Practice gce model questions vol1 mathematics 2021 Past Questions

Question 1 View Details

Find the smallest positive integer \(n>10\) such that the decimal representation of \(n^{2}\) ends with the digits of \(n\).

A. 30

B. 15

C. 36

Correct D. 25

Correct Answer: D

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Question 2 View Details

In triangle \(ABC\) we have \(\angle B = 2\angle A\). The side opposite \(A\) (\(BC\)) is 7 units long and the side opposite \(C\) (\(AB\)) is 5 units long. Find \(\sin \angle A\).

A. \frac{4\sqrt7}{7}

Correct B. \frac{2\sqrt7}{7}

C. \frac{2\sqrt7}{5}

D. \frac{\sqrt7}{7}

Correct Answer: B

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Question 3 View Details

What is the remainder when \(7^{2021}\) is divided by \(1000\)?

A. 0

Correct B. 7

C. 1

D. 13

Correct Answer: B

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Question 4 View Details

The region bounded by the curves \(y=\sqrt{x}\), \(y=2\) and the line \(x=0\) is rotated 360° about the \(y\)-axis. Find the volume of the solid of revolution, giving your answer in terms of \(\pi\).

Correct A. \frac{32\pi}{5}

B. \frac{24\pi}{5}

C. \frac{40\pi}{5}

D. \frac{32\pi}{3}

Correct Answer: A

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Question 5 View Details

A bag contains 5 red, 4 blue and 3 green balls (12 in total). Two balls are drawn sequentially without replacement. If the first ball drawn is red, an extra blue ball is added to the bag before a third ball is drawn. Find the probability that exactly two of the three drawn balls are blue.

A. \frac{470}{1980}

B. \frac{463}{1980}

Correct C. \frac{467}{1980}

D. \frac{467}{1970}

Correct Answer: C

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Question 6 View Details

In a school, let \(A\) be the set of students who study Mathematics and \(B\) the set of students who study Physics. It is known that \(|A| = 120\), \(|B| = 90\), the number of students who study neither subject is 30, and the number of students who study Mathematics only is twice the number who study Physics only. Find \(|A \cap B|\).

A. 45

B. 30

Correct C. 60

D. 90

Correct Answer: C

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Question 7 View Details

The quadratic equation \(x^{2}-(k+1)x+(k-2)=0\) has two rational roots whose difference is \(3\). Determine all possible values of \(k\).

A. k = -1 or k = 3

Correct B. k = 0 or k = 2

C. k = 0 or k = 3

D. k = 1 or k = 4

Correct Answer: B

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Question 8 View Details

The diagram shows a parabola passing through the points \((0,2)\), \((2,6)\) and \((4,2)\). Determine the equation of the parabola in the form \(y = ax^{2} + bx + c\).

Correct A. y = -x^{2} + 4x + 2

B. y = x^{2} + 4x + 2

C. y = -x^{2} - 4x + 2

D. y = -x^{2} + 4x - 2

Correct Answer: A

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Question 9 View Details

A closed cylindrical can is to hold exactly \(5000\text{ cm}^{3}\) of liquid. The material used for the curved side costs twice as much per square centimetre as the material used for each of the top and bottom. Determine the radius (in cm) and height (in cm) that minimise the total cost of the can. Give your answers correct to two decimal places.

A. radius ≈ 12.50 cm, height ≈ 10.00 cm

B. radius ≈ 10.00 cm, height ≈ 15.92 cm

C. radius ≈ 11.68 cm, height ≈ 12.00 cm

Correct D. radius ≈ 11.68 cm, height ≈ 11.67 cm

Correct Answer: D

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Question 10 View Details

In three‑dimensional space, vectors \(\mathbf{a}\) and \(\mathbf{b}\) satisfy \(|\mathbf{a}| = 5\), \(|\mathbf{b}| = 7\) and the angle between them is \(60^{\circ}\). Let \(\mathbf{c} = \mathbf{a} + \lambda\mathbf{b}\) be perpendicular to \(\mathbf{a} - 2\mathbf{b}\). Find the value of \(\lambda\) and the magnitude \(|\mathbf{c}|\) (give \(|\mathbf{c}|\) correct to two decimal places).

