Exam Body

Year

Subject

Paper Type

gce model questions vol1 2020 further_mathematics | Objective

Prepare for your exams with gce model questions vol1 questions? Reviewing past/model questions is one of the most effective ways to guarantee a high score. This practice hub features authentic 2020 further_mathematics (Objective) questions designed to simulate the real exam environment.

Practice these randomly selected questions to test your readiness.

Question 1 View Details

In the expansion of \( (x+1)^8 \), find the sum of the coefficients of the terms in which the power of \( x \) is a multiple of 3.

Correct A. 85

B. 90

C. 81

D. 75

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 2 View Details

Let \( C = \begin{pmatrix} 1 & 2 & 3 \\ 0 & k & 1 \\ 0 & 0 & 1 \end{pmatrix} \). Find the value of \( k \) for which the entry in the first row and third column of \( C^{-1} \) equals \( 5 \).

Correct A. \frac{1}{4}

B. \frac{1}{3}

C. \frac{3}{4}

D. \frac{1}{2}

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 3 View Details

A committee of 5 people is to be chosen from 8 men and 6 women. The chairperson must be a man and the committee must contain at least two women. How many different committees can be formed?

A. 5720

B. 3960

Correct C. 3760

D. 1526

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 4 View Details

In a right‑angled triangle \(ABC\) with \(\angle C = 90^{\circ}\), the side opposite \(\angle A\) is \(5\) cm and the side opposite \(\angle B\) is \(12\) cm. Find the exact value of \(\sin 2A\).

A. \frac{144}{169}

B. \frac{120}{144}

Correct C. \frac{120}{169}

D. \frac{60}{169}

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 5 View Details

For what real values of \(k\) does the matrix \(Q=\begin{pmatrix}k&1&1\\1&k&1\\1&1&k\end{pmatrix}\) have determinant zero?

A. k = 2 or k = -1

B. k = 0 or k = 3

Correct C. k = 1 or k = -2

D. k = -3 or k = 4

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 6 View Details

A particle of mass \(2\ \text{kg}\) moves in a straight line under a time‑dependent force \(F(t)=k t^{2}\) newtons, where \(k\) is a constant. The particle starts from rest at \(t=0\). After \(3\ \text{s}\) its speed is \(12\ \text{m s}^{-1}\). Find the distance travelled by the particle between \(t=2\ \text{s}\) and \(t=3\ \text{s}\).

A. 65/8 m

B. 55/9 m

C. 70/9 m

Correct D. 65/9 m

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 7 View Details

A complex number \(z\) satisfies \(|z|=5\) and the real part of \(z^{2}\) equals \(7\). Moreover the argument of \(z\) is acute. Determine the imaginary part of \(z\).

A. 2

B. 5

C. 4

Correct D. 3

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 8 View Details

The equation \(z^{4}-(8i)z^{2}-25=0\) has two distinct roots whose imaginary parts are positive. Let these roots be \(z_{1}\) and \(z_{2}\). Find \(|z_{1}+z_{2}|\).

A. 3\sqrt{3}

Correct B. 3\sqrt{2}

C. 2\sqrt{6}

D. 4\sqrt{2}

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 9 View Details

A rectangular garden is to be fenced on three sides; the side parallel to a river needs no fence. The garden must have an area of \(180\ \text{m}^{2}\). The cost per metre of fence along the side parallel to the river is three times the cost per metre of fence along each of the other two sides, and a fixed gate costing £50 is to be installed on one of the width sides. Determine the dimensions (length parallel to the river and width) that minimise the total cost.

A. length = 3\sqrt{30} m, width = 2\sqrt{30} m

Correct B. length = 2\sqrt{30} m, width = 3\sqrt{30} m

C. length = \sqrt{120} m, width = \sqrt{90} m

D. length = 4\sqrt{15} m, width = 3\sqrt{15} m

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 10 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 least one student. One of the teachers on the committee will be elected chairperson. How many different possible committees together with the choice of chairperson are there?

A. 5600

B. 4760

Correct C. 5320

D. 3640

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 11 View Details

A particle moves along a straight line with acceleration \(a(t)=6t-8\) (in \(\text{m s}^{-2}\)), where \(t\) is time in seconds. At \(t=0\) its velocity is \(5\ \text{m s}^{-1}\) and its position is \(2\ \text{m}\). Determine the position of the particle at the instant it first comes to rest, and state the total distance it has travelled up to that instant.

A. Position = 3 m; distance travelled = 1 m

B. Position = 2 m; distance travelled = 4 m

Correct C. Position = 4 m; distance travelled = 2 m

D. Position = 5 m; distance travelled = 3 m

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 12 View Details

Consider the system of linear equations \((k+1)x + 2y = 3\) and \(4x + (k-1)y = k\). (i) Find all real values of \(k\) for which the system has a unique solution. (ii) For \(k = 2\), compute the values of \(x\) and \(y\).

A. k \neq \pm3; for k=2, x = 0, y = 6/5

B. k \neq \pm3; for k=2, x = 1/5, y = 5/5

C. k \neq \pm3; for k=2, x = 2/5, y = 5/5

Correct D. k \neq \pm3; for k=2, x = 1/5, y = 6/5

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 13 View Details

Find the acute angle \(\theta\) (in degrees) satisfying \(\sin\theta + \sqrt{3}\cos\theta = \sqrt{2}\). Then give the exact value of \(\tan\theta\).

A. θ = 30°, tan θ = √3+1

B. θ = 45°, tan θ = 2+√3

C. θ = 75°, tan θ = 2-√3

Correct D. θ = 75°, tan θ = 2+√3

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 14 View Details

The curve is defined by \(x^{3}+y^{3}=6xy\). (i) Find \(\frac{dy}{dx}\) at the point \((3,3)\). (ii) Write the equation of the normal line to the curve at that point.

