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gce model questions vol1 2025 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 2025 further_mathematics (Objective) questions designed to simulate the real exam environment.

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Question 1 View Details

Let $z$ be a complex number such that $|z-(2+3i)|=5$ and $\operatorname{Arg}(z)=\frac{\pi}{6}$. Find $z$ and then evaluate $z^{2}+\overline{z}$. Give your answer in the form $a+bi$ rounded to two decimal places.

A. 38.70-51.06i

B. 38.70+46.06i

Correct C. 38.70+51.06i

D. 33.70+51.06i

Correct Answer: C

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Question 2 View Details

The circles $(x-1)^{2}+(y+2)^{2}=9$ and $(x+3)^{2}+(y-1)^{2}=16$ intersect at two points. Let $P$ be the intersection point that lies in the first quadrant. Find the equation of the line through $P$ that is perpendicular to $2x-y+5=0$, and then give the distance from the point $(4,-3)$ to this line, correct to two decimal places.

A. 2.42

B. 1.94

Correct C. 2.24

D. 3.14

Correct Answer: C

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Question 3 View Details

Let $A=\begin{pmatrix}k&1&0\\0&k&0\\0&0&k\end{pmatrix}$. Suppose $A$ satisfies $A^{2}=2A-I$, where $I$ is the $3\times3$ identity matrix. Find the value of $k$, and then compute $\det(A+I)$. Give the determinant as an integer.

Correct A. 8

B. 0

C. 12

D. -4

Correct Answer: A

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Question 4 View Details

Vectors $\mathbf{a}=\langle2,-1,3\rangle$ and $\mathbf{b}=\langle k,4,-2\rangle$ are given. Let $\mathbf{c}=\operatorname{proj}_{\mathbf{b}}\mathbf{a}+\mathbf{d}$, where $\mathbf{d}$ is perpendicular to $\mathbf{b}$ and has the same magnitude as $\operatorname{proj}_{\mathbf{b}}\mathbf{a}$. If $\mathbf{c}$ is orthogonal to $\mathbf{a}$, determine $k$ (to two decimal places) and then find the angle between $\mathbf{b}$ and $\mathbf{c}$ in degrees.

A. 6.58, 30

B. 9.15, 60

C. 7.92, 60

Correct D. 7.92, 45

Correct Answer: D

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Question 5 View Details

Consider the function $f(x)=\dfrac{x^{3}-3x}{x^{2}+1}$. Find the $x$‑coordinate of the point on the curve where the tangent is parallel to the line $y=x+4$, and then compute the second derivative $f''(x)$ at that $x$, giving the result to three decimal places.

A. 1.732

B. 2.236

C. 1.618

Correct D. 2.000

Correct Answer: D

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Question 6 View Details

The curve is given by $y = x^{3} - 6x^{2} + 9x + 2$. Find all values of $x$ at which the tangent line to the curve passes through the point $(0,5)$.

A. x = -1 or x = 2

B. x = 0 or x = 1

Correct C. x = -1 or x = 1

D. x = -2 or x = 2

Correct Answer: C

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Question 7 View Details

If $\log_{2} x + \log_{4} x = 5$, find the value of $x$.

A. 4\sqrt[3]{2}

Correct B. 8\sqrt[3]{2}

C. 16\sqrt[3]{2}

D. 8\sqrt{2}

Correct Answer: B

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Question 8 View Details

A particle is in equilibrium under three coplanar forces. Force $\mathbf{F}_{1}$ has magnitude 10 N and acts along the positive $x$-axis. Force $\mathbf{F}_{2}$ has magnitude 8 N and acts at an angle of $120^{\circ}$ measured anticlockwise from the positive $x$-axis. A third force $\mathbf{F}_{3}$ of unknown magnitude $P$ acts at an unknown angle $\theta$ (measured anticlockwise from the positive $x$-axis). Determine the magnitude $P$ (in newtons) and the angle $\theta$ (to one decimal place).

