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gce model questions vol1 2023 further_mathematics | Objective

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Question 1 View Details

Let \(A=\{1,2,3,4\}\) and \(B=\{a,b,c\}\). Define the relation \(R=\{(1,a),(2,b),(3,c),(4,a)\}\). Determine whether \(R\) is a function from \(A\) to \(B\). If it is, give the image of the subset \(\{2,4\}\) of \(A\).

A. Yes; \\{a,c\\}

Correct B. Yes; \{a,b\}

C. No; \\{a,b\\}

D. Yes; \\{b\\}

Correct Answer: B

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Question 2 View Details

Define \(f:\{1,2,3,4,5\}\to\{0,1\}\) by \(f(x)=1\) if \(x\) is odd and \(f(x)=0\) otherwise. Define \(g:\{0,1\}\to\{a,b\}\) by \(g(0)=b\) and \(g(1)=a\).\n(i) Is \(f\) onto?\n(ii) How many elements are in the pre‑image \(f^{-1}(\{0\})\)?\n(iii) Compute \((g\circ f)(3)\).

Correct A. Yes; 2; a

B. Yes; 2; b

C. Yes; 3; a

D. No; 2; a

Correct Answer: A

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Question 3 View Details

Simplify \(\displaystyle \frac{2^{3}\sqrt{50}}{5^{2}}\) and then evaluate \(\log_{10}\) of the simplified result, giving the answer to two decimal places.

Correct A. 0.35

B. 0.55

C. 0.45

D. 0.25

Correct Answer: A

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Question 4 View Details

Let \(A=\begin{pmatrix}2&k\\k&5\end{pmatrix}\) where \(k\) is an integer. Find all integer values of \(k\) for which \(A\) is invertible and the inverse matrix \(A^{-1}\) has only integer entries.

A. {-2, 2}

Correct B. {-3, 3}

C. {-3, 4}

D. {-4, 4}

Correct Answer: B

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Question 5 View Details

Solve \(\log_{3}(x)+\log_{3}(x-6)=2\) and then evaluate \(x^{2}-6x\).

A. 0

B. 18

Correct C. 9

D. -9

Correct Answer: C

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Question 6 View Details

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

A. 1 - \\ln\\frac{5}{2}

B. 1 - \\ln\\frac{3}{4}

Correct C. 1 - \ln\frac{3}{2}

D. \\ln\\frac{3}{2} - 1

Correct Answer: C

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Question 7 View Details

The circle \((x-2)^{2}+(y+1)^{2}=25\) and the point \(P(8,3)\) are given. A line through \(P\) is drawn tangent to the circle. Find the \(x\)-coordinate of the point where this tangent meets the \(x\)-axis.

A. \\frac{68-5\\sqrt{15}}{5}

B. \\frac{68-5\\sqrt{3}}{7}

Correct C. \frac{68-5\sqrt{15}}{7}

D. \\frac{68+5\\sqrt{15}}{7}

Correct Answer: C

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Question 8 View Details

Let \(w\) be a complex number satisfying \(w^{3}=8i\) with \(\operatorname{Re}(w)>0\). Compute \(\displaystyle \frac{w^{2}+\overline{w}}{w-i}\) and express your answer in the form \(a+bi\).

A. (1-2\/\sqrt{3})+i\left(2-1\/\sqrt{3}\right)

B. (1-2\/\sqrt{3})+i\left(2+1\/\sqrt{3}\right)

C. (1+2\/\sqrt{3})+i\left(2+1\/\sqrt{3}\right)

Correct D. (1+2/\sqrt{3})+i\left(2-1/\sqrt{3}\right)

Correct Answer: D

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Question 9 View Details

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

A. 4800

Correct B. 5600

C. 560

D. 6400

Correct Answer: B

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Question 10 View Details

A \(3\times3\) matrix \(A\) with integer entries satisfies \(A^{2}=2A\). Determine all possible values of \(\det(A)\).

