Practice gce model questions vol1 mathematics 2020 Past Questions

Question 1 View Details

The acute angle \( \theta \) satisfies \( \sin\theta + \sqrt{3}\cos\theta = 2 \). Find the exact value of \( \tan\theta \) and then evaluate \( \cos 3\theta \).

Correct A. tan\theta = \frac{\sqrt{3}}{3}, \; \cos 3\theta = 0

B. tan\\theta = 0, \\; \\cos 3\\theta = 0

C. tan\\theta = \\sqrt{3}, \\; \\cos 3\\theta = \\frac{1}{2}

D. tan\\theta = \\frac{\\sqrt{3}}{2}, \\; \\cos 3\\theta = -1

Correct Answer: A

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Question 2 View Details

In the diagram, points \(A(1,2)\) and \(B(5,4)\) are plotted on a coordinate plane. The perpendicular bisector of segment \(AB\) meets the x‑axis at point \(C\). Determine the coordinates of \(C\) and the distance \(BC\).

A. C\\left(\\frac{9}{2},0\\right), \\; BC = \\frac{\\sqrt{61}}{2}

B. C\\left(\\frac{5}{2},0\\right), \\; BC = \\frac{\\sqrt{41}}{2}

Correct C. C\left(\frac{9}{2},0\right), \; BC = \frac{\sqrt{65}}{2}

D. C\\left(\\frac{7}{2},0\\right), \\; BC = \\frac{\\sqrt{65}}{2}

Correct Answer: C

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Question 3 View Details

For a positive integer \(n\), define \(S_n = \sum_{k=1}^{n} k\,2^{k}\). Derive a closed‑form expression for \(S_n\) in terms of \(n\) and then evaluate \(S_5\).

A. S_n = (n)2^{\,n+1}; \; S_5 = 320

B. S_n = 2 + (n-1)2^{\,n}; \; S_5 = 130

Correct C. S_n = 2 + (n-1)2^{\,n+1}; \; S_5 = 258

D. S_n = 2 + (n-2)2^{\,n+1}; \; S_5 = 226

Correct Answer: C

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Question 4 View Details

Vectors \(\mathbf{u} = (x,2,-1)\) and \(\mathbf{v} = (1,y,3)\) satisfy \(\mathbf{u}\cdot\mathbf{v}=4\) and \(|\mathbf{u}| = 5\). Find all possible ordered pairs \((x,y)\) and, for each pair, determine the angle between \(\mathbf{u}\) and \(\mathbf{v}\) (to one decimal place).

A. (x,y) = (2\sqrt{5}, (7+2\sqrt{5})\/2) with angle ≈ 45.0°, or (x,y) = (-2\sqrt{5}, (7-2\sqrt{5})\/2) with angle ≈ 135.0°

Correct B. (x,y) = (2\sqrt{5}, (7-2\sqrt{5})/2) with angle ≈ 76.4°, or (x,y) = (-2\sqrt{5}, (7+2\sqrt{5})/2) with angle ≈ 82.9°

C. (x,y) = (\sqrt{5}, (7-\sqrt{5})\/2) with angle ≈ 70.0°, or (x,y) = (-\sqrt{5}, (7+\sqrt{5})\/2) with angle ≈ 85.0°

D. (x,y) = (3\sqrt{5}, (7-3\sqrt{5})\/2) with angle ≈ 60.0°, or (x,y) = (-3\sqrt{5}, (7+3\sqrt{5})\/2) with angle ≈ 90.0°

Correct Answer: B

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Question 5 View Details

A class of 20 students obtained the following test scores (grouped intervals). 0-9: 2 students, 10-19: 3, 20-29: 5, 30-39: 6, 40-49: 3, 50-59: 1. Assuming each student scored the midpoint of his/her interval, calculate the mean score for the class. A second class of 20 students has a mean of 28 and a standard deviation of 8. Which class has the higher coefficient of variation? State the percentages (to one decimal place).

A. First class mean = 28.5, CV ≈ 40.0%; second class CV ≈ 28.6%; therefore the first class has the higher coefficient of variation.

B. First class mean = 27.5, CV ≈ 45.0%; second class CV ≈ 28.6%; therefore the first class has the higher coefficient of variation.

