gce model questions vol1 2025 mathematics | Objective

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Question 1 View Details
In triangle \(ABC\), \(\angle A = 40^{\circ}\), \(\angle B = 70^{\circ}\) and \(\angle C = 70^{\circ}\). The side \(AB\) has length \(10\) cm. Point \(D\) lies on side \(BC\) such that \(BD:DC = 1:2\). Find the length of \(AD\) (give your answer to two decimal places).
A. 10.12
Correct B. 9.47
C. 8.23
D. 7.95

Correct Answer: B

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Question 2 View Details
In triangle \(ABC\) we have \(\angle BAC = 50^{\circ}\) and \(\angle ABC = 70^{\circ}\). Point \(D\) lies on side \(AB\) with \(AD = 3\) cm and \(DB = 5\) cm. A line through \(D\) parallel to \(AC\) meets \(BC\) at \(E\). Find the length of segment \(CE\) (to two decimal places).
A. 4.02
B. 1.87
Correct C. 2.65
D. 3.14

Correct Answer: C

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Question 3 View Details
Simplify the expression \(\displaystyle \frac{(2^{3}\cdot5^{2})^{\frac12}}{\sqrt{20}}\) and write your answer in simplest radical form.
A. \sqrt{20}
Correct B. \sqrt{10}
C. \sqrt{5}
D. \frac{\sqrt{10}}{2}

Correct Answer: B

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Question 4 View Details
Points \(A(1,2,3)\), \(B(4,0,-1)\) and \(C(2,-1,5)\) are given in space. Let \(\mathbf{p}\) be the projection of vector \(\overrightarrow{AB}\) onto vector \(\overrightarrow{AC}\). Find the magnitude of \(\mathbf{p}\) in simplest exact form.
A. \frac{\sqrt{28}}{14}
B. \frac{\sqrt{7}}{14}
Correct C. \frac{\sqrt{14}}{14}
D. \frac{\sqrt{14}}{7}

Correct Answer: C

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Question 5 View Details
Solve for \(x\): \(2^{x}=5^{\log_{5}(3^{\,x+1})}\). Give your answer correct to two decimal places.
Correct A. -2.71
B. -2.70
C. -2.68
D. -3.14

Correct Answer: A

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Question 6 View Details
The quadratic equation \(x^{2}+px+q=0\) has two distinct real roots \(r\) and \(s\). It is known that \(r+\frac{1}{s}=3\) and \(s+\frac{1}{r}=4\). Find the value of \(p+q\).
A. (9-5\sqrt{6})\/6
Correct B. (9+5\sqrt{6})/6
C. (8+5\sqrt{6})\/6
D. (9+4\sqrt{6})\/6

Correct Answer: B

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Question 7 View Details
For what real values of \(k\) does the system \(\begin{cases}2x+ky=7\\ kx-3y=4\end{cases}\) have a solution \((x,y)\) satisfying \(x-y=1\)?
A. (-3+\sqrt{97})\/2 or (-3-\sqrt{97})\/2
B. (-3+\sqrt{101})\/3 or (-3-\sqrt{101})\/3
C. (-3+\sqrt{105})\/2 or (-3-\sqrt{105})\/2
Correct D. (-3+\sqrt{101})/2 or (-3-\sqrt{101})/2

Correct Answer: D

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Question 8 View Details
In a class of 120 students, 68 study Mathematics, 54 study Physics and 48 study Chemistry. The numbers studying both Mathematics and Physics, Mathematics and Chemistry, and Physics and Chemistry are 30, 25 and 20 respectively. If exactly 38 students study only one of these subjects, how many students study all three subjects?
A. 8
B. 12
Correct C. 6
D. 10

Correct Answer: C

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Question 9 View Details
A sequence is defined by \(a_{1}=2\) and for \(n\ge 1\), \(a_{n+1}= \frac{3a_{n}+4}{a_{n}+2}\). Find \(a_{5}\) expressed as an irreducible fraction.
A. 475/187
Correct B. 479/187
C. 479/191
D. 483/187

Correct Answer: B

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Question 10 View Details
A survey of 200 people records their preferences for tea (T), coffee (C) and juice (J). The numbers are: \(|T|=120\), \(|C|=95\), \(|J|=80\); \(|T\cap C|=50\), \(|T\cap J|=50\), \(|C\cap J|=45\). The number of people who like none of the three drinks equals the number who like exactly two drinks. How many people like all three drinks?
A. 50
B. 55
C. 40
Correct D. 45

Correct Answer: D

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Question 11 View Details
A closed cylindrical tank with a hemispherical dome on top and a circular base has a total external surface area of \(1500\) cm\(^2\). The surface area includes the curved surface of the cylinder, the curved surface of the hemisphere, and the base of the cylinder. Determine the dimensions (radius \(r\) and cylinder height \(h\) in cm) that maximise the internal volume of the tank, and then give the maximum volume (to the nearest cm\(^3\)).
A. 4520 cm^3
B. 4750 cm^3
C. 5120 cm^3
Correct D. 4885 cm^3

Correct Answer: D

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Question 12 View Details
The circle \(x^{2}+y^{2}-6x+4y=0\) meets the line \(y=x\) at two points. A straight line passes through the point \((4,-2)\) and is perpendicular to the chord joining those intersection points. Find the equation of this line in the form \(ax+by+c=0\).
A. x - y + 2 = 0
B. 2x + y - 4 = 0
Correct C. x + y - 2 = 0
D. x + 2y - 2 = 0

