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gce model questions vol1 2018 further_mathematics | Objective

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Question 1 View Details

The function (f(x)=x^{3}-6x^{2}+kx+8\) satisfies that the tangent at \(x=1\) passes through the point \((3,10)\). Find the value of \(k\) and then give the exact value of the derivative \(f'(1)\).

A. -1/3

B. 2/3

C. -2

Correct D. -2/3

Correct Answer: D

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Question 2 View Details

Vectors \(\mathbf{a}\) and \(\mathbf{b}\) in three‑dimensional space satisfy \(|\mathbf{a}|=5\), \(|\mathbf{b}|=7\) and \(\mathbf{a}\cdot\mathbf{b}=14\). Let \(\mathbf{c}=\mathbf{a}\times\mathbf{b}\). Find the volume of the parallelepiped formed by \(\mathbf{a},\mathbf{b},\mathbf{c}\).

A. 1089

B. 1000

Correct C. 1029

D. 1024

Correct Answer: C

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Question 3 View Details

Evaluate the 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.

A. (1\/2)ln2 + (5π)\/8

B. (1\/4)ln2 + (3π)\/8

Correct C. (1/4)ln2 + (5π)/8

D. (1\/4)ln2 + (5π)\/4

Correct Answer: C

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Question 4 View Details

Let \(A=\begin{pmatrix}p&2&3\\4&p&6\\7&8&p\end{pmatrix}\).\n(a) Find the real value of \(p\) (to two decimal places) for which \(A\) is singular.\n(b) For \(p=5\) compute the determinant of the adjugate matrix \(\operatorname{adj}(A)\).

A. p≈7.21, 7200

Correct B. p≈7.21, 6400

C. p≈6.58, 6400

D. p≈8.00, 6400

Correct Answer: B

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Question 5 View Details

When the polynomial \(P(x)=2x^{4}-3x^{3}+5x^{2}-7x+6\) is divided by \(x^{2}-x+1\) the remainder is of the form \(ax+b\). Find \(a+b\).

A. -14

B. -10

C. -8

Correct D. -12

Correct Answer: D

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Question 6 View Details

A polynomial (P(x)=2x^{3}-ax^{2}+bx-6\) leaves a remainder of \(5\) when divided by \((x-1)\) and a remainder of \(-3\) when divided by \((x+2)\). Find the value of \(a+b\).

A. -\frac{8}{3}

B. -\frac{12}{5}

Correct C. -\frac{10}{3}

D. -\frac{14}{3}

Correct Answer: C

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Question 7 View Details

A block of mass \(1\ \text{kg}\) is attached to a spring of unknown constant \(k\) on a horizontal frictionless surface. It is pulled \(0.20\ \text{m}\) from its equilibrium position and released from rest. After \(0.50\ \text{s}\) the block is \(0.054\ \text{m}\) to the right of equilibrium and its speed is \(0.50\ \text{m s}^{-1}\). Determine the spring constant \(k\) (in \(\text{N m}^{-1}\)) and the maximum distance the block reaches from equilibrium.

A. k=6.72\ \text{N m}^{-1},\ \text{maximum distance}=0.25\ \text{m}

Correct B. k=6.72\ \text{N m}^{-1},\ \text{maximum distance}=0.20\ \text{m}

C. k=7.30\ \text{N m}^{-1},\ \text{maximum distance}=0.22\ \text{m}

D. k=5.60\ \text{N m}^{-1},\ \text{maximum distance}=0.18\ \text{m}

Correct Answer: B

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Question 8 View Details

In triangle \(ABC\) the angles satisfy \(\angle B = 2\angle A\). The side lengths opposite these angles are \(a = 5\) and \(b = 7\) respectively. Find \(\sin C\) and express your answer in simplest surd form.

A. \frac{9\sqrt{51}}{125}

Correct B. \frac{12\sqrt{51}}{125}

C. \frac{12\sqrt{45}}{125}

D. \frac{12\sqrt{51}}{150}

Correct Answer: B

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Question 9 View Details

Evaluate the exact value of \(\displaystyle \frac{\sin 75^{\circ}}{\sin 15^{\circ}}\) and write your answer in simplest surd form.

A. 2-\sqrt3

Correct B. 2+\sqrt3

C. 2+\frac{\sqrt3}{2}

D. 1+\sqrt3

Correct Answer: B

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Question 10 View Details

In a factory, machines \(A\) and \(B\) produce \(60\%\) and \(40\%\) of the output respectively. The defect rates are \(2\%\) for \(A\) and \(5\%\) for \(B\). An item is selected at random and found to be defective. The same machine then produces a second item. Before producing the second item, the machine has a \(10\%\) chance of being recalibrated, which halves its defect rate for that item only. What is the probability that the second item is also defective? Give your answer as a reduced fraction.

A. 3069/80000

B. 310/16000

C. 620/16000

Correct D. 589/16000

Correct Answer: D

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Question 11 View Details

Let \(\mathbf{u} = (2,-1,k)\) and \(\mathbf{v} = (1,3,4)\). It is known that \(\mathbf{u}\) is orthogonal to \(\mathbf{w} = (k,2,-1)\). Find the value of \(k\) and then compute the scalar triple product \([\mathbf{u},\mathbf{v},\mathbf{w}]\).

