Practice gce model questions vol1 further_mathematics 2024 Past Questions

Question 1 View Details

A 10‑question multiple‑choice test has four options per question, only one of which is correct. A student guesses each answer independently. Let \(X\) be the total number of correct answers. Given that the student answered at least one of the first five questions correctly, find the probability that the student obtains at least three correct answers in the whole test. Give your answer as a fraction in lowest terms.

A. 236847/399872

B. 235847/389872

Correct C. 235847/399872

D. 240000/399872

Correct Answer: C

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Question 2 View Details

For a real number \(a>-1\) define \(J(a)=\displaystyle\int_{0}^{1} x^{a}\ln x \;dx\). (i) Show that \(J(a) = -\dfrac{1}{(a+1)^{2}}\). (ii) Using this result, evaluate \(\displaystyle\int_{0}^{1} x^{2}\ln x \;dx\).

Correct A. -1/9

B. -1/10

C. -1/8

D. -2/9

Correct Answer: A

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Question 3 View Details

Let \(X\) and \(Y\) be independent exponential random variables with means \(2\) and \(3\) respectively. Find the conditional probability \(\Pr\bigl(X+Y>5 \mid X>1\bigr)\). Express your answer in exact form.

A. 3e^{-4\/3}+2e^{-2}

Correct B. 3e^{-4/3}-2e^{-2}

C. 3e^{-4\/3}-e^{-2}

D. 2e^{-4\/3}-2e^{-2}

Correct Answer: B

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Question 4 View Details

In the expansion of \((x^{2}+3x-2)^{5}\), determine the coefficient of the term \(x^{7}\).

Correct A. 150

B. 200

C. 120

D. 180

Correct Answer: A

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Question 5 View Details

A linear transformation \(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\) is represented in the standard basis by the matrix \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\). It satisfies \(T(1,2)=(3,5)\) and \(T(2,-1)=(0,k)\), where \(k\) is a real number. Moreover, \(\det A =4\). Find the value of \(k\).

A. -20/5

Correct B. -20/3

C. 20/3

D. -4/3

Correct Answer: B

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Question 6 View Details

Given that \( \sin \theta + \sqrt{3}\cos \theta = 2\) where \(0^\circ<\theta<90^\circ\), find the exact value of \(\sin 2\theta\).

A. \(\frac{\sqrt{3}}{4}\)

B. \(\frac{1}{2}\)

C. \(\frac{2\sqrt{3}}{3}\)

Correct D. \(\frac{\sqrt{3}}{2}\)

Correct Answer: D

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Question 7 View Details

A particle moves along a straight line with position function \(s(t)=t^{3}+3t\) metres, where \(t\) is measured in seconds. Determine the time at which the particle's speed is minimum and state that minimum speed.

A. Minimum speed occurs at \(t=1\) s and the minimum speed is \(4\) m s\(^{-1}\).

Correct B. Minimum speed occurs at \(t=0\) s and the minimum speed is \(3\) m s\(^{-1}\).

C. Minimum speed occurs at \(t=-1\) s and the minimum speed is \(3\) m s\(^{-1}\).

D. Minimum speed occurs at \(t=0\) s and the minimum speed is \(2\) m s\(^{-1}\).

Correct Answer: B

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Question 8 View Details

Let \(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\) be a linear transformation such that \(T(1,2)=(3,5)\) and \(T(2,-1)=(4,-2)\).\n\n(a) Find the matrix \(A\) of \(T\) with respect to the standard basis.\n(b) Compute the determinant and trace of \(A\).\n(c) State whether \(T\) is invertible and, if it is, give the matrix \(A^{-1}\).

