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gce model questions vol1 2017 further_mathematics | Objective

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Question 1 View Details

A smooth inclined plane of length 5\,m makes an angle \(\theta\) with the horizontal such that \(\tan\theta = \frac{3}{4}\). The coefficient of kinetic friction varies with the distance \(x\) (in metres) measured from the bottom as \(\mu(x)=0.1+0.02x\). A particle of mass \(m\) is projected up the plane from the bottom with speed \(v_0\). Using \(g=9.8\ \text{ms}^{-2}\), determine the minimum initial speed \(v_0\) required for the particle to just reach the top of the plane.

A. 10.0

B. 7.6

Correct C. 8.4

D. 9.2

Correct Answer: C

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Question 2 View Details

Consider the matrix \(B=\begin{pmatrix}1&2&3\\4&k&6\\7&8&9\end{pmatrix}\). Find all real values of \(k\) for which \(B\) is singular, and for the obtained value of \(k\) give the element in the second row, third column of the adjugate matrix \(\operatorname{adj}(B)\).

A. 3

Correct B. 6

C. 9

D. 0

Correct Answer: B

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Question 3 View Details

Integrate \(\displaystyle \int \frac{2x^{2}+3x+5}{(x-1)(x^{2}+4)}\,dx\) by expressing the integrand as a sum of partial fractions.

Correct A. 2\ln|x-1|+\frac{3}{2}\arctan\left(\frac{x}{2}\right)+C

B. 2\ln|x-1|+\frac{1}{2}\arctan\left(\frac{x}{2}\right)+C

C. \ln|x-1|+\frac{3}{2}\arctan\left(\frac{x}{2}\right)+C

D. 2\ln|x-1|-\frac{3}{2}\arctan\left(\frac{x}{2}\right)+C

Correct Answer: A

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Question 4 View Details

Let \(z\) be a complex number satisfying \(|z-2i|=3\) and \(\arg(z)=\frac{\pi}{6}\) (the argument is acute). Evaluate \(\displaystyle \frac{z^{2}+\overline{z}^{2}}{|z|^{2}}\).

A. 2

B. 0

C. -1

Correct D. 1

Correct Answer: D

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Question 5 View Details

The system of equations\n\\[\n\begin{cases}\n(p+1)x + 2y - z = 3\\\n4x + (p-2)y + 5z = p\\\n-2x + y + (p+3)z = 4\n\end{cases}\n\\]\ndepends on the real parameter \(p\). (a) State the condition on \(p\) for which the system has a unique solution. (b) For \(p=2\) find the value of \(x\).

A. unique for all real p except the real root of p^{3}+2p^{2}-20p-55=0 (≈4.718); for p=2, x = -5\/79

B. unique for all real p except the real root of p^{3}+2p^{2}-20p-55=0 (≈4.718); for p=2, x = 3\/79

Correct C. unique for all real p except the real root of p^{3}+2p^{2}-20p-55=0 (≈4.718); for p=2, x = -3/79

D. unique for all real p except the real root of p^{3}+2p^{2}-20p-55=0 (≈4.718); for p=2, x = -3\/78

Correct Answer: C

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Question 6 View Details

In the expansion of \((2x-3)^{7}\), find the sum of the coefficients of the terms in which the exponent of \(x\) is a multiple of 3.

A. 18000

Correct B. 19149

C. 19140

D. 19200

Correct Answer: B

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Question 7 View Details

In a plane, vectors \(\mathbf{a}\) and \(\mathbf{b}\) satisfy \(|\mathbf{a}|=5\) and \(|\mathbf{a}+\mathbf{b}|=8\). The angle between \(\mathbf{a}\) and \(\mathbf{a}+\mathbf{b}\) is \(30^{\circ}\). Find \(|\mathbf{b}|\).

A. \sqrt{89-20\sqrt{3}}

B. \sqrt{64-40\sqrt{3}}

C. \sqrt{89+40\sqrt{3}}

Correct D. \sqrt{89-40\sqrt{3}}

Correct Answer: D

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Question 8 View Details

Three non‑zero vectors \(\mathbf{a},\mathbf{b},\mathbf{c}\) in space satisfy \(|\mathbf{a}|=5\), \(|\mathbf{b}|=7\), \(\mathbf{a}\cdot\mathbf{b}=4\) and \(\mathbf{a}\times\mathbf{b}=3\mathbf{c}\). Find the volume of the parallelepiped formed by \(\mathbf{a},\mathbf{b},\mathbf{c}\).

A. 420

B. 315

Correct C. 403

D. 504

Correct Answer: C

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Question 9 View Details

Let vectors \(\mathbf{a}\) and \(\mathbf{b}\) lie in a plane with \(|\mathbf{a}|=6\) and \(|\mathbf{b}|=8\). If \((2\mathbf{a}-\mathbf{b})\) is perpendicular to \((\mathbf{a}+3\mathbf{b})\), find the angle between \(\mathbf{a}\) and \(\mathbf{b}\) in degrees.

A. 30

Correct B. 60

C. 90

D. 45

Correct Answer: B

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Question 10 View Details

Find the equation of the circle that passes through the points \(A(1,2)\) and \(B(4,6)\) and whose centre lies on the line \(2x-y=3\).

Correct A. (x-71/22)^2+(y-38/11)^2=3425/484

B. (x-71\/22)^2+(y-38\/11)^2=3500\/484

C. (x-71\/22)^2+(y-39\/11)^2=3425\/484

D. (x-70\/22)^2+(y-38\/11)^2=3425\/484

Correct Answer: A

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Question 11 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 \(T\) maps \((1,2)\) to \((3,5)\) and maps \((2,-1)\) to \((k,4)\). Determine the value of \(k\) for which \(A\) is invertible with determinant \(2\). Then find the image of the vector \((4,0)\) under the inverse transformation \(T^{-1}\).

