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gce model questions vol1 2019 further_mathematics | Objective

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Question 1 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 \(3\).

Correct A. 19149

B. 19148

C. 19249

D. 18149

Correct Answer: A

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Question 2 View Details

A particle moves in a straight line with acceleration \(a(t)=6t-4\) m s\(^{-2}\) for \(0\le t\le3\) s. At \(t=3\) s the particle then continues with constant speed for a further \(5\) s, and the total distance travelled from \(t=0\) to \(t=8\) s is \(200\) m. Find the particle's speed at \(t=0\).

Correct A. 14.5

B. 13.5

C. 12.5

D. 15.5

Correct Answer: A

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Question 3 View Details

Seven distinct books A, B, C, D, E, F, G are to be placed on a shelf. Book C must be at one of the two ends and books A and B must not be adjacent. In how many different arrangements can the books be placed?

Correct A. 960

B. 720

C. 1080

D. 840

Correct Answer: A

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Question 4 View Details

Let the function \(f:\{1,2,3,4\}\to\{a,b,c\}\) be defined by \(f(1)=a,\;f(2)=b,\;f(3)=a,\;f(4)=c\). Determine the pre‑image \(f^{-1}(\{a,c\})\).

A. {1,2,4}

Correct B. {1,3,4}

C. {2,3,4}

D. {1,3}

Correct Answer: B

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Question 5 View Details

Let \(A=\begin{pmatrix}k&2\\[2pt]3&5\end{pmatrix}\) and \(B=\begin{pmatrix}1&4\\[2pt]k&-1\end{pmatrix}\). The product \(AB\) is singular, but \(A\) is invertible. Find the value of \(k\) and then give \(\det A\) for that \(k\).

A. k = -\frac14, \det A = -\frac{19}{4}

B. k = \frac12, \det A = -\frac{31}{4}

C. k = \frac14, \det A = -\frac{29}{4}

Correct D. k = -\frac14, \det A = -\frac{29}{4}

Correct Answer: D

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Question 6 View Details

A line passes through the point (P(6,5)) and is tangent to the circle \((x-1)^2 + (y+2)^2 = 20\). Find, correct to two decimal places, the sum of the \(x\)-coordinates of the points of tangency.

Correct A. 4.70

B. 3.85

C. 5.12

D. 6.00

Correct Answer: A

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Question 7 View Details

Evaluate the integral \(\displaystyle \int_{0}^{1} \frac{x^{2}}{(x+1)(x^{2}+1)}\,dx\) by expressing the integrand as a sum of partial fractions.

Correct A. \frac{3}{4}\ln 2 - \frac{\pi}{8}

B. \frac{3}{4}\ln 2 + \frac{\pi}{8}

C. \frac{5}{4}\ln 2 - \frac{\pi}{8}

D. \frac{1}{2}\ln 2 - \frac{\pi}{6}

Correct Answer: A

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Question 8 View Details

On the set \(A=\{1,2,3,4\}\) define a binary operation \(*\) by \(a * b = (a+b)\bmod 5\), where a result of \(0\) is replaced by \(5\). Determine whether \(*\) has an identity element in \(A\), and compute \(3 * 4\).

A. Identity is 5; 2

B. No identity; 3

Correct C. No identity; 2

D. No identity; 5

Correct Answer: C

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Question 9 View Details

Line \(L_{1}\) passes through \(A(1,2,3)\) with direction vector \(\mathbf{d}_{1}=(2,-1,4)\). Line \(L_{2}\) passes through \(B(4,0,-1)\) with direction vector \(\mathbf{d}_{2}=(1,3,-2)\). Find the shortest distance between the two lines.

A. 73/\sqrt{213}

Correct B. 74/\sqrt{213}

C. 75/\sqrt{213}

D. 74/\sqrt{212}

Correct Answer: B

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Question 10 View Details

Given the matrix \(A=\begin{pmatrix}2&1&0\\-1&3&1\\4&0&5\end{pmatrix}\) and the column vector \(\mathbf{v}=\begin{pmatrix}7\\-2\\3\end{pmatrix}\), (a) compute \(\det A\); (b) if the inverse of \(A\) exists, find the vector \(\mathbf{x}=A^{-1}\mathbf{v}\).

A. (118/39, 36/39, -71/39)

B. (118/39, 37/39, 71/39)

Correct C. (118/39, 37/39, -71/39)

D. (119/39, 37/39, -71/39)

Correct Answer: C

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Question 11 View Details

The function \(f(x)=x^{3}-3x^{2}+kx+1\) has a tangent at \(x=1\) that passes through the point \((2,10)\). Determine the value of the derivative \(f'(1)\).

A. 5

Correct B. 4

C. 2

D. 6

Correct Answer: B

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Question 12 View Details

A committee of five members is to be chosen from eight men and six women. The chairperson must be a man and the committee must contain at least two women. In how many different ways can the committee be formed?

Correct A. 3760

B. 3840

C. 3520

D. 4000

Correct Answer: A

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Question 13 View Details

Let \(P(x)=x^{3}+ax^{2}+bx+c\) be a monic cubic polynomial satisfying \(P(1)=4\), \(P(2)=-1\) and \(P(3)=5\). Find the value of \(P(0)\).

Correct A. 14

B. 10

C. 12

D. 16

Correct Answer: A

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Question 14 View Details

In a factory, 60% of the items are produced by machine A and 40% by machine B. The proportion of defective items is 2% for machine A and 5% for machine B. An item selected at random is found to be defective. A second item is then selected from the same machine that produced the first item. What is the probability that the second item is also defective? Express your answer as a fraction in lowest terms.

