Practice neco further_mathematics 2023 Past Questions

Question 1 View Details

Find the equation of the straight line that passes through the point \(P(8,1)\) and is tangent to the circle \((x-3)^{2}+(y+2)^{2}=25\). Then determine the coordinates of the point of tangency.

Correct A. The tangent line is \(8x+15y=79\) and the point of tangency is \(\left(\frac{1547}{289},\frac{697}{289}\right)\).

B. The tangent line is \(8x-15y=79\) and the point of tangency is \(\left(\frac{1547}{289},-\frac{697}{289}\right)\).

C. The tangent line is \(8x+15y=80\) and the point of tangency is \(\left(\frac{1547}{289},\frac{698}{289}\right)\).

D. The tangent line is \(8x+15y=78\) and the point of tangency is \(\left(\frac{1546}{289},\frac{697}{289}\right)\).

Correct Answer: A

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Question 2 View Details

The centre of a circle lies on the line \(x - y = 2\) and the circle passes through the points \(A(1,2)\) and \(B(5,6)\). Find the equation of the circle and its radius.

A. The circle is \((x-\frac{9}{2})^{2}+(y-\frac{5}{2})^{2}=\frac{25}{2}\) and its radius is \(\frac{5\sqrt{5}}{2}\).

Correct B. The circle is \((x-\frac{9}{2})^{2}+(y-\frac{5}{2})^{2}=\frac{25}{2}\) and its radius is \(\frac{5\sqrt{2}}{2}\).

C. The circle is \((x-\frac{7}{2})^{2}+(y-\frac{3}{2})^{2}=\frac{25}{2}\) and its radius is \(\frac{5\sqrt{2}}{2}\).

D. The circle is \((x-\frac{9}{2})^{2}+(y-\frac{5}{2})^{2}=\frac{25}{4}\) and its radius is \(\frac{5}{2}\).

Correct Answer: B

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Question 3 View Details

Let \(f(x)=e^{2x}(3-2x)\). Find an antiderivative \(F(x)\) such that \(F(0)=5\). Then evaluate \(F(1)\).

Correct A. e^{2}+3

B. e^{3}+3

C. e^{2}+4

D. e^{2}+2

Correct Answer: A

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Question 4 View Details

Let the linear transformation \(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\) be represented by the matrix \(B=\begin{pmatrix}a&b\\c&d\end{pmatrix}\). It satisfies \(B\begin{pmatrix}1\\2\end{pmatrix}=\begin{pmatrix}5\\1\end{pmatrix}\) and \(B\begin{pmatrix}3\\-1\end{pmatrix}=\begin{pmatrix}k\\4\end{pmatrix}\). If additionally \(\det B =2\), determine the value of \(k\) and the trace of \(B\).

A. k = 22 and the trace is \\(\\frac{56}{7}\\).

B. k = 28 and the trace is \\(\\frac{68}{7}\\).

C. k = 40 and the trace is \\(\\frac{80}{7}\\).

Correct D. k = 34 and the trace is \(\frac{72}{7}\).

Correct Answer: D

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Question 5 View Details

On the set \(A=\{1,2,3,4,5\}\) define a binary operation \(*\) by \(a * b = ((a+b)\bmod 5)\) with the convention that a result of \(0\) is identified with \(5\). Find the identity element of \((A,*)\) and compute \((3 * 4) * 2\).

A. The identity element is 3 and \((3 * 4) * 2 = 3\).

Correct B. The identity element is 5 and \((3 * 4) * 2 = 4\).

C. The identity element is 2 and \((3 * 4) * 2 = 5\).

D. The identity element is 1 and \((3 * 4) * 2 = 2\).

Correct Answer: B

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Question 6 View Details

If \(2^{\log_{2}5}=5\), evaluate \(\sqrt{2^{4}\times5^{2}}\div2^{\log_{2}10}\).

A. 4

B. 1

C. 5

Correct D. 2

Correct Answer: D

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Question 7 View Details

A linear transformation \(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\) satisfies \(T(1,2)=(3,5)\) and \(T(4,-1)=(k,7)\). Find the value of \(k\) that makes \(T\) invertible and then give the determinant of the matrix representing \(T\).