A. (12/5, -24/5)

B. (12/5, -26/4)

C. (10/5, -26/5)

Correct D. (12/5, -26/5)

Correct Answer: D

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Question 12 View Details

From a group of 8 men and 6 women a committee of five is to be formed. The chair must be a woman and the secretary must be a man. Two particular persons, A (a woman) and B (a man), are not allowed to serve together on the committee. How many distinct committees (with the two offices filled) can be formed?

A. 9648

Correct B. 9638

C. 9538

D. 9630

Correct Answer: B

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Question 13 View Details

Expand \((2x-3)^8\). (i) Find the coefficient of the term containing \(x^{4}\). (ii) Using your expansion, determine the sum of the coefficients of all terms in which the power of \(x\) is even.

Correct A. 90720; 195313

B. 90720; 195312

C. 45360; 195313

D. 120960; 196830

Correct Answer: A

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Question 14 View Details

At 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) What is the probability that it was produced by machine A? (b) A second item, produced by the same machine as the first, is also found to be defective. What is the updated probability that the machine is A? (c) Based on the result of part (b), what is the expected number of items that must be inspected from that machine until a non‑defective item is found? Give your answer to three decimal places.

A. a) 3\/7; b) 4\/27; c) 1.060

B. a) 1\/3; b) 5\/24; c) 1.050

Correct C. a) 3/8; b) 6/31; c) 1.046

D. a) 2\/5; b) 7\/30; c) 1.040

Correct Answer: C

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Question 15 View Details

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

A. (1\/4)ln2 - (5π)\/8

Correct B. (1/4)ln2 + (5π)/8

C. (1\/4)ln2 + (π)\/8

D. (1\/2)ln2 + (5π)\/8

Correct Answer: B

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Question 16 View Details

A uniform rod OA of length \(10\) m and weight \(200\) N is hinged at O to a vertical wall. The rod makes an angle \(\theta\) with the wall. A cable is attached to the free end A and runs to a point D on the floor \(4\) m horizontally from the wall on the opposite side of the wall (i.e., D is \(4\) m to the left of the wall). Determine the value of \(\theta\) (to the nearest degree) for which the reaction at the hinge O is purely horizontal.

A. 45

B. 60

C. 70

Correct D. 53

Correct Answer: D

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Question 17 View Details

In the expansion of \((x^{2}+ \frac{3}{x})^{5}\,(1-2x)^{3}\), find the term that is independent of \(x\) and give its coefficient.

A. 5400

Correct B. 4860

C. 4320

D. 3780

Correct Answer: B

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Question 18 View Details

Simplify \(\displaystyle \frac{\sqrt{2^{6}\,3^{4}}}{3^{2}\,2^{2}}\) and give the final integer value.

A. 1

B. 3

Correct C. 2

D. 4

Correct Answer: C

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Question 19 View Details

Let the set \(A=\{1,2,3,4\}\). Define a binary operation \(*\) on \(A\) by \(a * b = (a+b)\bmod 5\). Find the value of \(3 * 4\).

A. 3

Correct B. 2

C. 1

D. 0

Correct Answer: B

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Question 20 View Details

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

A. 1 + \ln\left(\frac{3}{2}\right)

B. \ln\left(\frac{3}{2}\right) - 1

Correct C. 1 - \ln\left(\frac{3}{2}\right)

D. 1 - \ln\left(\frac{2}{3}\right)

Correct Answer: C

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Question 21 View Details

The curve is given by \(f(x)=x^{3}-6x^{2}+9x+2\). Find the y‑coordinate of the point on the curve where the tangent is parallel to the line \(y=3x+4\) and the x‑coordinate is greater than the y‑coordinate.

A. 4-2\sqrt{2}

Correct B. 4-\sqrt{2}

C. 4+\sqrt{2}

D. 2-\sqrt{2}

Correct Answer: B

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Question 22 View Details

Let \(A=\{1,2,3,4\}\) and \(B=\{a,b,c\}\). A function \(f:A\to B\) is defined by \(f(1)=a,\;f(2)=b,\;f(3)=a,\;f(4)=c\). Define a relation \(R\) on \(A\) by \(xRy\) iff \(f(x)=f(y)\). How many equivalence classes does \(R\) have?

A. 4

B. 5

C. 2

Correct D. 3

Correct Answer: D

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Question 23 View Details

A circle passes through the points \(P(1,2)\) and \(Q(4,6)\) and is tangent to the line \(3x-4y+5=0\). Determine the radius of the circle, expressed as an irreducible fraction.

A. 125/13

B. 130/14

Correct C. 125/14

D. 119/14

Correct Answer: C

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Question 24 View Details

Factorise \(2x^{3}-3x^{2}-8x+12\) completely and state the sum of all its real roots.

Correct A. 3/2

B. 1/2

C. 5/2

D. 4/3

Correct Answer: A

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Question 25 View Details

A rectangular garden is to have an area of \(200\ \text{m}^{2}\). One side of length \(x\) metres lies along a river and requires no fencing. The opposite side of length \(x\) costs £5 per metre to fence, while each of the two sides of length \(y\) metres costs £8 per metre. Find the dimensions \(x\) and \(y\) (in metres) that minimise the total cost of fencing. Give your answer in exact surd form.

A. x=8\sqrt{10},\ y=\frac{5\sqrt{10}}{4}

Correct B. x=8\sqrt{10},\ y=\frac{5\sqrt{10}}{2}

C. x=8\sqrt{5},\ y=\frac{5\sqrt{5}}{2}

D. x=5\sqrt{10},\ y=\frac{8\sqrt{10}}{2}

Correct Answer: B

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