A. 29/800

B. 33/800

C. 31/900

Correct D. 31/800

Correct Answer: D

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Question 15 View Details

A bag contains 5 red, 4 blue and 3 green balls. Two balls are drawn successively without replacement. Given that at least one of the two balls drawn is red, find the probability that the first ball drawn was red and the second ball was either blue or green. Express your answer as a fraction in lowest terms.

A. 1/3

B. 2/9

Correct C. 7/18

D. 5/12

Correct Answer: C

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Question 16 View Details

The function (f) is defined by (f(x)=\begin{cases} x^{3}+p x , & x\le 1,\\ q x^{2}+r , & x>1 .\end{cases}\) Find the values of the constants (p) and (q) that make (f) differentiable at (x=1) and such that (f'(2)=10\). Then give the value of (r\).

A. p = 2, q = 3, r = 1/2

B. p = 2, q = 5/2, r = 3/2

Correct C. p = 2, q = 5/2, r = 1/2

D. p = 3, q = 5/2, r = 1/2

Correct Answer: C

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Question 17 View Details

The 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 is known that \(T(1,2)=(3,5)\) and \(T(2,-1)=(k,4)\). Moreover, the area scaling factor of \(T\) is \(6\) (that is, \(|\det A|=6\)). Determine the value of \(k\) and the matrix \(A\).

A. k = 42/5, A = [[100/25, -12/25],[13/5, 6/5]]

B. k = 44/5, A = [[99/25, -12/25],[13/5, 6/5]]

Correct C. k = 42/5, A = [[99/25, -12/25],[13/5, 6/5]]

D. k = 42/5, A = [[99/25, -13/25],[13/5, 6/5]]

Correct Answer: C

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Question 18 View Details

The cubic polynomial \(P(x)=x^{3}+ax^{2}+bx-6\) has both \((x-2)\) and \((x+1)\) as factors. Find the coefficients \(a\) and \(b\).

Correct A. a = 2, b = -5

B. a = 2, b = 5

C. a = -2, b = -5

D. a = -2, b = 5

Correct Answer: A

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Question 19 View Details

Simplify the expression \(\sqrt{2^{6}\cdot 3^{4}}\) and write the result in the form \(2^{a}3^{b}\). State the values of \(a\) and \(b\).

Correct A. a = 3, b = 2

B. a = 4, b = 2

C. a = 2, b = 3

D. a = 3, b = 3

Correct Answer: A

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Question 20 View Details

Let \(M=\begin{pmatrix}1&2&3\\0&k&1\\4&5&6\end{pmatrix}\). Find the value of the parameter \(k\) for which the \((2,2)\) entry of the inverse matrix \(M^{-1}\) equals \(\frac13\). Then give the determinant of \(M\) for that value of \(k\).

A. k = 7/2, determinant = -20

B. k = 5/2, determinant = -18

Correct C. k = 7/2, determinant = -18

D. k = 3/2, determinant = -12

Correct Answer: C

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Question 21 View Details

A particle of mass \(2\ \text{kg}\) is projected up a plane that makes an angle of \(30^{\circ}\) with the horizontal. The coefficient of kinetic friction between the particle and the plane is \(0.2\). In addition, a constant resisting force of magnitude \(5\ \text{N}\) acts opposite to the motion along the plane. Determine the minimum initial speed (in \(\text{m s}^{-1}\), correct to two decimal places) required for the particle to just reach a vertical height of \(5\ \text{m}\) above its starting point.

Correct A. 13.49

B. 15.02

C. 12.34

D. 14.87

Correct Answer: A

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Question 22 View Details

Evaluate the definite integral \(\displaystyle \int_{0}^{1} \frac{2x^{2}+3x+5}{(x+1)(x^{2}+2)}\,dx\). Give your answer in exact form.

Correct A. ln2 + (1/3)ln3 + (7/(3\sqrt{2}))\arctan\left(\frac{1}{\sqrt{2}}\right)

B. ln2 + (1\/3)ln3 + (5\/(3\\sqrt{2}))\\arctan\\left(\\frac{1}{\\sqrt{2}}\\right)

C. ln2 + (2\/3)ln3 + (7\/(3\\sqrt{2}))\\arctan\\left(\\frac{1}{\\sqrt{2}}\\right)

D. ln2 + (1\/3)ln3 + (7\/(3\\sqrt{2}))\\arctan\\left(\\frac{2}{\\sqrt{2}}\\right)

Correct Answer: A

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Question 23 View Details

Let \(A=(1+2x)^{5}\) and \(B=(1-x)^{4}\). Find the coefficient of \(x^{3}\) in the expansion of the product \(A\cdot B\).

Correct A. -24

B. -12

C. -48

D. -30

Correct Answer: A

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Question 24 View Details

The circles \(C_{1}: x^{2}+y^{2}=25\) and \(C_{2}:(x-6)^{2}+y^{2}=16\) intersect. Determine the length of their common chord, giving your answer in simplest surd form.

A. 3\sqrt{7}/2

Correct B. 5\sqrt{7}/2

C. 5\sqrt{5}/2

D. 5\sqrt{7}/3

Correct Answer: B

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Question 25 View Details

Let \(z\) be the principal cube root of \(8i\) (i.e., the root with argument \(\pi/6\)). Compute \(\displaystyle S=\frac{1}{z}+\frac{1}{z^{2}}+\frac{1}{z^{4}}\) and express your answer in the form \(a+bi\) with exact surds.

A. (8\sqrt{3}+3)\/32 + i*(8+5\sqrt{3})\/32

B. (8\sqrt{3}+3)\/16 - i*(8+5\sqrt{3})\/16

Correct C. (8\sqrt{3}+3)/32 - i*(8+5\sqrt{3})/32

D. (8\sqrt{3}-3)\/32 - i*(8-5\sqrt{3})\/32

Correct Answer: C

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