A. -45

Correct B. -39

C. -27

D. -33

Correct Answer: B

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Question 12 View Details

The word \(\text{MATHEMATICS}\) consists of the letters M, A, T, H, E, M, A, T, I, C, S. How many distinct arrangements of all eleven letters have the first letter a vowel and the two T's not adjacent?

A. 145152

B. 1353024

C. 14515200

Correct D. 1451520

Correct Answer: D

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Question 13 View Details

Find all complex numbers \(z\) satisfying \((z-2i)^3 = 8(1-i)\). Then give the sum of the moduli of these solutions, correct to two decimal places.

A. 8.24

B. 8.64

Correct C. 8.44

D. 9.44

Correct Answer: C

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Question 14 View Details

A factory produces components of type A (60% of output) and type B (40%). A component of type A passes a quality test with probability 0.9, while a component of type B passes with probability 0.7. Five components are selected independently. The batch is accepted if at least four of the five components pass the test. Given that a batch is accepted, what is the probability that it contains exactly three components of type A? Give your answer to three decimal places.

A. 0.338

B. 0.428

Correct C. 0.348

D. 0.358

Correct Answer: C

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Question 15 View Details

A circle passes through the points \(A(1,2)\) and \(B(4,6)\) and is tangent to the line \(3x-4y+5=0\). Find the radius of the circle, expressed as a fraction in lowest terms.

A. 125/12

B. 125/16

C. 115/14

Correct D. 125/14

Correct Answer: D

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Question 16 View Details

The rational function \(\frac{x^{2}+5x+6}{x^{3}+2x^{2}-3x}\) can be expressed as \(\frac{A}{x}+\frac{B}{x+3}+\frac{C}{x-1}\). Find the value of \(A+B+C\).

A. 2

Correct B. 1

C. -1

D. 0

Correct Answer: B

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Question 17 View Details

Let \(A=\begin{pmatrix}2 & k\\ k & 5\end{pmatrix}\). If \(\det A =7\), find the trace of \(A^{2}\).

Correct A. 35

B. 40

C. 25

D. 30

Correct Answer: A

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Question 18 View Details

In a factory, 60% of the items are produced by machine \(M_{1}\) and 40% by machine \(M_{2}\). The defect rates are 2% for \(M_{1}\) and 5% for \(M_{2}\). An item is selected at random and 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? Give your answer as a fraction in lowest terms.

A. 29/800

Correct B. 31/800

C. 33/800

D. 31/900

Correct Answer: B

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Question 19 View Details

A linear transformation \(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\) is represented by the matrix \(A=\begin{pmatrix}1 & p\\ q & 4\end{pmatrix}\). It is known that \(T\) maps the vector \((2,-1)\) to \((3,5)\) and maps the vector \((1,1)\) to \((k,\,\tfrac{17}{2})\). Determine the value of \(k\) and state whether \(A\) is invertible.

A. k = -1; invertible

B. k = 2; invertible

Correct C. k = 0; invertible

D. k = 0; not invertible

Correct Answer: C

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Question 20 View Details

Solve the equation \((z-(1+i))^{2} = -2i\) for complex number \(z\). Then find the product of the moduli of the solutions. Give your final answer as an integer.

A. 8

Correct B. 4

C. 2

D. 16

Correct Answer: B

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Question 21 View Details

The curve is given by \(y = x^{3} - 3x\). Find the \(x\)-coordinate of the point on the curve at which the tangent line passes through the point \((0,2)\). Then state the gradient of the curve at that point.

A. x = -1, gradient = 2

B. x = 1, gradient = 0

Correct C. x = -1, gradient = 0

D. x = 2, gradient = 6

Correct Answer: C

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Question 22 View Details

Let the function \(f\) be defined from the set \{1,2,3,4,5\} to the set \{a,b,c\} by \(f(1)=a,\; f(2)=b,\; f(3)=c,\; f(4)=a,\; f(5)=b\). Determine whether \(f\) is one-to-one and whether it is onto. Also give the pre‑image of \(a\).

A. Not one-to-one; onto; pre‑image of a = {2,5}

Correct B. Not one-to-one; onto; pre‑image of a = {1,4}

C. Not one-to-one; not onto; pre‑image of a = {1,4}

D. One-to-one; not onto; pre‑image of a = {1,4}

Correct Answer: B

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Question 23 View Details

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

A. a = -3, b = -5, P(1) = -2

B. a = -2, b = -4, P(1) = 1

C. a = -1, b = -5, P(1) = 2

Correct D. a = -2, b = -5, P(1) = 0

Correct Answer: D

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Question 24 View Details

Vectors \(\mathbf{a}= (2,\,-1,\,k)\) and \(\mathbf{b}= (1,\,4,\,-2)\) are perpendicular. Determine the value of \(k\). Then give a unit vector that is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\).

Correct A. k = -1; unit vector = (2/√14, 1/√14, 3/√14)

B. k = 0; unit vector = (2\/√5, -1\/√5, 0\/√5)

C. k = 2; unit vector = (1\/√6, -2\/√6, 1\/√6)

D. k = -1; unit vector = (1\/√3, 1\/√3, -1\/√3)

Correct Answer: A

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Question 25 View Details

Evaluate the definite integral \(\displaystyle \int_{0}^{1} \frac{x^{3}}{x^{2}+1}\,dx\).

A. (1/2)·ln(2) - 1/2

Correct B. 1/2 - (1/2)·ln(2)

C. 1/2 - ln(2)

D. 1/2 + (1/2)·ln(2)

Correct Answer: B

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