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gce model questions vol1 2024 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 2024 mathematics (Objective) questions designed to simulate the real exam environment.

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Question 1 View Details

In an arithmetic progression \((a_n)\), the third term is twice the first term and the sum of the first six terms is \(84\). Find the ninth term \(a_9\).

A. 35/9

B. 140/9

Correct C. 280/9

D. 280/8

Correct Answer: C

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Question 2 View Details

Solve the inequality \(|2x - 5| < x^2 - 3x + 2\) and give the sum of all integer solutions that lie between \(-10\) and \(10\) inclusive.

A. 0

B. -3

C. -1

Correct D. -2

Correct Answer: D

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Question 3 View Details

A shop marks the price of an item at \(\frac{7}{9}\) of £\(180\). It then offers a discount of \(12.5\%\). What is the discounted price in pounds, correct to two decimal places?

A. 119.00

B. 130.00

Correct C. 122.50

D. 127.50

Correct Answer: C

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Question 4 View Details

If real numbers \(x\) and \(y\) satisfy \(x + y = 6\) and \(x^{2} + y^{2} = 20\), find the value of \(x^{3}y + xy^{3}\).

A. 180

B. 144

C. 200

Correct D. 160

Correct Answer: D

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Question 5 View Details

How many five‑digit numbers can be formed using each of the digits \(1,2,3,4,5\) exactly once that are divisible by \(3\)?

A. 60

B. 100

C. 240

Correct D. 120

Correct Answer: D

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Question 6 View Details

In right‑angled triangle \(ABC\) with \(\angle A = 90^{\circ}\), the sides are \(AB = 10\), \(AC = 24\) and \(BC = 26\). The altitude from \(A\) meets \(BC\) at \(D\). A circle centred at \(D\) passes through \(B\) and meets \(AB\) again at \(E\) (where \(E \neq B\)). Find the length \(AE\).

A. 1200/169

B. 1189/169

Correct C. 1190/169

D. 1190/168

Correct Answer: C

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Question 7 View Details

The graph of a function \(f\) is defined for \(0\le x\le5\) as follows: from \((0,2)\) to \((2,6)\) it is a straight line, and from \((2,6)\) to \((5,3)\) it is the quadratic curve \(y = -\tfrac12 x^{2}+\tfrac52 x+3\). Calculate the exact value of the definite integral \(\displaystyle\int_{0}^{5} f(x)\,dx\).

A. 127/3

B. 125/4

Correct C. 127/4

D. 128/4

Correct Answer: C

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Question 8 View Details

The quadratic equation \(x^{2}-(k+1)x+(k^{2}-5)=0\) has real roots \(r_{1},r_{2}\). Given that \(r_{1}+\dfrac{1}{r_{2}}=3\), determine all possible values of \(k\) (to two decimal places).

Correct A. k≈2.96 or k≈-2.16

B. k≈1.00 or k≈-3.00

C. k≈2.50 or k≈-2.50

D. k≈3.14 or k≈-1.80

Correct Answer: A

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Question 9 View Details

A 250 g mixture contains 40 % sugar by mass. How many grams of pure sugar must be added to raise the sugar concentration to 55 %? Give your answer as a fraction in lowest terms.

A. 125/2

Correct B. 250/3

C. 75/1

D. 100/3

Correct Answer: B

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Question 10 View Details

Let \(U=\{1,2,\dots ,20\}\). Define \(A=\{x\in U\mid x\text{ divides }60\}\), \(B=\{x\in U\mid x\text{ is a multiple of }3\}\) and \(C=\{x\in U\mid x\text{ is a perfect square}\}\). Find the number of elements in \((A\cap B)\cup(C\setminus A)\).

A. 9

Correct B. 6

C. 5

D. 7

Correct Answer: B

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Question 11 View Details

Given the expression \(A = \frac{2x^{3} - 6x^{2} + 4x}{x^{2} - 3x + 2}\). For which integer values of \(x\) does \(A = 4\) hold? State all such integers.

Correct A. no solution

B. no integer solution

C. x = 2

D. x = 1

Correct Answer: A

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Question 12 View Details

Find the integer value of \(k\) such that the expression \(\frac{x^{2} - kx + 12}{x-3}\) is an integer for every integer \(x\) with \(1\le x\le5\) and \(x\neq3\).

Correct A. 7

B. 5

C. 8

D. 10

Correct Answer: A

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Question 13 View Details

Consider the system of equations \(2y - x = 3\) and \(3x + ky = 12\). Find all integer values of the parameter \(k\) with \(-10\le k\le 10\) for which the solution \((x,y)\) has \(x\) an integer.

A. -9, -8, -7, -5, -4, -3, 0, 1, 7

Correct B. -9, -8, -7, -5, -4, -3, 0, 1, 8

C. -9, -8, -6, -5, -4, -3, 0, 1, 8

D. -9, -8, -7, -5, -4, -3, 0, 2, 8

Correct Answer: B

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Question 14 View Details

For the quadratic equation \(x^{2} - (a+1)x + a = 0\) the two roots differ by exactly \(2\). Determine all real values of the parameter \(a\).