Correct C. First class mean = 28.5, CV ≈ 46.3%; second class CV ≈ 28.6%; therefore the first class has the higher coefficient of variation.

D. First class mean = 28.5, CV ≈ 46.3%; second class CV ≈ 30.0%; therefore the second class has the higher coefficient of variation.

Correct Answer: C

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Question 6 View Details

Three numbers have a sum of \(48\). The first number exceeds the second by \(4\), and the third number is half the sum of the first and second. Find the first number.

A. 24

B. 20

C. 14

Correct D. 18

Correct Answer: D

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Question 7 View Details

In a school of 120 students, the numbers studying Mathematics (set A), Physics (set B) and Chemistry (set C) are as follows: \(|A|=70\), \(|B|=65\), \(|C|=50\). The pairwise intersections are \(|A\cap B|=30\), \(|A\cap C|=25\), \(|B\cap C|=20\). No number is given for the three‑way intersection. Using the Venn diagram, find how many students study exactly one of the three subjects.

A. 70

Correct B. 65

C. 55

D. 80

Correct Answer: B

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Question 8 View Details

Find the smallest positive integer that leaves a remainder of \(5\) when divided by \(7\) and a remainder of \(2\) when divided by \(9\).

A. 110

Correct B. 47

C. 68

D. 23

Correct Answer: B

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Question 9 View Details

A rectangular swimming pool has length \(5\) m more than twice its width and a surface area of \(180\) m². A uniform deck of width \(d\) metres surrounds the pool, and the total area (pool plus deck) is \(300\) m². Determine the width \(d\) of the deck (give your answer to two decimal places).

A. 2.05

B. 1.95

Correct C. 1.79

D. 1.23

Correct Answer: C

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Question 10 View Details

Solve the inequality \(\frac{2x-5}{x+3}\le 1\), stating the solution set in interval notation.

A. (-∞, 8]

B. [-3, 8]

Correct C. (-3, 8]

D. (-3, 8)

Correct Answer: C

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Question 11 View Details

A straight line passes through the point \( (2,5) \) and together with the coordinate axes encloses a triangle of area \(25\) square units. Find the possible gradients of the line, giving your answers correct to two decimal places.

Correct A. -0.96 and -6.55

B. -1.10 and -7.20

C. -0.50 and -4.00

D. -0.85 and -5.90

Correct Answer: A

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Question 12 View Details

The time \(T\) (in seconds) taken by a particle to travel a distance \(s\) (in metres) under a constant acceleration \(a\) (in \(\text{m s}^{-2}\)) varies directly with \(s\) and inversely with \(\sqrt{a}\). A particle travels \(200\) m in \(10\) s when the acceleration is \(a_{1}\). Find the time required to travel \(450\) m if the acceleration is increased to \(a_{2}=4a_{1}\). Give your answer in seconds to two decimal places.

A. 12.80

B. 10.50

C. 9.75

Correct D. 11.25

Correct Answer: D

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Question 13 View Details

In triangle \(ABC\) the side lengths are \(AB = 13\) cm, \(AC = 15\) cm and \(BC = 14\) cm. A circle passes through points \(B\) and \(C\) and is tangent to side \(AB\) at \(B\). Find the radius of the circle, giving your answer as an exact fraction and also to two decimal places.

A. \(\frac{91}{10}\) cm (≈9.10 cm)

B. \(\frac{85}{12}\) cm (≈7.08 cm)

Correct C. \(\frac{91}{12}\) cm (≈7.58 cm)

D. \(\frac{13}{2}\) cm (≈6.50 cm)

Correct Answer: C

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Question 14 View Details

In a school, 40 % of students belong to the Drama club, 35 % to the Science club and 30 % to the Sports club. The percentages of students belonging to both clubs are: 15 % in Drama & Science, 10 % in Drama & Sports, and 12 % in Science & Sports. Five percent of the students belong to all three clubs. If a student is known to belong to at least one of these clubs, what is the probability that the student belongs to exactly one club? Express your answer as a fraction in lowest terms.

A. 46\/72

Correct B. 46/73

C. 45\/73

D. 47\/73

Correct Answer: B

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Question 15 View Details

Vectors \(\mathbf{a}\) and \(\mathbf{b}\) lie in a plane with magnitudes \(|\mathbf{a}| = 5\) and \(|\mathbf{b}| = 8\). It is given that the vector \(\mathbf{a} + 2\mathbf{b}\) is perpendicular to the vector \(3\mathbf{a} - \mathbf{b}\). Find the angle \(\theta\) between \(\mathbf{a}\) and \(\mathbf{b}\) (where \(0^{\circ} < \theta < 180^{\circ}\)). Give your answer in degrees correct to one decimal place.

A. 82.1°

B. 90.0°

Correct C. 74.5°

D. 65.3°

Correct Answer: C

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Question 16 View Details

The volume \(V\) of a gas varies directly with the amount of gas \(n\) and the absolute temperature \(T\), and varies inversely with the pressure \(P\). When \(V=500\) cm\(^3\), \(n=2\) mol, \(T=300\) K and \(P=1\) atm, the gas is changed so that the amount of gas increases by 50 %, the temperature rises by 20 % and the pressure rises by 25 %. Find the new volume (in cm\(^3\)).

Correct A. 720

B. 900

C. 660

D. 800

Correct Answer: A

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Question 17 View Details

A box contains red, blue and green marbles. The probability of drawing a red marble is twice the probability of drawing a blue marble, and the probability of drawing a green marble is 0.2. Two marbles are drawn with replacement. Given that at least one of the drawn marbles is red, find the probability that both marbles are red. Give your answer as a fraction in lowest terms.

A. 3/11

Correct B. 4/11

C. 5/11

D. 2/5

Correct Answer: B

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Question 18 View Details

The system of equations \(\begin{cases}2x + y = 5\\ kx - 3y = 4\end{cases}\) has a solution \((x,y)\). Find the sum of all integer values of the parameter \(k\) for which the solution satisfies \(x > y\).

A. 2

B. -1

C. 1

Correct D. 0

Correct Answer: D

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Question 19 View Details

Find the smallest positive integer that leaves a remainder of 3 when divided by 5 and a remainder of 4 when divided by 7.

Correct A. 18

B. 17

C. 23

D. 19

Correct Answer: A

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Question 20 View Details

A number is increased by 50 % and then reduced by 20 % of the original number. The final result is 65. Find the original number.

A. 55

B. 45

Correct C. 50

D. 70

Correct Answer: C

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Question 21 View Details

Simplify the expression \( \frac{2^{3}\times\sqrt{8}}{4^{\frac{3}{2}}}\) and give your answer in simplest surd form.

A. \sqrt{2}

Correct B. 2\sqrt{2}

C. 2\sqrt{3}

D. 4\sqrt{2}

Correct Answer: B

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Question 22 View Details

In the coordinate plane, points A(2,3) and B(8,-1) are given. Let P be the point on line AB that divides the segment internally in the ratio AP:PB = 1:2. A line through P is drawn perpendicular to AB and meets the x‑axis at Q. Find the distance \(PQ\) expressed in simplest surd form.

A. \frac{5\sqrt{7}}{9}

B. \frac{\sqrt{13}}{9}

C. \frac{5\sqrt{13}}{3}

Correct D. \frac{5\sqrt{13}}{9}

Correct Answer: D

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Question 23 View Details

A closed rectangular box has a square base of side \(x\) cm and height \(h\) cm. Its total surface area (including top and bottom) is \(1500\) cm\(^2\). Determine the dimensions that give the greatest possible volume and state that maximum volume in simplest surd form.

A. 1250\sqrt{15}

Correct B. 1250\sqrt{10}

C. 1250\sqrt{5}

D. 2500\sqrt{10}

Correct Answer: B

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Question 24 View Details

The quadratic equation \(x^{2} - kx + 12 = 0\) has two real roots whose difference is \(2\). Find the value of \(k\) and then give the product of the squares of the roots.

A. 169

B. 121

Correct C. 144

D. 196

Correct Answer: C

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Question 25 View Details

The time \(T\) (in minutes) required for a machine to process a batch varies directly as the number of items \(n\) and inversely as the square of the speed setting \(s\). If processing 200 items at speed setting 5 takes 40 minutes, how many minutes will be needed to process 500 items at speed setting 8? Express your answer as an improper fraction.

A. 100/1

B. 125/16

Correct C. 625/16

D. 625/2

Correct Answer: C

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