Correct A. A=\begin{pmatrix}\frac{11}{5}&\frac{2}{5}\\[2pt]\frac{1}{5}&\frac{12}{5}\end{pmatrix}, \;\det A=\frac{26}{5}, \;\operatorname{tr}A=\frac{23}{5}, \;A^{-1}=\begin{pmatrix}\frac{6}{13}&-\frac{1}{13}\\-\frac{1}{26}&\frac{11}{26}\end{pmatrix}

B. A=\begin{pmatrix}\frac{11}{5}&\frac{1}{5}\\[2pt]\frac{2}{5}&\frac{12}{5}\end{pmatrix}, \;\det A=\frac{27}{5}, \;\operatorname{tr}A=\frac{24}{5}, \;A^{-1}=\begin{pmatrix}\frac{6}{13}&-\frac{2}{13}\\-\frac{1}{26}&\frac{11}{26}\end{pmatrix}

C. A=\begin{pmatrix}\frac{10}{5}&\frac{2}{5}\\[2pt]\frac{1}{5}&\frac{13}{5}\end{pmatrix}, \;\det A=\frac{128}{25}, \;\operatorname{tr}A=\frac{23}{5}, \;A^{-1}=\begin{pmatrix}\frac{5}{12}&-\frac{1}{12}\\-\frac{1}{24}&\frac{9}{24}\end{pmatrix}

D. A=\begin{pmatrix}\frac{11}{5}&\frac{2}{5}\\[2pt]\frac{1}{5}&\frac{12}{5}\end{pmatrix}, \;\det A=\frac{25}{5}, \;\operatorname{tr}A=\frac{22}{5}, \;A^{-1}=\begin{pmatrix}\frac{5}{13}&-\frac{2}{13}\\-\frac{1}{26}&\frac{10}{26}\end{pmatrix}

Correct Answer: A

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Question 9 View Details

Define the function \(f:\mathbb{Z}\to\mathbb{Z}\) by \(f(x)=2x+3\).\n\n(i) Is \(f\) injective?\n(ii) Is \(f\) surjective onto \(\mathbb{Z}\)?\n(iii) Find the pre‑image of the integer \(11\).

Correct A. Injective: yes. Surjective onto \(\mathbb{Z}\): no (only odd integers are attained). Pre‑image of 11 is \(4\).

B. Injective: yes. Surjective onto \(\mathbb{Z}\): yes. Pre‑image of 11 is \(4\).

C. Injective: no. Surjective onto \(\mathbb{Z}\): no (only odd integers are attained). Pre‑image of 11 is \(4\).

D. Injective: yes. Surjective onto \(\mathbb{Z}\): no (only even integers are attained). Pre‑image of 11 is \(5\).

Correct Answer: A

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Question 10 View Details

Points \(A(1,2)\), \(B(4,6)\) and \(C(5,2)\) lie in the plane.\n\n(a) Determine the equation of the circle passing through \(A\), \(B\) and \(C\) in the form \((x-h)^{2}+(y-k)^{2}=r^{2}\).\n(b) From the external point \(P(8,5)\) a tangent is drawn to this circle. Find the length of the tangent segment \(PT\).

A. Circle: \((x-3)^{2}+(y-\frac{27}{8})^{2}=\frac{425}{64}\). Tangent length \(PT=\frac{9}{2}\) units.

B. Circle: \((x-3)^{2}+(y-\frac{29}{8})^{2}=\frac{425}{64}\). Tangent length \(PT=\frac{5}{2}\) units.

C. Circle: \((x-3)^{2}+(y-\frac{29}{8})^{2}=\frac{441}{64}\). Tangent length \(PT=\frac{9}{2}\) units.

Correct D. Circle: \((x-3)^{2}+(y-\frac{29}{8})^{2}=\frac{425}{64}\). Tangent length \(PT=\frac{9}{2}\) units.

Correct Answer: D

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Question 11 View Details

A circle has centre (C(3, -2)) and radius (5\). Find the equation of the line that passes through the point (P(8, 1)\) and is tangent to the circle. Also give the coordinates of the point of tangency.

Correct A. Equation: 8x+15y-79=0; Point of tangency: (1547/289, 697/289)

B. Equation: 8x-15y+79=0; Point of tangency: (1547/289, -697/289)

C. Equation: -8x+15y-79=0; Point of tangency: (-1547/289, 697/289)

D. Equation: 8x+15y-81=0; Point of tangency: (1545/289, 695/289)

Correct Answer: A

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Question 12 View Details

In the expansion of \((2x-3)^{7}\), find the coefficient of \(x^{3}\). Then give the sum of the coefficients of all terms in which the power of \(x\) is odd.

A. Coefficient of x^3: 22680; Sum of odd‑power coefficients: 39162

B. Coefficient of x^3: 22680; Sum of odd‑power coefficients: 38062

Correct C. Coefficient of x^3: 22680; Sum of odd‑power coefficients: 39062

D. Coefficient of x^3: 22560; Sum of odd‑power coefficients: 39062

Correct Answer: C

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Question 13 View Details

Find all complex numbers \(z\) that satisfy \(|z-2i|=3\) and \(\arg(z+1)=\frac{\pi}{4}\). Give the value of \(z\) and then compute \(z^{2}\).

A. z = (1+√17)\/2 - i(3+√17)\/2; z^2 = (-2-√17) - i(10+2√17)

Correct B. z = (1+√17)/2 + i(3+√17)/2; z^2 = (-2-√17) + i(10+2√17)

C. z = (1-√17)\/2 + i(3+√17)\/2; z^2 = (-2+√17) + i(10-2√17)

D. z = (1+√17)\/2 + i(3-√17)\/2; z^2 = (-2-√17) + i(10-2√17)

Correct Answer: B

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Question 14 View Details

Let \(A=\{1,2,3,4,5\}\). Define a relation \(R\) on \(A\) by \(aRb\) iff \(a\equiv b\pmod{2}\). Determine whether \(R\) is an equivalence relation and, if it is, list its equivalence classes.

A. Yes, R is an equivalence relation. Classes: {1,4,5} and {2,3}

B. No, R is not an equivalence relation.

C. Yes, R is an equivalence relation. Classes: {1,2,3,4,5}

Correct D. Yes, R is an equivalence relation. Classes: {1,3,5} and {2,4}

Correct Answer: D

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Question 15 View Details

A linear transformation \(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\) satisfies \(T(1,2)=(3,5)\) and \(T(2,-1)=(4,-2)\).\n(a) Find the matrix \(A\) of \(T\) with respect to the standard basis.\n(b) Determine the eigenvalues of \(A\).\n(c) State whether \(A\) is diagonalizable over the real numbers.

Correct A. A = [[11/5, 2/5],[1/5, 12/5]]; Eigenvalues: 13/5 and 2; Diagonalizable over ℝ: Yes

B. A = [[11/5, 2/5],[1/5, 12/5]]; Eigenvalues: 13/5 and 2; Diagonalizable over ℝ: No

C. A = [[12/5, 2/5],[1/5, 12/5]]; Eigenvalues: 13/5 and 2; Diagonalizable over ℝ: Yes

D. A = [[11/5, 2/5],[1/5, 12/5]]; Eigenvalues: 3 and 4; Diagonalizable over ℝ: Yes

Correct Answer: A

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Question 16 View Details

A particle of mass \(2\ \text{kg}\) moves along a straight line under a force \(F(x)=4(x+1)\ \text{N}\), where \(x\) is the displacement (in metres) from the origin. The particle starts from rest at \(x=0\).\n\n(i) Find its speed when it reaches \(x=3\ \text{m}\).\n\n(ii) Assuming the force continues to act, determine the time taken to travel from \(x=0\) to \(x=3\ \text{m}\). Give your answers to two decimal places.

A. Speed = 5.00 m/s, Time = 1.60 s

B. Speed = 6.20 m/s, Time = 1.20 s

Correct C. Speed = 5.48 m/s, Time = 1.46 s

D. Speed = 4.90 m/s, Time = 1.80 s

Correct Answer: C

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Question 17 View Details

Let \(a = 2^{\log_{2}3}\) and \(b = 3^{\log_{3}4}\). Find the value of \(\displaystyle \frac{ab}{12}\).

A. 3

B. 2

Correct C. 1

D. 0

Correct Answer: C

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Question 18 View Details

Solve for \(x\) in the equation \(3^{2x}=27^{\,x-1}\).

Correct A. 3

B. 9

C. 1/3

D. 6

Correct Answer: A

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Question 19 View Details

A particle of mass \(0.5\ \text{kg}\) is attached to a light string and moves in a vertical circle of radius \(1.2\ \text{m}\). It is given a speed of \(8\ \text{m s}^{-1}\) at the lowest point of the circle.\n\n(i) Determine the speed of the particle at the highest point of the circle (neglecting air resistance).\n\n(ii) Find the tension in the string when the particle is at the highest point. Give your answer to two decimal places.

A. Speed = 6.00 m/s, Tension = 4.00 N

Correct B. Speed = 4.12 m/s, Tension = 2.17 N

C. Speed = 5.00 m/s, Tension = 3.00 N

D. Speed = 3.00 m/s, Tension = 1.00 N

Correct Answer: B

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Question 20 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 10 widgets is inspected, each widget being independent of the others.\n\nWhat is the probability that exactly two widgets in the sample are defective and that at least one of the defective widgets is of type B? Give your answer as a decimal correct to four significant figures.

A. 0.02890

Correct B. 0.03045

C. 0.0355

D. 0.02500

Correct Answer: B

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Question 21 View Details

A committee of five members is to be formed from 8 men and 6 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?

A. 3360

Correct B. 3760

C. 3120

D. 4200

Correct Answer: B

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Question 22 View Details

Vectors \(\mathbf{u}\) and \(\mathbf{v}\) in \(\mathbb{R}^3\) satisfy \(|\mathbf{u}| = 5\), \(\mathbf{u}\cdot\mathbf{v}=8\) and \(|\mathbf{u}+\mathbf{v}| = \sqrt{61}\). (i) Find \(|\mathbf{v}|\). (ii) Determine the angle between \(\mathbf{u}\) and \(\mathbf{v}\) (to the nearest degree). (iii) Find the magnitude of the cross‑product \(|\mathbf{u}\times\mathbf{v}|\).

A. |v| = 3√5, angle ≈ 80°, |u×v| = 5√13

B. |v| = √20, angle ≈ 45°, |u×v| = √200

C. |v| = √13, angle ≈ 30°, |u×v| = √85

Correct D. |v| = 2√5, angle ≈ 69°, |u×v| = 2√109

Correct Answer: D

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Question 23 View Details

The cubic polynomial \(P(x)=x^{3}+ax^{2}+bx+6\) has \((x-1)\) as a factor and satisfies \(P(2)=20\). Find the integers \(a\) and \(b\).

A. a = 8, b = -15

B. a = 12, b = -19

Correct C. a = 10, b = -17

D. a = 10, b = -15

Correct Answer: C

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Question 24 View Details

A uniform rod \(AB\) of length \(6\text{ m}\) is hinged at \(A\) to a vertical wall. The rod makes an angle of \(30^{\circ}\) with the horizontal. A weight of \(200\text{ N}\) is attached at the midpoint of the rod. A cable joins the end \(B\) to a point \(C\) on the wall directly above \(A\) at a height of \(4\text{ m}\). Determine (i) the tension in the cable and (ii) the horizontal and vertical components of the reaction at the hinge \(A\).

A. T = 55√7 N (≈146 N); R_x = 80√3 N (≈138 N) rightward, R_y = 185 N upward

Correct B. T = 50√7 N (≈132 N); R_x = 75√3 N (≈130 N) rightward, R_y = 175 N upward

C. T = 50√8 N (≈141 N); R_x = 70√3 N (≈121 N) rightward, R_y = 180 N upward

D. T = 45√7 N (≈119 N); R_x = 70√3 N (≈121 N) rightward, R_y = 165 N upward

Correct Answer: B

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Question 25 View Details

Ten people, consisting of five men and five women, are to be seated round a circular table. The men must occupy every alternate seat. Two particular women, Alice and Brenda, must not be seated with exactly one man between them. How many distinct seating arrangements are possible?

A. 240

B. 432

Correct C. 288

D. 360

Correct Answer: C

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