Correct A. 0 or 8

B. 0 or 4

C. 2 or 8

D. 0 or 16

Correct Answer: A

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Question 11 View Details

A particle moves in a straight line with a time‑dependent acceleration \(a(t)=\frac{k}{t}\) for \(2\le t\le5\) seconds. At \(t=2\) s its velocity is \(3\ \text{m s}^{-1}\) and at \(t=5\) s its velocity is \(9\ \text{m s}^{-1}\). Determine the distance travelled by the particle between \(t=2\) s and \(t=5\) s. Give your answer in metres, correct to two decimal places.

A. 17.80 m

B. 22.10 m

Correct C. 19.35 m

D. 21.40 m

Correct Answer: C

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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 is known that \(T\bigl(1,2\bigr)=(3,5)\) and \(T\bigl(2,-1\bigr)=(k,4)\). Moreover, the transformation scales areas by a factor of \(3\). Find the value of \(k\) and give the matrix \(A\).

Correct A. k = 27/5; A = [[69/25, 3/25], [13/5, 6/5]]

B. k = 23/5; A = [[65/25, 4/25], [12/5, 7/5]]

C. k = 31/5; A = [[71/25, 2/25], [14/5, 5/5]]

D. k = 27/4; A = [[68/25, 3/20], [13/5, 6/5]]

Correct Answer: A

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Question 13 View Details

Consider the curve \(y=\ln\bigl(x^{2}+1\bigr)-x\). Find all real values of \(x\) at which the tangent to the curve passes through the point \((0,2)\). Give your answers correct to two decimal places.

A. -8.00, 8.00

B. -7.50, 7.50

Correct C. -7.21, 7.21

D. -6.85, 6.85

Correct Answer: C

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Question 14 View Details

Evaluate the integral \(I=\displaystyle\int_{0}^{\ln 2}\frac{e^{2x}}{1+e^{x}}\,dx\). Then determine the constant \(C\) such that the function \(F(x)=\displaystyle\int_{0}^{x}\frac{e^{2t}}{1+e^{t}}\,dt+C\) satisfies \(F(\ln 2)=5\). Give \(I\) and \(C\) in exact form.

A. I = 1-ln(2/3); C = 4+ln(3/2)

B. I = 2+ln(2/3); C = 3+ln(3/2)

Correct C. I = 1+ln(2/3); C = 4+ln(3/2)

D. I = 1+ln(3/2); C = 4+ln(2/3)

Correct Answer: C

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Question 15 View Details

The word "STATISTICS" consists of the letters S(3), T(3), A(1), I(2) and C(1). In how many distinct arrangements of all ten letters are (i) no two S's adjacent and (ii) the three T's are not all together? Give the total number of arrangements satisfying both conditions.

Correct A. 22320

B. 23220

C. 21320

D. 22420

Correct Answer: A

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Question 16 View Details

The polynomial (P(x)=x^{3}+a x^{2}+b x+6) satisfies that \((x-2)\) is a factor and that the remainder when \(P(x)\) is divided by \((x+1)\) is \(4\). Find the value of \(a+b\).

A. -14/3

B. 13/3

Correct C. -13/3

D. -13/2

Correct Answer: C

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Question 17 View Details

If \(2^{x}= \dfrac{8^{1/3}\,\sqrt{32}}{2^{3/2}}\), determine the value of \(x\).

A. 1

B. 3

Correct C. 2

D. 4

Correct Answer: C

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Question 18 View Details

Let \(\mathbf{a}= (2,-1,3)\) and \(\mathbf{b}= (k,4,-2)\). Define \(\mathbf{c}= \mathbf{a}+ \lambda \mathbf{b}\). The vector \(\mathbf{c}\) is orthogonal to \(\mathbf{d}= (1,2,k)\) and has magnitude \(|\mathbf{c}|= \sqrt{14}\). Find the values of \(k\) and \(\lambda\).