A. P = 2\sqrt{20} N (≈8.9 N), \theta = 229.1^{\circ}

B. P = 3\sqrt{21} N (≈13.7 N), \theta = 229.1^{\circ}

C. P = 2\sqrt{21} N (≈9.2 N), \theta = 210.0^{\circ}

Correct D. P = 2\sqrt{21} N (≈9.2 N), \theta = 229.1^{\circ}

Correct Answer: D

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Question 9 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 most three students. How many different committees can be formed?

Correct A. 1876

B. 1870

C. 1885

D. 1902

Correct Answer: A

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Question 10 View Details

A linear transformation $T:\mathbb{R}^{2}\to\mathbb{R}^{2}$ satisfies $T(1,2) = (3,5)$ and $T(2,-1) = (0,4)$. Find the matrix $A$ representing $T$ with respect to the standard basis, then determine the determinant of $A^{-1}$. Express $A$ in fractional form.

A. A = [[3\/5, 6\/5],[12\/5, 6\/5]], det(A^{-1}) = -5\/12

Correct B. A = [[3/5, 6/5],[13/5, 6/5]], det(A^{-1}) = -5/12

C. A = [[3\/5, 6\/5],[13\/5, 6\/5]], det(A^{-1}) = 5\/12

D. A = [[3\/5, 6\/5],[13\/5, 7\/5]], det(A^{-1}) = -5\/13

Correct Answer: B

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Question 11 View Details

Simplify the expression $ \frac{2^{5}\times 4^{3}}{8^{2}}\times\sqrt{16}$ and give the final numerical value.

A. 64

Correct B. 128

C. 256

D. 144

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 $2\times2$ matrix $A$. It satisfies $A\begin{pmatrix}1\\2\end{pmatrix}=\begin{pmatrix}3\\5\end{pmatrix}$ and $A\begin{pmatrix}2\\1\end{pmatrix}=\begin{pmatrix}k\\7\end{pmatrix}$. If the determinant of $A$ is $2$, find the value of $k$ and state the determinant of $A$.

A. 27/5, 3

B. 23/5, 2

Correct C. 27/5, 2

D. 29/5, 2

Correct Answer: C

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Question 13 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) What is the probability that a second independently selected item from the same batch is non‑defective? (b) Given that the first item was defective, what is the expected total number of defective items in a sample of 10 items drawn from the batch? Express your answers as fractions in lowest terms.

A. 771/800, 1081/800

B. 773/800, 1083/800

C. 779/800, 1089/800

Correct D. 769/800, 1079/800

Correct Answer: D

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Question 14 View Details

Consider the matrix $A=\begin{pmatrix}2&x&1\\0&3&y\\4&0&6\end{pmatrix}$. It is known that $\det A =0$ and that $x$ and $y$ are integers with $|x|\le5$ and $|y|\le5$. List all possible ordered pairs $(x,y)$ that satisfy these conditions.

A. (-2,3), (2,-3), (-3,2), (2,2)

B. (-2,3), (2,-3), (-3,-2), (3,2)

C. (-2,-3), (2,3), (-3,2), (3,-2)

Correct D. (-2,3), (2,-3), (-3,2), (3,-2)

Correct Answer: D

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Question 15 View Details

A block of mass $2\;\text{kg}$ slides down a rough inclined plane that makes an angle of $30^{\circ}$ with the horizontal and has a length of $5\;\text{m}$. The coefficient of kinetic friction is $0.2$. At the bottom of the incline the block compresses a spring of spring constant $150\;\text{N\,m}^{-1}$ and comes to rest after maximum compression. (a) Determine the speed of the block at the bottom of the incline (to two decimal places). (b) Find the maximum compression of the spring (to two decimal places).

Correct A. 5.66 m/s, 0.65 m

B. 4.80 m/s, 0.80 m

C. 5.00 m/s, 0.70 m

D. 6.20 m/s, 0.55 m

Correct Answer: A

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Question 16 View Details

In the expansion of $(2x-3)^7$, find the sum of the coefficients of the terms in which the power of $x$ is even.