A. λ = -7/20, |c| ≈ 6.00

Correct B. λ = -20/161, |c| ≈ 4.63

C. λ = -5/41, |c| ≈ 5.12

D. λ = 3/8, |c| ≈ 3.87

Correct Answer: B

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Question 11 View Details

The system of linear equations (kx + 4y = 12) (2x + ky = 8) has no solution for certain values of the parameter (k). (a) Find all such values of (k). (b) Then, for (k = 4), solve the system and give the ordered pair \((x, y)\).

A. k = \pm 2; for k = 4, (x, y) = (2, 1)

B. k = \pm 3; for k = 4, (x, y) = (2, 1)

Correct C. k = \pm 2\sqrt{2}; for k = 4, (x, y) = (2, 1)

D. k = \pm 2\sqrt{2}; for k = 4, (x, y) = (1, 2)

Correct Answer: C

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Question 12 View Details

The two classes A and B have the following statistics for a test: - Class A: 15 students, mean \(72\), standard deviation \(6\). - Class B: 20 students, mean \(68\), standard deviation \(5\). When the two classes are combined, what is the overall mean (to 1 decimal place) and the combined standard deviation (to 2 decimal places)?

Correct A. Overall mean \approx 69.7; combined standard deviation \approx 5.80

B. Overall mean \approx 70.2; combined standard deviation \approx 5.50

C. Overall mean \approx 69.7; combined standard deviation \approx 6.00

D. Overall mean \approx 68.9; combined standard deviation \approx 6.10

Correct Answer: A

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Question 13 View Details

A circle with centre at the origin O\((0,0)\) has radius 5. A chord AB of this circle has slope \(-\frac{1}{2}\) and its midpoint M lies on the line \(y = -2x + 3\). Find the exact coordinates of points A and B.

Correct A. A\(\bigl(\frac{3+2\sqrt{71}}{4},\frac{6-\sqrt{71}}{4}\bigr)\), B\(\bigl(\frac{3-2\sqrt{71}}{4},\frac{6+\sqrt{71}}{4}\bigr)\)

B. A\bigl(\frac{3+2\sqrt{71}}{4},\frac{6+\sqrt{71}}{4}\bigr), B\bigl(\frac{3-2\sqrt{71}}{4},\frac{6-\sqrt{71}}{4}\bigr)

C. A\bigl(\frac{3+\sqrt{71}}{4},\frac{6-2\sqrt{71}}{4}\bigr), B\bigl(\frac{3-\sqrt{71}}{4},\frac{6+2\sqrt{71}}{4}\bigr)

D. A\bigl(\frac{5+2\sqrt{71}}{4},\frac{4-\sqrt{71}}{4}\bigr), B\bigl(\frac{5-2\sqrt{71}}{4},\frac{4+\sqrt{71}}{4}\bigr)

Correct Answer: A

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Question 14 View Details

Simplify the expression \(\frac{2^{5}\cdot 2^{-2}}{2^{-3}}\) and give your answer as an integer.

A. 8

B. 128

C. 32

Correct D. 64

Correct Answer: D

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Question 15 View Details

The time \(t\) (in seconds) for a stone to fall from rest is directly proportional to the square root of the height \(h\) (in metres): \(t = k\sqrt{h}\). A stone dropped from a height of 20 m takes 2.0 s. Because of air resistance, the actual time for a drop from 45 m is 1.5 s longer than the time predicted by the direct‑variation law. Find the measured time (in seconds) for the 45 m drop, correct to two decimal places.

A. 4.00

Correct B. 4.50

C. 3.00

D. 5.00

Correct Answer: B

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Question 16 View Details

A bag contains red, blue and green balls. The probability that a ball drawn at random is red is \(\frac13\). If a red ball is drawn first and not replaced, the probability that the second ball drawn is blue is \(\frac27\). Two balls are drawn without replacement and it is known that at least one of them is blue. Find the probability that the other ball is red.

A. 3/5

Correct B. 2/5

C. 2/3

D. 1/3

Correct Answer: B

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Question 17 View Details

In triangle \(ABC\) the side lengths are \(AB=10\,\text{cm},\; AC=13\,\text{cm},\; BC=15\,\text{cm}.\) Let \(M\) be the midpoint of \(BC\). The line \(AM\) meets the circumcircle of \(ABC\) again at \(D\) (where \(D\neq A\)). Find the length of \(AD\) correct to three significant figures.

A. 13.9

Correct B. 15.2

C. 14.8

D. 16.5

Correct Answer: B

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Question 18 View Details

In a survey of 170 students, the numbers studying Mathematics (M), Physics (P) and Chemistry (C) are as follows: \(n(M)=90\), \(n(P)=80\), \(n(C)=70\). The numbers studying each pair of subjects are \(n(M\cap P)=45\), \(n(M\cap C)=40\), \(n(P\cap C)=35\). Let \(x\) be the number of students who study all three subjects. It is known that the number of students who study none of the three subjects equals the number who study exactly two subjects. Find \(x\).

A. 55

Correct B. 40

C. 30

D. 45

Correct Answer: B

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Question 19 View Details

A solid consists of a right circular cylinder of radius \(r\) cm and height \(h\) cm, topped by a right circular cone of the same base radius \(r\) and height \(\frac{h}{2}\). The total surface area (including the base of the cylinder) is \(600\,\text{cm}^2\). It is also given that the height of the cylinder equals twice its radius (\(h = 2r\)). Find the volume of the solid, correct to the nearest cubic centimetre.

Correct A. 958

B. 1020

C. 1125

D. 850

Correct Answer: A

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Question 20 View Details

For a certain gas the pressure \(P\) (in kPa) and volume \(V\) (in L) are related by \(P = k V^{n}\), where \(k\) and \(n\) are constants. Experimental data give \(P=150\) kPa when \(V=2\) L and \(P=60\) kPa when \(V=5\) L. Determine the exponent \(n\) and the constant \(k\). Then predict the pressure when the volume is \(3\) L. Give your final answer for the pressure to the nearest kilopascal.

A. 90

B. 110

C. 130

Correct D. 100

Correct Answer: D

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Question 21 View Details

The sequence \((a_n)\) is arithmetic. It satisfies that the sum of the first five terms equals twice the fifth term, and the sum of the first five terms is 80. Find the common difference \(d\) and the first term \(a_1\), then give the sum of the first twelve terms.

Correct A. 696

B. 680

C. 720

D. 750

Correct Answer: A

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Question 22 View Details

The time \(t\) (seconds) taken for a particle to travel a distance \(s\) (metres) under constant acceleration \(a\) (m\,s\(^{-2}\)) varies directly with \(\sqrt{s}\) and inversely with \(\sqrt{a}\). The particle's final speed \(v\) (m\,s\(^{-1}\)) varies directly with \(\sqrt{as}\). When \(s = 200\) m and \(a = 4\) m s\(^{-2}\), the time is \(10\) s and the final speed is \(20\) m s\(^{-1}\). Find the time and final speed when \(s = 450\) m and \(a = 9\) m s\(^{-2}\).

Correct A. 10 s and 45 m/s

B. 8 s and 50 m\/s

C. 12 s and 40 m\/s

D. 10 s and 40 m\/s

Correct Answer: A

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Question 23 View Details

Simplify the expression \(E = \frac{x^{3} - 6x^{2} + 11x - 6}{x^{2} - 5x + 6}\). Then find the sum of all positive integer values of \(x\) for which the simplified value of \(E\) is less than \(10\).

A. 45

B. 55

C. 60

Correct D. 50

Correct Answer: D

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Question 24 View Details

Simplify \(\displaystyle \frac{2^{3}\cdot5^{2}}{10^{2}}\times\sqrt{50}\) and give your answer in simplest surd form.

Correct A. 10\sqrt{2}

B. 10\sqrt{5}

C. 20\sqrt{2}

D. 5\sqrt{2}

Correct Answer: A

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Question 25 View Details

The line \(L_{1}\) is given by \(3x - ky = 12\). A second line \(L_{2}\) passes through the point \((2,-1)\) and is perpendicular to \(L_{1}\). Find the value of \(k\) for which the intersection point of \(L_{1}\) and \(L_{2}\) has integer coordinates, and state those coordinates.

A. k = 6, intersection point (3, -2)

B. k = 4, intersection point (3, -2)

Correct C. k = 6, intersection point (2, -1)

D. k = 8, intersection point (1, 0)

Correct Answer: C

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