A. k=0, \lambda=2

Correct B. k=0, \lambda=0

C. k=2, \lambda=-1

D. k=1, \lambda=0

Correct Answer: B

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Question 19 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 \(30^{\circ}\) above the horizontal. A weight of \(800\text{ N}\) hangs from its end \(B\). A cable is attached to the beam at point \(D\) located \(2\text{ m}\) from \(A\) along the beam and runs vertically upward, exerting a tension \(T\). Determine the tension \(T\) in the cable and the magnitude and direction of the reaction force at the hinge \(A\).

A. T = 3000 N; hinge reaction = 1600 N upwards

Correct B. T = 2400 N; hinge reaction = 1600 N downwards

C. T = 2400 N; hinge reaction = 1200 N downwards

D. T = 1800 N; hinge reaction = 2000 N upwards

Correct Answer: B

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Question 20 View Details

Evaluate the indefinite integral \(\displaystyle \int \frac{2x+3}{(x-1)(x^{2}+4)}\,dx\).

A. ln|x-1| - (1\/2)ln(x^2+4) + (1\/3)arctan(x\/2) + C

Correct B. ln|x-1| - (1/2)ln(x^2+4) + (1/2)arctan(x/2) + C

C. ln|x-1| - (1\/2)ln(x^2+4) - (1\/2)arctan(x\/2) + C

D. ln|x-1| + (1\/2)ln(x^2+4) + (1\/2)arctan(x\/2) + C

Correct Answer: B

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Question 21 View Details

Find the integer value of \(k\) such that the polynomial \(2x^{3}-5x^{2}+kx+10\) is exactly divisible by \(x+2\).

A. 13

B. -28

C. 7

Correct D. -13

Correct Answer: D

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Question 22 View Details

Let \(A=\begin{pmatrix}2& a& 1\\0& b& 3\\4& 5& c\end{pmatrix}\) where \(a,b,c\) are positive integers with \(a<b<c\). If \(\det A = 24\), find the minimum possible value of \(a+b+c\).

A. 14

B. 13

Correct C. 12

D. 15

Correct Answer: C

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Question 23 View Details

A factory produces components of type A (60% of output) and type B (40%). Defect rates are 5% for A and 2% for B. An inspection detects a defect with probability 0.9 for a defective A component and 0.8 for a defective B component; non‑defective components always pass. After a component is randomly selected and *passes* inspection, it is used in a device. If the component is non‑defective, the device fails with probability 0.01 for type A and 0.02 for type B. If the component is defective (but undetected), the device fails with probability 0.5 for type A and 0.7 for type B. What is the probability that the device fails, given that the component passed inspection? Give your answer as a decimal rounded to four significant figures.

A. 0.0150

Correct B. 0.0167

C. 0.0185

D. 0.0123

Correct Answer: B

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Question 24 View Details

In a plane, vectors \(\mathbf{a}\) and \(\mathbf{b}\) satisfy \(|\mathbf{a}|=3\), \(|\mathbf{b}|=4\) and the angle between them is \(60^{\circ}\). Define \(\mathbf{c}=2\mathbf{a}-\mathbf{b}\) and \(\mathbf{d}=\mathbf{a}\times\mathbf{b}\) (a vector perpendicular to the plane). Find the volume of the parallelepiped formed by the three vectors \(\mathbf{c},\ \mathbf{a}+2\mathbf{b},\ \mathbf{d}\).

Correct A. 540

B. 600

C. 480

D. 720

Correct Answer: A

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Question 25 View Details

Evaluate the integral \(\displaystyle \int \frac{3x^{2}+5x+2}{(x+1)(x^{2}+4)}\,dx\) by first decomposing the integrand into partial fractions.

A. (3\/2)\ln\bigl(x^{2}+5\bigr)+\arctan\frac{x}{2}+C

B. (1\/2)\ln\bigl(x^{2}+4\bigr)+\arctan\frac{x}{2}+C

Correct C. (3/2)\ln\bigl(x^{2}+4\bigr)+\arctan\frac{x}{2}+C

D. (3\/2)\ln\bigl(x^{2}+4\bigr)-\arctan\frac{x}{2}+C

Correct Answer: C

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