \theta\) and then evaluate \(\sin 2\theta\).

A. tan\\theta = \\frac{\\sqrt{3}}{3}, \\; \\sin 2\\theta = \\frac{1}{2}

B. tan\\theta = 1, \\; \\sin 2\\theta = \\frac{\\sqrt{3}}{2}

C. tan\\theta = \\frac{\\sqrt{3}}{2}, \\; \\sin 2\\theta = \\frac{\\sqrt{3}}{2}

Correct D. tan\theta = \frac{\sqrt{3}}{3}, \; \sin 2\theta = \frac{\sqrt{3}}{2}

Correct Answer: D

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Question 7 View Details

Let \(A=\begin{pmatrix}2&1\\k&3\end{pmatrix}\) where \(k\) is an integer. It is known that \(A\) is invertible and that \(A^{-1}\) has only integer entries. Determine all possible values of \(k\) and then find the sum of all possible values of \(\det\bigl(A^{2}+I\bigr)\), where \(I\) is the \(2\times2\) identity matrix.

Correct A. 54

B. 60

C. 48

D. 72

Correct Answer: A

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Question 8 View Details

Let \(A=\{1,2,3,4\}\) and \(B=\{a,b,c\}\). A function \(f:A\to B\) is defined by \(f(1)=a,\; f(2)=b,\; f(3)=c,\; f(4)=a\). Determine the image of the subset \(\{2,4\}\) under \(f\) and state whether \(f\) is onto.

A. Image = \\{b\\}; \\; f \\text{ is onto}

B. Image = \\{a,b,c\\}; \\; f \\text{ is not onto}

C. Image = \\{a,c\\}; \\; f \\text{ is not onto}

Correct D. Image = \{a,b\}; \; f \text{ is onto}

Correct Answer: D

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Question 9 View Details

Find constants \(A\) and \(B\) such that \(\displaystyle \frac{2x+3}{x^{2}-x-6}=\frac{A}{x-3}+\frac{B}{x+2}\). Then evaluate the indefinite integral \(\displaystyle \int \frac{2x+3}{x^{2}-x-6}\,dx\).

A. (\\frac{9}{5})\\ln|x-3| - (\\frac{1}{5})\\ln|x+2| + C

B. (\\frac{9}{5})\\ln|x-3| + (\\frac{2}{5})\\ln|x+2| + C

Correct C. (\frac{9}{5})\ln|x-3| + (\frac{1}{5})\ln|x+2| + C

D. (\\frac{1}{5})\\ln|x-3| + (\\frac{9}{5})\\ln|x+2| + C

Correct Answer: C

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Question 10 View Details

A circle passes through the point \((1,2)\) and is tangent to both lines \(x+y=4\) and \(x-y=2\). If the centre of the circle lies in the first quadrant, find the radius of the circle.

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

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

C. 2\\sqrt{3} - \\sqrt{2}

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

Correct Answer: D

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Question 11 View Details

The curve is given by \( y = \ln\left(x^{2}+1\right) \). Determine the positive value of \( a \) for which the tangent to the curve at \( x = a \) passes through the origin. Then, using that value of \( a \), find the second derivative \( \frac{d^{2}y}{dx^{2}} \) at \( x = a \). Give your answer for \( a \) correct to two decimal places and for the second derivative correct to three decimal places.

Correct A. a = 1.98, y'' = -0.241

B. a = 2.10, y'' = -0.200

C. a = 2.05, y'' = -0.215

D. a = 1.85, y'' = -0.260

Correct Answer: A

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Question 12 View Details

In three‑dimensional space let \( \mathbf{a}= (3,\,4,\,0) \). A vector \( \mathbf{b}= (x,\,y,\,z) \) satisfies \( |\mathbf{b}| = 7 \) and \( \mathbf{a}\cdot\mathbf{b}=14 \).\n(a) Find the magnitude of \( \mathbf{c}=\mathbf{a}\times\mathbf{b} \).\n(b) Determine a unit vector \( \mathbf{u} \) that is orthogonal to \( \mathbf{a} \) and makes an angle of \(30^{\circ}\) with \( \mathbf{c} \). Give the components of \( \mathbf{u} \) correct to two decimal places.

A. (-0.53, -0.40, -0.75)

B. (0.53, -0.40, 0.75)

Correct C. (0.53, -0.40, -0.75)

D. (0.60, -0.45, -0.70)

Correct Answer: C

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Question 13 View Details

A committee of five members is to be chosen from eight boys and six girls. The chairperson must be a girl and the secretary must be a boy. Moreover, a particular boy \(B_{1}\) and a particular girl \(G_{1}\) are not allowed to serve together on the same committee. How many different committees can be formed under these conditions?

Correct A. 9330

B. 9120

C. 9450

D. 10560

Correct Answer: A

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Question 14 View Details

Consider the matrix \( A = \begin{pmatrix} k & 2 & -1 \\ 3 & k & 4 \\ 5 & -2 & k \end{pmatrix} \).\n(a) Find all real values of \( k \) for which \( A \) is invertible.\n(b) For \( k = 1 \), compute the determinant of \( B = A^{2} - 3A + 2I \), where \( I \) is the \(3\times3\) identity matrix.

A. 1740

B. 1648

C. 1758

Correct D. 1748

Correct Answer: D

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Question 15 View Details

Find the remainder when the polynomial \( P(x)=2x^{4}-3x^{3}+x^{2}-5x+7 \) is divided by \( x^{2}-x+1 \). Express the remainder in the form \( ax+b \).

