Practice neco further_mathematics 2019 Past Questions

Question 1 View Details

Matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) maps vector \( (1,2) \) to \( (3,5) \) and maps vector \( (2,-1) \) to \( (k,4) \). Find the integer value of \(k\) for which the trace of \(A\) equals its determinant, and then give that determinant.

A. k = 5, determinant = \\frac{15}{5}

Correct B. k = 7, determinant = \frac{23}{5}

C. k = 8, determinant = \\frac{27}{5}

D. k = 6, determinant = \\frac{19}{5}

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 2 View Details

Given the cubic polynomial \(P(x)=2x^{3}-ax^{2}+bx-6\), where \((x-1)\) is a factor of \(P(x)\) and \(P(2)=10\), determine the value of \(a+b\).

A. 10

B. 8

C. 14

Correct D. 12

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 3 View Details

Solve for the positive real number \(x\) satisfying \(\log_{2}(x^{2})-\log_{x}8 = 1\).

Correct A. x = \frac{1}{2} \text{ or } x = 2\sqrt{2}

B. x = \\frac{1}{2} \\text{ or } x = \\sqrt{2}

C. x = \\frac{1}{4} \\text{ or } x = 2\\sqrt{2}

D. x = 1 \\text{ or } x = 2\\sqrt{2}

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 4 View Details

A uniform beam AB of length 6 m is hinged at A to a vertical wall. The beam is supported by a cable CD attached at B, the cable making an angle of \(30^{\circ}\) above the beam. The beam carries a downward point load of 800 N located 2 m from A and its own weight of 300 N acting at its centre. Determine (i) the tension in the cable, (ii) the horizontal reaction at the hinge A, and (iii) the magnitude of the resultant reaction at A. Give your final answers to the nearest newton.

A. Tension \\(T \approx 900\\ \text{N};\\) Horizontal reaction \\(A_{x} \approx 650\\ \text{N};\\) Resultant \\(R \approx 1100\\ \text{N}.\\)

B. Tension \\(T \approx 750\\ \text{N};\\) Horizontal reaction \\(A_{x} \approx 800\\ \text{N};\\) Resultant \\(R \approx 1150\\ \text{N}.\\)

Correct C. Tension \(T \approx 833\ \text{N};\) Horizontal reaction \(A_{x} \approx 722\ \text{N};\) Resultant \(R \approx 994\ \text{N}.\)

D. Tension \\(T \approx 820\\ \text{N};\\) Horizontal reaction \\(A_{x} \approx 710\\ \text{N};\\) Resultant \\(R \approx 980\\ \text{N}.\\)

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 5 View Details

Points \(A(0,4)\) and \(B(6,4)\) lie on a circle that is tangent to the \(x\)-axis. Find the radius of the circle, giving your answer as an exact fraction.

A. \frac{9}{2}

B. \frac{27}{8}

C. \frac{15}{4}

Correct D. \frac{25}{8}

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 6 View Details

Given vectors \(\mathbf{a} = (2, -1, k)\) and \(\mathbf{b} = (1, 3, 4)\) are perpendicular. Let \(\mathbf{c} = \mathbf{a} + \mathbf{b}\). Find the value of \(k\) and then determine the unit vector in the direction of \(\mathbf{c}\).

Correct A. (0.539, 0.359, 0.763)

B. (0.538, 0.359, 0.764)

C. (0.600, 0.400, 0.690)

D. (0.450, 0.250, 0.860)

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 7 View Details

A fair coin is tossed. If it lands heads, a ball is drawn from Urn A containing 4 red and 6 green balls; if it lands tails, a ball is drawn from Urn B containing 5 red and 5 green balls. The first ball drawn is observed to be red. (i) Find the probability that the coin showed heads. (ii) Given that the coin showed heads, a second ball is drawn from the same urn without replacement. Find the probability that this second ball is green. (iii) Compute the overall probability that the second ball is green given that the first drawn ball was red.

A. 48\/81

B. 45\/81

C. 50\/81

Correct D. 49/81

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 8 View Details

The matrix \(A=\begin{pmatrix}2 & k\\3 & 5\end{pmatrix}\) has an inverse whose entries are all integers. Determine the integer value of \(k\) and then find \(A^{-1}\).

