Practice neco further_mathematics 2025 Past Questions

Question 1 View Details

The matrix \(A = \begin{pmatrix} 1 & 2 & k \\ 0 & 1 & 3 \\ 4 & 0 & 1 \end{pmatrix}\) satisfies \(\det A = 5\). Find the value of \(k\) and then compute the trace of \(A^{2}\).

Correct A. k = 5; trace of A^2 = 43

B. k = 4; trace of A^2 = 35

C. k = 6; trace of A^2 = 51

D. k = 5; trace of A^2 = 39

Correct Answer: A

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Question 2 View Details

The cubic polynomial \(P(x)=x^{3}-6x^{2}+ax+b\) has \((x-2)\) as a factor and leaves a remainder of \(5\) when divided by \((x+1)\). Determine the value of \(a\).

A. 4\/5

B. 3\/4

C. 5\/3

Correct D. 4/3

Correct Answer: D

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Question 3 View Details

A committee of five members is to be chosen from eight men and six women. The committee must contain at least two women and at most three men. How many different committees can be formed?

Correct A. 1526

B. 1946

C. 1584

D. 1520

Correct Answer: A

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Question 4 View Details

A uniform horizontal beam \(AB\) 6 m long and weighing 120 N is hinged at \(A\). A cable \(BD\) is attached to the free end \(B\) and anchored to a wall at point \(D\) which is 4 m horizontally from \(B\) and 3 m above \(B\). Determine the tension in the cable and the horizontal reaction at the hinge \(A\).

A. Tension = 120 N; horizontal reaction at A = 60 N (to the right)

Correct B. Tension = 100 N; horizontal reaction at A = 80 N (to the right)

C. Tension = 100 N; horizontal reaction at A = 80 N (to the left)

D. Tension = 80 N; horizontal reaction at A = 100 N (to the right)

Correct Answer: B

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Question 5 View Details

Evaluate the integral \(\displaystyle \int \frac{2x^{2}+3x+5}{(x-1)(x^{2}+4)}\,dx\) by expressing the integrand as a sum of partial fractions.

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

B. 2\ln|x-1| + \frac{5}{2}\arctan\left(\frac{x}{2}\right) + C

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

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

Correct Answer: C

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Question 6 View Details

In the expansion of \( (2x^{2} - \frac{3}{x})^{5} \), find the sum of the coefficients of all terms in which the power of \(x\) is an even integer.

A. 1462

B. 1526

C. 1625

Correct D. 1562

Correct Answer: D

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Question 7 View Details

Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) lie in the same plane with magnitudes \(|\mathbf{a}| = 5\) cm and \(|\mathbf{b}| = 7\) cm. The vector \(\mathbf{c}=\mathbf{a}+2\mathbf{b}\) has magnitude \(13\) cm. Determine the angle between \(\mathbf{a}\) and \(\mathbf{b}\).

A. \cos^{-1}\left(\frac{13}{35}\right)

B. \cos^{-1}\left(-\frac{12}{35}\right)

Correct C. \cos^{-1}\left(-\frac{13}{35}\right)

D. \cos^{-1}\left(-\frac{13}{30}\right)

Correct Answer: C

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Question 8 View Details

In the expansion of \((3x - 2)^{9}\), find the sum of the coefficients of all terms in which the exponent of \(x\) is a multiple of \(3\).

A. -326565

B. -325555

C. -324565

Correct D. -325565

Correct Answer: D

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Question 9 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. A constant force of \(60\ \text{N}\) is applied parallel to the plane upward. The coefficient of kinetic friction between the block and the plane is \(0.20\). The block starts from rest at the bottom and moves \(4\ \text{m}\) along the plane. Using the work‑energy principle, determine the speed of the block when it reaches the top of the incline. Give your answer to two decimal places in \(\text{m s}^{-1}\).

A. 7.31

B. 5.12

C. 8.45

Correct D. 6.57

Correct Answer: D

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Question 10 View Details

Let \(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\) be the linear transformation that first reflects a vector about the line \(y=x\), then rotates the result \(90^{\circ}\) counter‑clockwise, and finally scales the vector by a factor of \(3\). Find the standard matrix of \(T\) and state its determinant.