Correct A. 3, -1

B. 3, 1

C. 2, -1

D. -3, -1

Correct Answer: A

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Question 15 View Details

A shop sells two types of pens. Type A costs £\(p\) per pen and Type B costs £\(p+2\) per pen. A customer buys a total of 20 pens for £150. The number of Type B pens is twice the number of Type A pens minus 4. Find the price \(p\) (in pounds) of a Type A pen.

A. 8.0

Correct B. 6.3

C. 5.5

D. 7.2

Correct Answer: B

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Question 16 View Details

In triangle ABC, the side opposite angle A has length 5 cm and the side opposite angle B has length 3 cm. If angle A is twice angle B, find \(\sin C\).

A. \frac{8\sqrt{11}}{30}

B. \frac{7\sqrt{11}}{27}

Correct C. \frac{8\sqrt{11}}{27}

D. \frac{8\sqrt{13}}{27}

Correct Answer: C

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Question 17 View Details

The quadratic equation \(x^{2}-kx+12=0\) has roots whose ratio equals the ratio of the two roots of \(2x^{2}-7x+5=0\). Find \(k\).

Correct A. \frac{7\sqrt{30}}{5}

B. \frac{7\sqrt{31}}{5}

C. \frac{7\sqrt{30}}{6}

D. \frac{6\sqrt{30}}{5}

Correct Answer: A

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Question 18 View Details

Find all real values of \(x\) that satisfy \(\frac{2x-3}{x+4}=\frac{k-5}{k+2}\) together with the relation \(3k-7=2x\).

Correct A. -5\pm\frac{\sqrt{114}}{2}

B. -5\pm\frac{\sqrt{114}}{3}

C. -5\pm\frac{\sqrt{144}}{2}

D. -5\pm\frac{\sqrt{124}}{2}

Correct Answer: A

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Question 19 View Details

A right circular cone is inscribed in a sphere of radius \(R\). The cone's height is twice its base radius. Determine the ratio of the volume of the cone to the volume of the sphere, expressed in simplest fractional form.

A. \frac{32}{135}

Correct B. \frac{32}{125}

C. \frac{27}{125}

D. \frac{64}{125}

Correct Answer: B

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Question 20 View Details

The set of points \(P(x,y)\) satisfies \(PA^{2}+PB^{2}=2\,d^{2}\), where \(A(1,2)\) and \(B(4,6)\) and \(d\) is the perpendicular distance from \(P\) to the line \(2x-y+3=0\). (i) Derive the Cartesian equation of the locus. (ii) Find the coordinates of the point on this locus that is nearest to the origin, giving your answer correct to two decimal places.

A. (3.10, 0.90)

B. (3.50, 1.20)

Correct C. (3.37, 1.33)

D. (2.90, 1.80)

Correct Answer: C

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Question 21 View Details

Find the remainder when (7^{2021}) is divided by 100.

A. 3

B. 9

Correct C. 7

D. 1

Correct Answer: C

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Question 22 View Details

A quantity (y) varies directly as (x) and inversely as (z), and also varies directly as \(\sqrt{w}\). When (y = 30), (x = 5), (z = 2) and (w = 9). Find the value of (y) when (x = 8), (z = 4) and (w = 16).

A. 48

Correct B. 32

C. 40

D. 24

Correct Answer: B

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Question 23 View Details

Solve for (x): \(\log_{2}(x^{2} - 5x + 6) = \log_{2}(3x - 9)\). Then evaluate \(\sqrt{x + 4}\).

A. 2

Correct B. 3

C. 4

D. 5

Correct Answer: B

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Question 24 View Details

In a plane, vectors \(\mathbf{a}\) and \(\mathbf{b}\) satisfy \(|\mathbf{a}| = 5\) and \(|\mathbf{b}| = 7\). The vector \(\mathbf{c} = 2\mathbf{a} - \mathbf{b}\) is perpendicular to \(\mathbf{d} = \mathbf{a} + 3\mathbf{b}\). Find the angle between \(\mathbf{a}\) and \(\mathbf{b}\) in degrees (to one decimal place).

A. 63.2

B. 45.7

Correct C. 56.3

D. 70.0

Correct Answer: C

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Question 25 View Details

Solve the inequality \(\displaystyle \frac{x^{2} - 5x + 6}{x^{2} - 4x - 12} \ge 0\) and express the solution set in interval notation.

A. (-\infty, -2) \cup [2,3] \cup [6, \infty)

Correct B. (-\infty, -2) \cup [2,3] \cup (6, \infty)

C. (-\infty, -2) \cup [2,3) \cup (6, \infty)

D. (-\infty, -2] \cup (2,3) \cup (6, \infty)

Correct Answer: B

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