Correct A. k = 3; A^{-1}=\begin{pmatrix}5 & -3\\-3 & 2\end{pmatrix}

B. k = 2; A^{-1}=\begin{pmatrix}5 & -2\\-3 & 2\end{pmatrix}

C. k = 4; A^{-1}=\begin{pmatrix}5 & -3\\-4 & 2\end{pmatrix}

D. k = 3; A^{-1}=\begin{pmatrix}5 & -3\\-3 & 3\end{pmatrix}

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 9 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. \ln|x-1| + \frac{3}{2}\arctan\left(\frac{x}{2}\right) + C

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

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

D. 2\ln|x-1| + \frac{3}{2}\arctan\left(\frac{x}{4}\right) + C

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 10 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. How many different committees can be formed?

A. 1820

B. 1456

Correct C. 1876

D. 1316

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 11 View Details

Let \(\mathbf{u}= (a,2,-1)\) and \(\mathbf{v}= (3,b,4)\) be vectors in \(\mathbb{R}^3\) such that \(\mathbf{u}\) is perpendicular to \(\mathbf{v}\) and \(|\mathbf{u}| = 6\). Given that \(a>0\), find the ordered pair \((a,b)\).

A. (\\sqrt{31}, \\frac{4+3\\sqrt{31}}{2})

B. (\\-\\sqrt{31}, \\frac{4-3\\sqrt{31}}{2})

C. (\\sqrt{30}, \\frac{4-3\\sqrt{30}}{2})

Correct D. (\sqrt{31}, \frac{4-3\sqrt{31}}{2})

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 12 View Details

A linear transformation \(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\) is represented by a matrix \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\). It satisfies \(T(1,2)=(3,5)\) and \(T(2,-1)=(k,4)\). Find the value of \(k\) for which \(\det A =2\). Then compute \(T(4,1)\).

A. (k = \frac{22}{5},\; T(4,1)=\left(\frac{244}{25},\;\frac{60}{5}\right))

Correct B. (k = \frac{22}{5},\; T(4,1)=\left(\frac{244}{25},\;\frac{58}{5}\right))

C. (k = \frac{24}{5},\; T(4,1)=\left(\frac{244}{25},\;\frac{58}{5}\right))

D. (k = \frac{22}{5},\; T(4,1)=\left(\frac{240}{25},\;\frac{58}{5}\right))

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 13 View Details

Consider the matrix \(M=\begin{pmatrix}2& p&1\\4&5& q\\6&7&8\end{pmatrix}\). Find the relationship between \(p\) and \(q\) that makes \(\det M =0\). For any such pair \((p,q)\), determine the rank of \(M\).

A. q = \\frac{32p-78}{6p-12}\\;(p\\neq\\tfrac{2}{1}),\\; \\operatorname{rank}(M)=2

B. q = \\frac{30p-78}{6p-14}\\;(p\\neq\\tfrac{7}{3}),\\; \\operatorname{rank}(M)=2

C. q = \\frac{32p-78}{6p-14}\\;(p\\neq\\tfrac{7}{3}),\\; \\operatorname{rank}(M)=3

Correct D. q = \frac{32p-78}{6p-14}\;(p\neq\tfrac{7}{3}),\; \operatorname{rank}(M)=2

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 14 View Details

The curve is defined by \(x^{2}y + y^{2}=12\).\n(a) Find \(\dfrac{dy}{dx}\) at the point \((2,2)\).\n(b) Write the equation of the tangent line at that point.\n(c) Determine the x‑intercept of this tangent line.

Correct A. dy/dx = -1; tangent: y = -x + 4; x‑intercept = 4

B. dy\/dx = -1; tangent: y = -x + 4; x‑intercept = 5

C. dy\/dx = -2; tangent: y = -2x + 6; x‑intercept = 3

D. dy\/dx = -1; tangent: y = -x + 5; x‑intercept = 5

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 15 View Details

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

A. 1 + \\ln\\left(\\frac{3}{2}\\right)

B. 1 - \\ln\\left(\\frac{2}{3}\\right)

C. \\ln\\left(\\frac{3}{2}\\right) - 1

Correct D. 1 - \ln\left(\frac{3}{2}\right)

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 16 View Details

In the expansion of \((2x - 3)^7\), find the sum of the coefficients of the terms in which the power of \(x\) is a multiple of 3.

A. 20493

B. 26211

C. 23523

Correct D. 19149

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 17 View Details

A vector \(\mathbf{a}\) has magnitude 5, a vector \(\mathbf{b}\) has magnitude 7, and the angle between them is \(60^{\circ}\). A third vector \(\mathbf{c}\) is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\) and its magnitude equals the area of the parallelogram formed by \(\mathbf{a}\) and \(\mathbf{b}\). Find the magnitude of \(\mathbf{S}= \mathbf{a}+ \mathbf{b}+ \mathbf{c}\).