A. \begin{pmatrix}-3&0\\0&-3\end{pmatrix}; determinant = 9

B. \begin{pmatrix}3&0\\0&-3\end{pmatrix}; determinant = -9

Correct C. \begin{pmatrix}-3&0\\0&3\end{pmatrix}; determinant = -9

D. \begin{pmatrix}0&-3\\3&0\end{pmatrix}; determinant = 9

Correct Answer: C

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Question 11 View Details

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

A. \( \frac{1}{6}\bigl(3\ln2+\ln3\bigr)+\frac{7}{3\sqrt2}\arctan\!\left(\frac1{\sqrt2}\right) \)

B. \( \frac{1}{6}\bigl(3\ln2-\ln3\bigr)+\frac{5}{3\sqrt2}\arctan\!\left(\frac1{\sqrt2}\right) \)

C. \( \frac{1}{6}\bigl(2\ln2-\ln3\bigr)+\frac{7}{3\sqrt2}\arctan\!\left(\frac2{\sqrt2}\right) \)

Correct D. \( \frac{1}{6}\bigl(3\ln2-\ln3\bigr)+\frac{7}{3\sqrt2}\arctan\!\left(\frac1{\sqrt2}\right) \)

Correct Answer: D

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Question 12 View Details

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

A. \( (x-11)^{2}+(y-18)^{2}=400 \)

B. \( (x+11)^{2}+(y+18)^{2}=400 \)

Correct C. \( (x+11)^{2}+(y-18)^{2}=400 \)

D. \( (x+11)^{2}+(y-18)^{2}=225 \)

Correct Answer: C

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Question 13 View Details

A circle passes through the points \(A(2,3)\) and \(B(8,7)\) and its centre lies on the line \(2x-y=1\). Determine the centre and the radius of the circle.

A. \(\text{Centre }\left(\frac{27}{7},\frac{47}{7}\right),\;\text{radius }\frac{12\sqrt5}{7}\)

B. \(\text{Centre }\left(\frac{26}{7},\frac{48}{7}\right),\;\text{radius }\frac{13\sqrt5}{7}\)

C. \(\text{Centre }\left(\frac{47}{7},\frac{27}{7}\right),\;\text{radius }\frac{13\sqrt5}{7}\)

Correct D. \(\text{Centre }\left(\frac{27}{7},\frac{47}{7}\right),\;\text{radius }\frac{13\sqrt5}{7}\)

Correct Answer: D

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Question 14 View Details

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

Correct A. \( \frac{\pi^{2}}{4} \)

B. \\( \\frac{\\pi}{4} \\)

C. \\( \\frac{\\pi^{2}}{6} \\)

D. \\( \\frac{\\pi^{2}}{2} \\)

Correct Answer: A

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Question 15 View Details

Determine the locus of the centre \((x,y)\) of a circle that passes through the fixed point \((2,0)\) and is tangent to the line \(y=x+2\).

Correct A. \( (x+y)^{2}-12x+4y+4=0 \)

B. \( (x+y)^{2}-12x-4y+4=0 \)

C. \( (x+y)^{2}+12x+4y+4=0 \)

D. \( (x+y)^{2}-12x+4y-4=0 \)

Correct Answer: A

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Question 16 View Details

A committee of five is to be formed from \\(8\\) boys and \\(7\\) girls. The chairperson must be a girl and the secretary a boy. In addition the committee must contain at least two girls and at most three boys. How many different committees can be formed?

A. 13056

Correct B. 14056

C. 14064

D. 14560

Correct Answer: B

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Question 17 View Details

For \\(0\\le x\\le \\pi\\) the function is \\(f(x)=e^{2x}\sin x\\). (a) Find all values of \\(x\\) in this interval where the tangent to the curve is horizontal. (b) State whether each such point is a local maximum, minimum or point of inflection.

A. x = \\arctan\\left(\\frac{1}{2}\\right) (approximately 0.464); the point is a local minimum

B. x = \\frac{\\pi}{2} (approximately 1.571); the point is a local maximum

Correct C. x = \pi - \arctan\left(\frac{1}{2}\right) (approximately 2.678); the point is a local maximum

D. x = \\pi - \\arctan\\left(2\\right) (approximately 2.034); the point is a point of inflection

Correct Answer: C

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Question 18 View Details

Simplify \\( \log_{3} 27 + \log_{3} \sqrt{9} - \log_{3} \frac{1}{3} \).

A. 4

Correct B. 5

C. 6

D. 3

Correct Answer: B

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Question 19 View Details

Let \\(z\\) be a complex number satisfying \\(z^{2}=1+i\sqrt{3}\\). Evaluate \\( \dfrac{z^{6}+1}{z^{3}-1} \\) and give all possible values.

A. \frac{5}{9} \pm \frac{14\sqrt{2}}{9} i

Correct B. \frac{7}{9} \pm \frac{14\sqrt{2}}{9} i

C. \frac{7}{9} \pm \frac{14\sqrt{3}}{9} i

D. \frac{7}{9} \pm \frac{7\sqrt{2}}{9} i

Correct Answer: B

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Question 20 View Details

A linear transformation \\(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\\) is represented by the matrix \\(M=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\\). It is known that \(T\) maps \((1,2)\) to \((3,4)\) and maps \((2,-1)\) to \((k,5)\). Moreover the determinant of \(M\) equals \(6\). Find the value of \(k\) and then give the trace \(a+d\) of \(M\).

A. k = \\frac{9}{4}; \\; a+d = \\frac{57}{10}

B. k = \\frac{45}{4}; \\; a+d = \\frac{57}{8}

C. k = \\frac{45}{2}; \\; a+d = \\frac{57}{5}

Correct D. k = \frac{45}{4}; \; a+d = \frac{57}{10}

Correct Answer: D

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Question 21 View Details

The universal set is \(U=\{1,2,3,4,5,6,7,8\}\). Subset \(A=\{2,4,6,8\}\) and subset \(B=\{1,3,5,7\}\). Find the complement of the union of \(A\) and \(B\), i.e., \((A\cup B)'\).

A. \(\{0\}\)

B. \(\{1\}\)

C. \(\{2,3\}\)

Correct D. \(\varnothing\)

Correct Answer: D

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Question 22 View Details

The vectors \(\mathbf{a},\mathbf{b},\mathbf{c}\) in three‑dimensional space satisfy \(|\mathbf{a}|=3\), \(|\mathbf{b}|=4\), \(|\mathbf{c}|=5\), \(\mathbf{a}\cdot\mathbf{b}=6\) and \(\mathbf{b}\cdot\mathbf{c}=8\). It is also given that \(\mathbf{a}+\mathbf{b}+\mathbf{c}\) is orthogonal to \(\mathbf{a}\).\n\n(a) Find the value of \(\mathbf{a}\cdot\mathbf{c}\).\n(b) Let \(\mathbf{d}= \mathbf{a}+2\mathbf{b}\). Determine the magnitude \(|\mathbf{d}|\) and the angle between \(\mathbf{d}\) and \(\mathbf{c}\) (to one decimal place in degrees).

A. |\\d| = \\sqrt{101}, angle \\approx 90.0^{\\circ}

B. |\\d| = \\sqrt{105}, angle \\approx 80.5^{\\circ}

C. |\\d| = \\sqrt{89}, angle \\approx 85.0^{\\circ}

Correct D. |\d| = \sqrt{97}, angle \approx 88.8^{\circ}

Correct Answer: D

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Question 23 View Details

A box contains red, blue and green balls. The probability of drawing a red ball on the first draw (without replacement) is \(\frac13\). After a red ball has been drawn, the probability that the next ball drawn is blue is \(\frac25\). If the total number of balls in the box is 21, determine the number of green balls.

A. 5

Correct B. 6

C. 9

D. 7

Correct Answer: B

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Question 24 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 length of the plane is \(10\ \text{m}\). The coefficient of kinetic friction between the block and the plane is \(0.2\). A constant horizontal force \(F\) is applied to the block. The block starts from rest at the bottom and reaches the top with a speed of \(2\ \text{m s}^{-1}\). Determine the magnitude of the horizontal force \(F\) (to one decimal place in newtons).

A. 30.5

B. 42.7

C. 28.9

Correct D. 35.2

Correct Answer: D

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Question 25 View Details

In the plane, vectors \(\mathbf{u}\) and \(\mathbf{v}\) have magnitudes \(|\mathbf{u}|=5\) and \(|\mathbf{v}|=12\) respectively, and the angle between them is \(60^{\circ}\).\n\nLet \(\mathbf{w}=2\mathbf{u}-\mathbf{v}\).\n\n(a) Find the magnitude \(|\mathbf{w}|\).\n(b) Determine the angle between \(\mathbf{w}\) and \(\mathbf{u}\) (to the nearest degree).

A. |\\mathbf{w}| = 2\\sqrt{31}, angle \\approx 60^{\\circ}

B. |\\mathbf{w}| = \\sqrt{31}, angle \\approx 69^{\\circ}

Correct C. |\mathbf{w}| = 2\sqrt{31}, angle \approx 69^{\circ}

D. |\\mathbf{w}| = 2\\sqrt{30}, angle \\approx 69^{\\circ}

Correct Answer: C

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