A. k = 4; determinant = -1/9

B. k = 5; determinant = 4/9

C. k = 2; determinant = -11/9

Correct D. k = 3; determinant = -2/3

Correct Answer: D

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Question 8 View Details

Let vectors \(\mathbf{A}= (1,2,-2)\) and \(\mathbf{B}= (x, y, z)\) satisfy the conditions: \(\mathbf{A}\cdot\mathbf{B}=6\), \(\mathbf{B}\) is perpendicular to \((2,-1,1)\), and \(|\mathbf{B}|=7\). Find the value of \(x\) and then determine the angle between \(\mathbf{A}\) and \(\mathbf{B}\) to the nearest degree.

Correct A. x = 6/5; angle = 73°

B. x = 6/5; angle = 65°

C. x = 4/5; angle = 68°

D. x = 7/5; angle = 80°

Correct Answer: A

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Question 9 View Details

Let \(f(x)=x^{3}-3x^{2}+ax+b\). If \(f(1)=5\) and \(f(2)=8\), determine the constants \(a\) and \(b\) and then compute \(f(3)\).

A. a = 4; b = 3; f(3) = 15

B. a = 5; b = 3; f(3) = 18

Correct C. a = 5; b = 2; f(3) = 17

D. a = 6; b = 1; f(3) = 19

Correct Answer: C

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Question 10 View Details

A real \(3\times3\) matrix \(A\) satisfies the polynomial equation \(A^{3}-3A^{2}+3A-I=0\), where \(I\) is the identity matrix. Find the determinant of \(A\).

A. 3

Correct B. 1

C. -1

D. 0

Correct Answer: B

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Question 11 View Details

If \( \int_{0}^{2} \frac{x}{x^{2}+k}\,dx = \ln 5 \), find \(k\).

Correct A. 1/6

B. 2/5

C. 1/3

D. 1/2

Correct Answer: A

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Question 12 View Details

Let \(P(x) = (x^{2}+\frac{2}{x})^{6}\). Define \(Q(x) = (x-1)P(x)\). Find the coefficient of \(x^{3}\) in \(Q(x)\).

A. -80

B. -120

Correct C. -160

D. -200

Correct Answer: C

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Question 13 View Details

The matrix \(A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & k \\ 7 & 8 & 9 \end{pmatrix}\). Find the value of \(k\) for which the rows of \(A\) are linearly dependent, and give one non‑zero vector \(\mathbf{v}\) such that \(A\mathbf{v}=\mathbf{0}\).

Correct A. k=6, vector (1,-2,1)

B. k=8, vector (2,-3,1)

C. k=6, vector (1,0,-1)

D. k=5, vector (1,-2,1)

Correct Answer: A

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Question 14 View Details

Given that \(\int_{0}^{\pi} (a\sin x + \cos x)^{2}\,dx = \frac{3\pi}{2}\), determine all possible values of \(a\) and then evaluate \(\int_{0}^{\pi} (a\sin x + \cos x)\,dx\).

A. a = \\pm\\sqrt{2}, integral = \\mp 2\\sqrt{2}

Correct B. a = \pm\sqrt{2}, integral = \pm 2\sqrt{2}

C. a = \\pm\\sqrt{3}, integral = \\pm 2\\sqrt{3}

D. a = \\pm 1, integral = \\pm \\sqrt{2}

Correct Answer: B

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Question 15 View Details

Find the remainder when \(x^{5} - 3x^{4} + 2x^{3} - x + 5\) is divided by \(x - 2\).

A. 5

B. 7

Correct C. 3

D. -1

Correct Answer: C

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Question 16 View Details

An uniform block of mass \(5\ \text{kg}\) slides down a rough inclined plane that makes an angle of \(30^{\circ}\) with the horizontal. The length of the plane is \(10\ \text{m}\). The coefficient of kinetic friction is unknown. At the foot of the plane the block compresses a spring of stiffness \(k = 200\ \text{N m}^{-1}\) by a distance of \(0.20\ \text{m}\) and then comes to rest. Determine the coefficient of kinetic friction \(\mu\) (give your answer to two decimal places).