A. 6x + 9

Correct B. -6x + 9

C. -5x + 9

D. -6x - 9

Correct Answer: B

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Question 16 View Details

Let \(A = \{1,2,3,4\}\) and \(B = \{a,b,c\}\). A function \(f: A \to B\) is defined by \(f(1)=a,\; f(2)=b,\; f(3)=a,\; f(4)=c\). Another function \(g: B \to A\) is defined by \(g(a)=2,\; g(b)=3,\; g(c)=1\). Find the set of all elements \(x\in A\) for which \((g\circ f)(x)=x\).

A. \{2,3\}

B. \{1,2,3,4\}

Correct C. \emptyset

D. \{1\}

Correct Answer: C

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Question 17 View Details

A block of mass \(5\ \text{kg}\) is pulled up a rough inclined plane that makes an angle of \(30^{\circ}\) with the horizontal. The coefficient of kinetic friction between the block and the plane is \(0.2\). The block starts from rest at the bottom and moves \(4\ \text{m}\) up the plane, reaching a speed of \(3\ \text{m s}^{-1}\). The pulling force is applied parallel to the plane. Determine the magnitude of the pulling force (in newtons, to one decimal place).

A. 45.0 N

B. 41.2 N

C. 34.5 N

Correct D. 38.6 N

Correct Answer: D

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Question 18 View Details

A right circular cylindrical tank is to hold exactly \(5000\ \text{cm}^{3}\) of water. Find the radius \(r\) (in cm) and height \(h\) (in cm) of the tank that minimize the total surface area (including top and bottom). Give your answers correct to one decimal place.

A. r \approx 10.0 cm, h \approx 15.5 cm

Correct B. r \approx 9.2 cm, h \approx 18.6 cm

C. r \approx 7.9 cm, h \approx 25.2 cm

D. r \approx 8.5 cm, h \approx 21.0 cm

Correct Answer: B

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Question 19 View Details

Simplify the expression \\displaystyle \frac{2^{5}\cdot 5^{3}}{10^{2}} and write the result in the form \(2^{m}\cdot 5^{n}\) where \(m\) and \(n\) are integers.

Correct A. 2^{3}\cdot5^{1}

B. 2^{5}\\cdot5^{1}

C. 2^{2}\\cdot5^{1}

D. 2^{3}\\cdot5^{2}

Correct Answer: A

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Question 20 View Details

On the set of integers \(\mathbb{Z}\) define a binary operation \(*\) by \(a * b = a + b - 3\) for all \(a,b\in\mathbb{Z}\). (i) State whether \(*\) is associative. (ii) Find the identity element of \(*\) in \(\mathbb{Z}\).

Correct A. Associative; identity element = 3

B. Associative; identity element = -3

C. Associative; identity element = 0

D. Non-associative; identity element = 3

Correct Answer: A

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Question 21 View Details

Given that \\( \\sin \\theta + \\sqrt{3}\\cos \\theta = 2 \\) and \\(0^{\\circ} \\le \\theta < 360^{\\circ}\\), find the value of \\( \\theta \\) and then compute \\( \\tan \\frac{\\theta}{2} \\).

A. 2+\\sqrt{3}

B. \\sqrt{3}-2

C. 2-\\frac{\\sqrt{3}}{2}

Correct D. 2-\\sqrt{3}

Correct Answer: D

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Question 22 View Details

Let \\( A \\) be a real \\(2\\times2\\) matrix satisfying \\( A + A^{-1} = 3I \\), where \\( I \\) is the identity matrix. Find \\( \\det A \\).

A. -1

Correct B. 1

C. 3

D. 0

Correct Answer: B

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Question 23 View Details

Each of 5 multiple‑choice questions is attempted independently. For any question the probability that the student knows the answer is 0.6; if known the answer is correct, otherwise the student guesses randomly among 4 options. The student passes if at least 4 questions are answered correctly. (a) What is the probability that the student passes the exam? (b) If a randomly selected answered question is known to be correct, what is the probability that the student actually knew the answer to that question?

Correct A. Pass probability = 26411/50000; Knowledge probability = 6/7

B. Pass probability = 27000\/50000; Knowledge probability = 2\/3

C. Pass probability = 26000\/50000; Knowledge probability = 5\/7

D. Pass probability = 25000\/50000; Knowledge probability = 7\/8

Correct Answer: A

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Question 24 View Details

Simplify the expression \\( \\log_{2} 8^{\\log_{3} 27} - \\log_{3} 9^{\\log_{2} 4} \\).

A. 23

B. 13

Correct C. 5

D. 7

Correct Answer: C

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Question 25 View Details

In three‑dimensional space let \\( \\mathbf{a} = \\langle 2, -1, 3 \\rangle \\). A vector \\( \\mathbf{b} \\) satisfies \\( \\mathbf{a} \\times \\mathbf{b} = \\langle 4, 8, -2 \\rangle \\) and \\( \\mathbf{a} \\cdot \\mathbf{b} = 10 \\). Find the magnitude \\( |\\mathbf{b}| \\).

A. \\sqrt{161}\/7

B. 2\\sqrt{162}\/7

Correct C. 2\\sqrt{161}/7

D. 2\\sqrt{161}\/5

Correct Answer: C

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