Correct A. -39063

B. -39062

C. -39064

D. -39163

Correct Answer: A

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Question 17 View Details

Let $R = \{(1,x),(2,x),(2,y)\}$ be a relation from the set $A=\{1,2,3\}$ to the set $B=\{x,y\}$. Determine the domain and range of $R$ and state whether $R$ is a function.

A. Domain = {1,2,3}; Range = {x,y}; not a function

Correct B. Domain = {1,2}; Range = {x,y}; not a function

C. Domain = {1,2}; Range = {x,y}; is a function

D. Domain = {1,2}; Range = {x}; not a function

Correct Answer: B

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Question 18 View Details

The polynomial $x^{4}-5x^{3}+7x^{2}-5x+6$ can be expressed as the product of two monic quadratic polynomials with integer coefficients. One of the factors is $x^{2}+px+q$. Find all possible ordered pairs $(p,q)$.

A. (1,1) and (-5,7)

Correct B. (0,1) and (-5,6)

C. (0,2) and (-5,5)

D. (1,2) and (-4,5)

Correct Answer: B

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Question 19 View Details

Evaluate the indefinite integral $\displaystyle \int \frac{2x^{2}+3x+5}{(x-1)(x^{2}+1)}\,dx$ by first decomposing the integrand into partial fractions.

A. 5\ln|x-1| - \frac{1}{2}\ln(x^{2}+1) + C

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

C. 5\ln|x-1| + \frac{3}{2}\ln(x^{2}+1) + C

Correct D. 5\ln|x-1| - \frac{3}{2}\ln(x^{2}+1) + C

Correct Answer: D

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Question 20 View Details

A committee of five members is to be chosen from eight men and six women. Two particular men, A and B, are not allowed to serve together on the committee. Moreover, the committee must contain at least two women. How many different committees can be formed?

A. 1526

B. 1436

C. 1506

Correct D. 1416

Correct Answer: D

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Question 21 View Details

A block of mass $2\,\text{kg}$ is pulled up a $20^{\circ}$ incline by a constant force of $10\,\text{N}$ acting $30^{\circ}$ above the horizontal. The coefficient of kinetic friction is $\mu$. The block starts from rest at the bottom and moves $5\,\text{m}$ up the incline, coming to rest at the top. Determine the minimum value of $\mu$ that allows this motion.

A. 0.200

Correct B. 0.235

C. 0.250

D. 0.300

Correct Answer: B

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Question 22 View Details

Let $z$ be a complex number satisfying $|z+1|=|z-2i|$ and $\operatorname{Arg}(z-1)=\frac{\pi}{4}$. Find the exact value of $z$ and then evaluate $z^{2}+\overline{z}$.

A. 5/2 + i/9

B. 5/2 - 2i/9

Correct C. 5/2 + 2i/9

D. 3/2 + 2i/9

Correct Answer: C

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Question 23 View Details

Find the acute angle $\theta$ (in radians) that satisfies $\sin\theta+\sqrt{3}\cos\theta=2$. Then give the exact value of $\tan\frac{\theta}{2}$.

A. 2 - 2*sqrt(3)

B. sqrt(3) - 2

Correct C. 2 - sqrt(3)

D. 2 + sqrt(3)

Correct Answer: C

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Question 24 View Details

Evaluate the definite integral $\displaystyle\int_{0}^{1}\frac{x^{2}}{\sqrt{1-x^{2}}}\,dx$.

Correct A. pi/4

B. pi/6

C. 1/4

D. pi/2

Correct Answer: A

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Question 25 View Details

Find the remainder when the polynomial $x^{5}-3x^{4}+2x^{3}-x+5$ is divided by $x-2$.

A. -1

Correct B. 3

C. 5

D. 2

Correct Answer: B

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