A. \\frac{\\sqrt{4100}}{2}

Correct B. \\frac{\\sqrt{4111}}{2}

C. \\frac{\\sqrt{4000}}{2}

D. \\frac{\\sqrt{4112}}{2}

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 18 View Details

The circle \(x^{2}+y^{2}-6x+8y+9=0\) has centre \(C\) and radius \(r\). A line passes through the point \(P(10,0)\) and is tangent to the circle with a positive slope. Determine the equation of the line in the form \(ax+by+c=0\) and give the coordinates of the point of tangency.

A. 60x-35y-600=0; (425\/65, -400\/65)

B. 48x-27y-480=0; (408\/65, -384\/65)

C. 56x-33y-560=0; (419\/65, -380\/65)

Correct D. 56x-33y-560=0; (419/65, -392/65)

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 19 View Details

A 4\,kg block rests on a horizontal surface with coefficient of kinetic friction \(\mu_k =0.2\). It is attached to a light spring of constant \(k=50\ \text{N\,m}^{-1}\) whose other end is fixed to a wall. The block is pulled 0.30\,m to the right of the equilibrium position and released from rest. At the same instant a constant horizontal force of 10\,N is applied to the block in the direction of its motion. Determine the maximum compression of the spring beyond the equilibrium position during the subsequent motion (in metres, correct to three decimal places).

Correct A. 0.386

B. 0.452

C. 0.250

D. 0.306

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 20 View Details

In the expansion of \((3x - \frac{2}{x})^{8}\), find the sum of the coefficients of the terms in which the exponent of \(x\) is a multiple of 4.

A. 195057

B. 104593

C. 113665

Correct D. 195313

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 21 View Details

Evaluate the definite integral \( \displaystyle \int_{0}^{1} \frac{x^{2}}{\sqrt{4 - x^{2}}}\,dx \).

Correct A. \frac{\pi}{3} - \frac{\sqrt{3}}{2}

B. \frac{\pi}{3} - \frac{\sqrt{6}}{2}

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

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

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 22 View Details

If \( \log_{3} x = 2 - \log_{3} 9\), find the value of \(x\).

A. 2/9

B. 1/3

Correct C. 1

D. 0

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 23 View Details

The curve is defined implicitly by \(x^{3} + y^{3} = 6xy\). Find \(\displaystyle \frac{dy}{dx}\) and \(\displaystyle \frac{d^{2}y}{dx^{2}}\) at the point \((3,3)\).

A. dy\/dx = -1, d^2y\/dx^2 = \frac{16}{3}

B. dy\/dx = -1, d^2y\/dx^2 = -\frac{8}{3}

Correct C. dy/dx = -1, d^2y/dx^2 = -\frac{16}{3}

D. dy\/dx = 1, d^2y\/dx^2 = -\frac{16}{3}

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 24 View Details

Express \( \displaystyle \frac{2x+5}{(x-1)(x+2)^{2}} \) as a sum of partial fractions.

A. \frac{5}{9}\cdot\frac{1}{x-1} - \frac{7}{9}\cdot\frac{1}{x+2} - \frac{1}{3}\cdot\frac{1}{(x+2)^{2}}

B. \frac{7}{9}\cdot\frac{1}{x-1} + \frac{7}{9}\cdot\frac{1}{x+2} - \frac{1}{3}\cdot\frac{1}{(x+2)^{2}}

Correct C. \frac{7}{9}\cdot\frac{1}{x-1} - \frac{7}{9}\cdot\frac{1}{x+2} - \frac{1}{3}\cdot\frac{1}{(x+2)^{2}}

D. \frac{7}{9}\cdot\frac{1}{x-1} - \frac{7}{9}\cdot\frac{1}{x+2} - \frac{2}{3}\cdot\frac{1}{(x+2)^{2}}

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 25 View Details

A uniform beam \(AB\) of length \(6\text{ m}\) is hinged at \(A\) to a vertical wall. The beam makes an angle of \(30^{\circ}\) with the horizontal. A cable \(CD\) is attached to the free end \(B\) and to a point \(D\) on the wall \(4\text{ m}\) above \(A\). The beam carries a downward load of \(800\text{ N}\) applied at its midpoint, and its own weight is \(200\text{ N}\) acting at the centre. Determine the tension in the cable \(CD\) (in newtons).

A. 250\sqrt{5}

Correct B. 250\sqrt{7}

C. 250\sqrt{6}

D. 200\sqrt{7}

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

neco 2019 further_mathematics | Objective

Master the Exam!

Help others prepare! Share this practice hub: