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gce model questions vol1 2022 further_mathematics | Objective

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Question 1 View Details

A game uses a fair six‑sided die. If the result is 1 or 2 you win £5. If the result is 3, 4 or 5 you win an amount in pounds equal to the face value. If the result is 6 you lose £x. The expected gain from a single throw is £2. (i) Find the value of x. (ii) Using this value, determine the variance of the gain. (iii) Finally, calculate the probability that a single throw yields a gain of at least £4.

A. x = 10; variance = 90/3; probability = 2/3

B. x = 12; variance = 92/3; probability = 3/4

C. x = 8; variance = 80/3; probability = 1/2

Correct D. x = 10; variance = 88/3; probability = 2/3

Correct Answer: D

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Question 2 View Details

Solve \((z^{2}+1)^{3}=8i\) for the complex number \(z\). Then find the sum of the moduli of all distinct solutions.

A. 2\big((5-2\sqrt3)^{1\/4}+(5+2\sqrt3)^{1\/4}+5^{1\/2}\big)

Correct B. 2\big((5-2\sqrt3)^{1/4}+(5+2\sqrt3)^{1/4}+5^{1/4}\big)

C. 2\big((5-2\sqrt3)^{1\/4}+(5+2\sqrt3)^{1\/4}-5^{1\/4}\big)

D. 2\big((5-2\sqrt3)^{1\/2}+(5+2\sqrt3)^{1\/4}+5^{1\/4}\big)

Correct Answer: B

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Question 3 View Details

Let \(A\) be a \(2\times2\) matrix with integer entries satisfying \(A^{2}=3A-2I\), where \(I\) is the identity matrix. (i) Determine \(\operatorname{tr}(A)\) and \(\det(A)\). (ii) Express \(A^{5}\) in the form \(pA+qI\) and give the integers \(p\) and \(q\).

A. tr(A)=4, det(A)=2; A^5 = 31A - 29I

B. tr(A)=3, det(A)=1; A^5 = 30A - 30I

C. tr(A)=2, det(A)=2; A^5 = 31A - 31I

Correct D. tr(A)=3, det(A)=2; A^5 = 31A - 30I

Correct Answer: D

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Question 4 View Details

Consider the matrix \(M=\begin{pmatrix}2&-1&0\\ k&1&1\\ 0&0&m\end{pmatrix}\) with integers \(k,m\). Suppose \(M\) is invertible and its inverse has only integer entries. (i) Find all possible integer pairs \((k,m)\). (ii) For each pair, state \(\det(M)\).

A. (k,m)=(-1,2), (-1,-1), (-3,2), (-3,-1); det(M)=4 when m=2 and -4 when m=-2

Correct B. (k,m)=(-1,1), (-1,-1), (-3,1), (-3,-1); det(M)=2 when m=1 and -2 when m=-1

C. (k,m)=(-1,1), (-2,1), (-3,1), (-3,-1); det(M)=2 when m=1 and -2 when m=-1

D. (k,m)=(-1,1), (-1,-1), (-2,1), (-2,-1); det(M)=2 when m=1 and -2 when m=-1

Correct Answer: B

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Question 5 View Details

In the expansion of \((1+p)^{n}\) the coefficient of \(p^{3}\) is 20 and the coefficient of \(p^{4}\) is 15. (i) Find the integer \(n\). (ii) Using this value of \(n\), compute the sum of the coefficients of the terms \(p^{0},p^{1},\dots ,p^{5}\) in the expansion.

A. n = 7; sum = 120

Correct B. n = 6; sum = 63

C. n = 8; sum = 219

D. n = 5; sum = 32

Correct Answer: B

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Question 6 View Details

Find the positive constant \(a\) such that \(\displaystyle \int_{0}^{\ln 2} \frac{dx}{a+\sin x}= \frac{\pi}{4}\).

Correct A. \sqrt{2}

B. \frac{1}{\sqrt{2}}

C. \sqrt{3}

D. 2

Correct Answer: A

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Question 7 View Details

Given that \(\sin\theta + \sin 3\theta = \frac{3}{2}\), determine the exact value of \(\cos 2\theta\).

A. 3\/4

Correct B. 1/2

C. -1\/2

D. 0\/1

Correct Answer: B

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Question 8 View Details

A car starts from rest and accelerates along a straight road with acceleration \(a(t)=3t\) (in \(\text{m s}^{-2}\)) for \(t\) seconds. It then continues at the speed attained for exactly \(5\) s, after which it brakes with constant deceleration and comes to rest after travelling a further \(40\) m. If the total distance covered from start to stop is \(192\) m, find the total time of the journey in seconds, expressed as an exact fraction.

A. 13/2

Correct B. 37/3

C. 35/3

D. 38/3

Correct Answer: B

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Question 9 View Details

Evaluate the integral \(\displaystyle I=\int_{0}^{1}\frac{\ln(1+x)}{1+x^{2}}\,dx\) and give your answer in exact form.

A. (π/6)ln 2

Correct B. (\pi/8)\ln 2

C. (π/8)ln 3

D. (π/4)ln 2

Correct Answer: B

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Question 10 View Details

A particle of mass \(m\) moves along a straight line under the action of a force \(F(x)=A-Bx\) (with \(A,B>0\)) and a constant resistive force of magnitude \(R\) opposing the motion. The particle starts from rest at \(x=0\). Determine (i) the distance \(x_{\max}\) at which the particle's speed is maximum, and (ii) the corresponding maximum speed \(v_{\max}\), in terms of \(A,B,R\) and \(m\).

Correct A. x_max = (A - R)/B, v_max = (A - R)/sqrt(B m)

B. x_max = (A + R)\/B, v_max = (A + R)\/sqrt(B m)

C. x_max = (A - R)\/B, v_max = (A - R)\/(B sqrt(m))

D. x_max = (A - R)\/(2B), v_max = (A - R)\/sqrt(2 B m)

Correct Answer: A

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Question 11 View Details

The curve is given by \(y = x^{3} - 6x^{2} + 9x + 2\). The line through the points \((1,4)\) and \((4,10)\) has a certain slope. Find the x‑coordinate of the point on the curve where the tangent line is parallel to this line and has x > 2.

A. 2+\frac{\sqrt{12}}{3}

Correct B. 2+\frac{\sqrt{15}}{3}

C. 2+\frac{\sqrt{15}}{2}

D. 2-\frac{\sqrt{15}}{3}

Correct Answer: B

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Question 12 View Details

A linear transformation \(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\) is represented by a matrix \(A\). It is known that \(T(1,2) = (3,5)\) and \(T(2,-1) = (0,-3)\). Determine the matrix \(A\), then find the image of the vector \((4,1)\) under the inverse transformation \(T^{-1}\).

A. (44\/9,7\/9)

Correct B. (46/9,7/9)

C. (46\/9,8\/9)

D. (46\/9,5\/9)

Correct Answer: B

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Question 13 View Details

In three‑dimensional space let \(\mathbf{a} = (2,-1,3)\) and \(\mathbf{b} = (1,4,-2)\). Let \(\mathbf{c} = (5,0,1)\). A unit vector \(\mathbf{u}\) is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\). Compute the absolute value of the dot product \(|\mathbf{u}\cdot\mathbf{c}|\).

A. 41/\sqrt{231}

B. 41/\sqrt{115}

C. \sqrt{230}/41

Correct D. 41/\sqrt{230}

Correct Answer: D

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Question 14 View Details

Let the function \(f:\{1,2,3,4\}\to\{a,b,c\}\) be defined by \(f(1)=a,\;f(2)=b,\;f(3)=c,\;f(4)=b\). Let \(g:\{a,b,c\}\to\{0,1\}\) be defined by \(g(a)=0,\;g(b)=1,\;g(c)=0\). Find the value of the composition \((g\circ f)(3)\).

A. 1

Correct B. 0

C. -1

D. 2

Correct Answer: B

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Question 15 View Details

The line \(L\) passes through the point \(P_{0}=(1,2,3)\) and has direction vector \(\mathbf{d}=(4,-2,1)\). Determine the shortest distance from the point \(Q=(5,-1,4)\) to the line \(L\).

A. \sqrt{\frac{18}{21}}

Correct B. \sqrt{\frac{17}{21}}

C. \sqrt{\frac{17}{20}}

D. \sqrt{\frac{16}{21}}

Correct Answer: B

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Question 16 View Details

In three dimensions, vectors \(\mathbf{a}\) and \(\mathbf{b}\) satisfy \(|\mathbf{a}| = 5\), \(|\mathbf{b}| = 7\) and the projection of \(\mathbf{a}\) onto \(\mathbf{b}\) is \(3\). Find the magnitude of \((\mathbf{a}\times\mathbf{b})\times(\mathbf{a}+\mathbf{b})\).

Correct A. 112\sqrt{2}

B. 112\sqrt{5}

C. 112\sqrt{3}

D. 64\sqrt{2}

Correct Answer: A

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Question 17 View Details

Let \(P(x)=x^{3}+ax^{2}+bx+6\). The factor theorem tells us that \((x+1)\) is a factor of \(P(x)\) and the remainder when \(P(x)\) is divided by \((x-2)\) is \(5\). Determine the values of \(a\) and \(b\).

Correct A. a = -19/6, b = 11/6

B. a = -17\/6, b = 13\/6

C. a = -19\/6, b = 12\/6

D. a = -20\/6, b = 10\/6

Correct Answer: A

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Question 18 View Details

A uniform beam \(AB\) 6 m long is hinged at \(A\) and rests on a roller at \(B\). A downward point load of 10 kN acts 2 m from \(A\). A cable attached at point \(C\) (4 m from \(A\)) makes an angle of \(30^{\circ}\) above the horizontal and carries tension \(T\). Determine (i) the tension \(T\) in the cable, (ii) the vertical reaction at \(B\), and (iii) the horizontal reaction at the hinge \(A\).

A. T = 12 kN; vertical reaction at B = 4 kN upward; horizontal reaction at A = 6\sqrt{3} kN

Correct B. T = 10 kN; vertical reaction at B = 5 kN upward; horizontal reaction at A = 5\sqrt{3} kN

C. T = 10 kN; vertical reaction at B = 5 kN downward; horizontal reaction at A = 5\sqrt{3} kN

D. T = 8 kN; vertical reaction at B = 6 kN upward; horizontal reaction at A = 4\sqrt{3} kN

Correct Answer: B

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Question 19 View Details

Consider the matrix \(A = \begin{pmatrix} 2 & k & 1 \\ k & 3 & 2 \\ 1 & 2 & k \end{pmatrix}\). Find, to three significant figures, the value of \(k\) for which \(A\) is singular (i.e., \(\det A = 0\)).

A. 2.14

Correct B. 2.28

C. 2.45

D. 1.96

Correct Answer: B

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Question 20 View Details

Let \(f(x)=2x-3\) and \(g(x)=x^{2}+1\). Find the value of the composition \((f\circ g)(2)\).

A. 5

Correct B. 7

C. 50

D. 13

Correct Answer: B

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Question 21 View Details

Evaluate the definite integral \(\displaystyle \int_{0}^{1} \frac{2x+3}{(x+1)(x^{2}+1)}\,dx\) by first expressing the integrand as a sum of partial fractions.

Correct A. (1/4)ln 2 + (5π)/8

B. (1/2)ln 2 + (5π)/8

C. (3/4)ln 2 + (5π)/8

D. (1/4)ln 2 - (5π)/8

Correct Answer: A

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Question 22 View Details

The equation \(\sin x + \sqrt{3}\cos x = 2\) holds for an angle \(x\) in the interval \([0,2\pi)\). Find the exact value of \(\tan\frac{x}{2}\).

A. 1 - sqrt(3)

B. 2 + sqrt(3)

C. sqrt(3) - 2

Correct D. 2 - sqrt(3)

Correct Answer: D

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Question 23 View Details

If \(\cos 2\theta = \frac{1}{3}\) and \(\theta\) is an acute angle, determine the exact value of \(\sin\theta + \cos\theta\).

A. (1-sqrt(2))/sqrt(3)

B. (sqrt(2)-1)/sqrt(3)

Correct C. (1+sqrt(2))/sqrt(3)

D. (1+sqrt(2))/3

Correct Answer: C

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Question 24 View Details

Find the sum of the infinite series \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{n\,(n+2)}\).

A. 1/2

Correct B. 3/4

C. 5/6

D. 2/3

Correct Answer: B

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Question 25 View Details

Calculate \(\log_{2} 8 + \log_{4} 16\).

Correct A. 5

B. 4

C. 6

D. 7

Correct Answer: A

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