Correct Answer: C

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Question 13 View Details
The quadratic equation \(x^{2}-(k-1)x+(k-6)=0\) has two real roots whose product equals twice their sum. Determine the value of \(k\) and then give the product of the roots for that value of \(k\).
A. k = 11, product = 4
B. k = 9, product = 5
Correct C. k = 11, product = 5
D. k = 13, product = 5

Correct Answer: C

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Question 14 View Details
Solve the inequality \(|2x-5| \le x^{2}-4x+3\) for integer values of \(x\) in the interval \(-10 \le x \le 10\). Then find the sum of all such integer solutions.
A. -5
B. 0
Correct C. -6
D. -8

Correct Answer: C

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Question 15 View Details
In the Venn diagram below, three sets \(A, B, C\) have the following numbers of elements in each region: \(A\) only: 12, \(B\) only: 8, \(C\) only: 5, \(A\cap B\) only: 7, \(A\cap C\) only: 4, \(B\cap C\) only: 6, \(A\cap B\cap C\): 3. Using this information, determine (i) the number of elements that belong to exactly two of the sets, and (ii) the total number of elements that belong to at least one of the sets.
A. Exactly two sets: 15; at least one set: 42
Correct B. Exactly two sets: 17; at least one set: 45
C. Exactly two sets: 16; at least one set: 44
D. Exactly two sets: 18; at least one set: 46

Correct Answer: B

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Question 16 View Details
The sum of the digits of a two‑digit number is 12 and the tens digit is 2 more than the units digit. What is the number?
A. 66
Correct B. 75
C. 84
D. 93

Correct Answer: B

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Question 17 View Details
The time \(T\) (in hours) taken to paint a wall varies directly as the area \(A\) (in \(\text{m}^2\)) and inversely as the number of painters \(P\). If 4 painters can paint a 120\,\(\text{m}^2\) wall in 6 hours, how many hours will it take 6 painters to paint a 180\,\(\text{m}^2\) wall?
A. 8
B. 9
Correct C. 6
D. 5

Correct Answer: C

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Question 18 View Details
A bag contains red, blue and green balls. The numbers of red and blue balls are 5 and 4 respectively, while the number of green balls is unknown. A ball is drawn at random, its colour noted, and the ball is replaced. The experiment is repeated once more. It is known that the probability that the two drawn balls are of the same colour is exactly \(\frac{25}{72}\). (a) Determine the number of green balls in the bag. (b) Using this number, find the probability that at least one of the two drawn balls is red.
Correct A. 3; 95/144
B. 4; 13/24
C. 2; 7/12
D. 5; 85/144

Correct Answer: A

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Question 19 View Details
Write \(\frac{7}{8}\) as a decimal.
A. 0.887
B. 0.857
C. 0.785
Correct D. 0.875

Correct Answer: D

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Question 20 View Details
The quadratic equation \(x^2-(m+3)x+2m=0\) has two real roots that differ by 3. Find all possible values of \(m\).
A. -1 or 5
B. 0 or 3
Correct C. 0 or 2
D. 1 or 3

Correct Answer: C

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Question 21 View Details
The sum of the first \(n\) terms of an arithmetic sequence is given by \(S_n = 3n^{2}+5n\). The sum of the first \(2n\) terms is \(S_{2n}=12n^{2}+10n\). Determine the common difference of the sequence.
A. 8
B. 10
C. 4
Correct D. 6

Correct Answer: D

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Question 22 View Details
A right circular cylinder is inscribed in a cone whose height is \(12\) cm and base radius is \(6\) cm. Find the radius and height of the cylinder that give the maximum possible volume, and state that maximum volume.
Correct A. radius 4 cm, height 4 cm, maximum volume 64\pi cm^3
B. radius 5 cm, height 3 cm, maximum volume 75\\pi cm^3
C. radius 3 cm, height 5 cm, maximum volume 60\\pi cm^3
D. radius 2 cm, height 6 cm, maximum volume 48\\pi cm^3

Correct Answer: A

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Question 23 View Details
Simplify \(\displaystyle \frac{2^{3}\sqrt{50}}{5\sqrt{2}}\) and give your answer in lowest integer form.
A. 12
Correct B. 8
C. 6
D. 10

Correct Answer: B

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Question 24 View Details
An infinite geometric series has sum \(12\). Its third term equals \(\frac{3}{8}\). Find the first term and common ratio (to three decimal places), and then compute the sum of the first five terms (to two decimal places).
A. first term 9.632, common ratio 0.250, sum of first five terms 12.50
Correct B. first term 9.632, common ratio 0.197, sum of first five terms 12.00
C. first term 9.500, common ratio 0.200, sum of first five terms 11.80
D. first term 8.900, common ratio 0.197, sum of first five terms 11.90

Correct Answer: B

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Question 25 View Details
In the coordinate plane the points \(A(2,3)\) and \(B(8,k)\) are joined by a straight line. The point \(C(5,5)\) lies on this line. Determine the value of \(k\).
A. 8
Correct B. 7
C. 6
D. 9

Correct Answer: B

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