A. dy\/dx = 0; normal line: y = x

B. dy\/dx = -1; normal line: y = -x

Correct C. dy/dx = -1; normal line: y = x

D. dy\/dx = 2; normal line: y = x + 6

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 15 View Details

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

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

Correct B. ln|x-1| - \frac{1}{2}ln(x^{2}+4) + \frac{1}{2}\arctan\frac{x}{2} + C

C. ln|x-1| - \\frac{1}{2}ln(x^{2}+4) - \\frac{1}{2}\\arctan\\frac{x}{2} + C

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

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 16 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 the vector \((1,2)\) to \((3,5)\) and maps the vector \((2,-1)\) to \((k,4)\). Moreover, the transformation has determinant \(1\). Find the value of \(k\) and then compute the trace of \(A\) (i.e. \(a+d\)).

Correct A. k = 17/5; trace = 79/25

B. k = 3\/5; trace = 79\/25

C. k = 17\/5; trace = 79\/20

D. k = 17\/5; trace = 81\/25

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 17 View Details

A bag contains 4 white balls and 6 black balls. A ball is drawn at random. If the first ball drawn is white, a further white ball is added to the bag before the second draw; otherwise no ball is added. A second ball is then drawn. If the second ball drawn is black, a third ball is drawn from the remaining balls without replacement. Find the probability that the third ball drawn is white.

Correct A. 12/25

B. 7\/25

C. 12\/30

D. 13\/25

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 18 View Details

Evaluate the integral \( I = \int_{0}^{\pi/4} \frac{\tan x}{1+\sin x}\,dx \) and give your answer in exact form.

A. (1\/2) * ln(3+2*sqrt(2)) + (1 - sqrt(2))\/2

B. (1\/4) * ln(3-2*sqrt(2)) + (1 - sqrt(2))\/2

C. (1\/4) * ln(3+2*sqrt(2)) + (1 + sqrt(2))\/2

Correct D. (1/4) * ln(3+2*sqrt(2)) + (1 - sqrt(2))/2

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 19 View Details

Let \(P(x)=ax^{3}+bx^{2}+cx+d\) be a cubic polynomial with leading coefficient \(a=1\). It satisfies \(P(1)=2\), \(P(-2)=-3\) and \(P(2)=10\). Find the value of \(P(0)\).

Correct A. 7/6

B. 5\/6

C. 13\/6

D. 7\/5

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 20 View Details

The curve is defined implicitly by \(y^{3}-3xy+x^{3}=0\). Determine the coordinates of the point(s) on the curve where the tangent line is parallel to the line \(y=x\).

A. (-1,0)

B. (1,1)

C. (0,1)

Correct D. (0,0)

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 21 View Details

Let \(A = \{1,2,3,4\}\) and define a function \(f: A \to A\) by \(f(x) = x^{2} \bmod 5\). Determine the set of values taken by \(f\) (the range of \(f\)).

Correct A. {1,4}

B. {0,1,4}

C. {2,3}

D. {1}

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 22 View Details

On the set \(S = \{0,1,2\}\) define a binary operation \(*\) by \(a * b = (a + b) \bmod 3\). (i) State whether \(*\) is commutative. (ii) Find the identity element of \(*\) (if it exists).

A. commutative; identity element 1

B. non-commutative; no identity element

C. non-commutative; identity element 0

Correct D. commutative; identity element 0

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 23 View Details

Simplify the surd expression \(\sqrt{12} + \sqrt{27} - \sqrt{48}\) and write the result in simplest radical form.

A. \(2\sqrt{3}\)

B. \(\sqrt{6}\)

Correct C. \(\sqrt{3}\)

D. \(\sqrt{12}\)

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 24 View Details

Let \(A = \begin{pmatrix}2 & 1\\ k & 3\end{pmatrix}\) where \(k\) is an integer. (i) Find all integer values of \(k\) for which \(A\) is invertible and its inverse has only integer entries. (ii) For each such \(k\), give \(A^{-1}\).

A. k = 4 with A^{-1}=\begin{pmatrix}3 & -1\\ -4 & 2\end{pmatrix}; k = 6 with A^{-1}=\begin{pmatrix}-3 & 1\\ 6 & -2\end{pmatrix}

B. k = 1 with A^{-1}=\begin{pmatrix}3 & -1\\ -1 & 2\end{pmatrix}; k = 9 with A^{-1}=\begin{pmatrix}-3 & 1\\ 9 & -2\end{pmatrix}

Correct C. k = 5 with A^{-1}=\begin{pmatrix}3 & -1\\ -5 & 2\end{pmatrix}; k = 7 with A^{-1}=\begin{pmatrix}-3 & 1\\ 7 & -2\end{pmatrix}

D. k = 5 with A^{-1}=\begin{pmatrix}2 & -1\\ -5 & 3\end{pmatrix}; k = 8 with A^{-1}=\begin{pmatrix}-2 & 1\\ 8 & -3\end{pmatrix}

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 25 View Details

Consider the function \(f\) on the set \{1,2,3,4\} defined by \(f(1)=2,\; f(2)=3,\; f(3)=4,\; f(4)=1\). Compute \(f^{3}(2)\) (the third iterate of \(f\) applied to 2).

A. 3

B. 4

C. 2

Correct D. 1

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Master the Exam!

You've seen a preview, but there are thousands more questions plus AI tutor to break down complex solutions.

Unlock Full Access Available for Android & Windows

Help others prepare! Share this practice hub:

Chat with Support