A. 0.52

Correct B. 0.57

C. 0.47

D. 0.67

Correct Answer: B

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Question 17 View Details

Simplify the expression \(\frac{(2^{3}\cdot5^{2})^{\frac{1}{2}}}{\sqrt{20}}\) and give your answer in simplest radical form.

A. \sqrt{2}

B. \sqrt{5}

C. 2\\sqrt{5}

Correct D. \sqrt{10}

Correct Answer: D

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Question 18 View Details

A uniform beam AB of length \(6\) m and weight \(120\) N is hinged at A to a vertical wall. The beam is supported by a cable attached to B and anchored to the wall above A, the cable making an angle of \(30^{\circ}\) with the horizontal. A downward point load of \(80\) N acts at a point \(2\) m from A. Determine the tension in the cable (in newtons, correct to the nearest newton).

A. 190

B. 150

Correct C. 173

D. 210

Correct Answer: C

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Question 19 View Details

Vectors \(\mathbf{a}= (2,-1,3)\) and \(\mathbf{b}= (k,4,-2)\) are such that the angle between them is \(60^{\circ}\). (i) Find the value of \(k\). (ii) Find a vector \(\mathbf{c}\) that is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\) and has magnitude \(5\). Give \(\mathbf{c}\) in component form, rounded to two decimal places.

A. k = 40 + 2\\sqrt{385} \\approx 79.24; c \\approx (3.40, -2.14, -2.98)

B. k = 40 - 2\\sqrt{400} \\approx 0; c \\approx (-3.00, 2.00, 3.00)

Correct C. k = 40 - 2\sqrt{385} \approx 0.76; c \approx (-3.40, 2.14, 2.98)

D. k = 20 - \\sqrt{385} \\approx -0.62; c \\approx (-2.50, 1.80, 2.10)

Correct Answer: C

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Question 20 View Details

From a group of 8 teachers and 6 students, a committee of 5 members is to be formed. The committee must contain at least 2 teachers and at least 1 student. Moreover, two particular teachers, A and B, cannot both be on the committee. How many different committees can be formed?

A. 1600

B. 1680

Correct C. 1620

D. 1820

Correct Answer: C

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Question 21 View Details

From 7 mathematics teachers and 5 physics teachers, a panel of 4 teachers is to be formed containing exactly 2 mathematics teachers. One of the four members will be appointed chairperson. In how many different ways can the panel and the chairperson be chosen?

Correct A. 840

B. 210

C. 420

D. 1050

Correct Answer: A

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Question 22 View Details

Find the sum of the coefficients of the terms in which the power of \( x \) is odd in the expansion of \( (2x - 3)^7 \).

A. 7812

B. 39063

C. -39063

Correct D. 39062

Correct Answer: D

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Question 23 View Details

For the curve \( y = \ln(x) + x^{2} \), find the point(s) where the tangent line is parallel to the line through \( A(1,3) \) and \( B(4,12) \).

Correct A. (1,1) and (1/2, -\ln 2 + 1/4)

B. (2,4) and (1\/2, -\ln 2 + 1\/4)

C. (1,2) and (1\/2, -\ln 2 + 1\/4)

D. (1,1) and (1\/3, -\ln 3 + 1\/9)

Correct Answer: A

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Question 24 View Details

Let \( A = \begin{pmatrix} 2 & k \\ k & 5 \end{pmatrix} \) where \( k \) is an integer. Determine all integer values of \( k \) for which \( A \) is invertible and its inverse has integer entries.

Correct A. k = \pm 3

B. k = 0

C. k = \pm 2

D. k = \pm 4

Correct Answer: A

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Question 25 View Details

In a factory, 60% of the widgets produced are type A and 40% are type B. A type A widget is defective with probability 0.02, while a type B widget is defective with probability 0.05. A random sample of 5 widgets is inspected and exactly 2 are found to be defective. What is the probability that the sample contains exactly 3 type A widgets? Give your answer to three decimal places.

A. 0.291

B. 0.357

Correct C. 0.331

D. 0.415

Correct Answer: C

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