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igcse model questions vol1 2023 mathematics | Objective

Prepare for your exams with igcse 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 2023 mathematics (Objective) questions designed to simulate the real exam environment.

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Question 1 View Details

A number is increased by \(15\%\) and then the resulting amount is decreased by \(15\%\). The final amount is 2.25 less than the original number. Find the original number.

A. 85

B. 125

C. 115

Correct D. 100

Correct Answer: D

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Question 2 View Details

A manufacturing process produces defective items with probability 0.02 independently. A batch of 200 items is inspected. Using an appropriate approximation, find the probability that at most 3 items are defective. Give your answer correct to three decimal places.

A. 0.453

B. 0.423

C. 0.400

Correct D. 0.433

Correct Answer: D

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Question 3 View Details

A box contains red, blue and green balls. The probability of drawing a green ball at random is 0.2. If two balls are drawn successively without replacement, the probability that the first ball is red and the second ball is blue is 0.14. Find the number of red balls in the box.

A. 12

Correct B. 14

C. 15

D. 18

Correct Answer: B

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Question 4 View Details

The function \(f\) is defined for all real numbers by \(f(x)=2x+3\). Another function \(g\) is defined by \(g(x)=kx-5\), where \(k\) is a constant. It is known that the compositions \(f\circ g\) and \(g\circ f\) are identical functions; that is, \((f\circ g)(x)=(g\circ f)(x)\) for every real \(x\).\n\n(a) Determine the value of \(k\).\n\n(b) Using this value of \(k\), find the value of \(x\) for which \((f\circ g)(x)=20\).

A. -82/4

B. -79/4

C. -81/5

Correct D. -81/4

Correct Answer: D

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Question 5 View Details

Solve the compound inequality \(5x-3<2x+7\le 5x+1\) and give the sum of all integer values of \(x\) that satisfy it.

A. 6

Correct B. 5

C. 4

D. 8

Correct Answer: B

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Question 6 View Details

If \(2^{x}= \sqrt{8}\) and \(3^{y}=27\), find the value of \(x+y\).

Correct A. 9/2

B. 5\/2

C. 11\/2

D. 7\/2

Correct Answer: A

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Question 7 View Details

In triangle \(ABC\), side \(AB = 8\) cm, side \(AC = 6\) cm, and \(\angle BAC = \theta\). The line \(AD\) is the internal bisector of \(\angle BAC\) meeting side \(BC\) at \(D\). Given that \(\sin\theta = \frac{3}{5}\), calculate the length of segment \(BD\) (in cm).

A. 8√145\/30

Correct B. 8√145/35

C. 6√145\/35

D. 8√125\/35

Correct Answer: B

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Question 8 View Details

If \(a = \log_{2}\!\left(\sqrt{32}\right)\) and \(b = \log_{2} 8\), find the value of \(a - b\).

Correct A. -1/2

B. 1/2

C. 3/2

D. -3/2

Correct Answer: A

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Question 9 View Details

Let \(p = \frac{2x - 3}{x + 4}\). If \(p = \frac{1}{2}\), evaluate \(\frac{x^{2} - 5x + 6}{x - 2}\).

Correct A. 1/3

B. -1/3

C. 2

D. 1/2

Correct Answer: A

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Question 10 View Details

The number \(k\) satisfies \(k^{2} - 4k + 3 = k - 1\). Find the value of the expression \(k^{2} - 5k + 6\).

A. 3

B. 1

C. 4

Correct D. 2

Correct Answer: D

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Question 11 View Details

A line L passes through the point A(2, -1) and is perpendicular to the line through B(5, 3) and C(k, 7). The line L also passes through a point D whose coordinates satisfy \(x = 2y\) and \(y = x - 4\). Find the value of k and the coordinates of D.

Correct A. k = 5/3; D(8,4)

B. k = 5\/2; D(8,5)

C. k = 2; D(10,5)

D. k = 4\/3; D(6,3)

Correct Answer: A

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Question 12 View Details

In the diagram the points A(0,0), B(8,0) and C(3,6) are plotted on a coordinate plane. The line through B and C is extended to meet the y‑axis at point D. (i) Find the coordinates of D. (ii) Find the length of CD, expressed in simplest surd form. (iii) Find the area of triangle ABD.

Correct A. D(0,48/5); CD = 3√61/5; area = 192/5

B. D(0,48\/5); CD = 2√61\/5; area = 192\/5

C. D(0,10); CD = 3√61\/5; area = 192\/5

D. D(0,48\/5); CD = 3√61\/5; area = 180\/5

Correct Answer: A

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Question 13 View Details

Find the sum of all integer values of \(x\) with \(-10 \le x \le 10\) that satisfy \(\displaystyle \frac{|2x-5|}{x-3} < 2\).

A. -50

B. -48

C. -54

Correct D. -52

Correct Answer: D

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Question 14 View Details

Let the universal set be \(U = \{1,2,3,4,5,6,7,8,9,10\}\). Define \(A = \{x\in U \mid x\text{ is a multiple of }2\text{ or }3\}\) and \(B = \{x\in U \mid x>5\}\). Determine the number of elements in \((A\cap B)\,\cup\,(U\setminus (A\cup B))\).

A. 7

B. 5

C. 8

Correct D. 6

Correct Answer: D

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Question 15 View Details

In three‑dimensional space let \(\mathbf{a} = (2,\,-1,\,3)\) and \(\mathbf{b} = (k,\,4,\,-2)\). Let \(\mathbf{c} = \mathbf{a}\times\mathbf{b}\). It is given that \(|\mathbf{c}| = 13\) and that the vector \(\mathbf{d} = \mathbf{a} + 2\mathbf{b}\) is orthogonal to \(\mathbf{c}\). If \(k < -1\), find the exact value of \(k\) and then give the angle between \(\mathbf{a}\) and \(\mathbf{b}\) to the nearest degree.

A. k = -3 - \\frac{\\sqrt{290}}{10}; angle ≈ 150°

B. k = -2 - \\frac{\\sqrt{250}}{10}; angle ≈ 143°

Correct C. k = -2 - \frac{\sqrt{290}}{10}; angle ≈ 143°

D. k = -2 + \\frac{\\sqrt{290}}{10}; angle ≈ 37°

Correct Answer: C

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Question 16 View Details

The time \(T\) (in minutes) required to fill a tank varies directly with the volume \(V\) (in litres) and inversely with the flow rate \(R\) (in litres per minute). However, if the flow rate exceeds \(20\) L/min, turbulence increases the effective filling time by \(10\%\). When \(V=300\) L and \(R=15\) L/min, the tank is filled in \(20\) min. What is the minimum flow rate (in L/min) that can be used to fill a \(450\) L tank in at most \(18\) min?

A. 30

B. 25

Correct C. 27.5

D. 22.5

Correct Answer: C

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Question 17 View Details

Solve the equation \(\frac{2}{x-1}-\frac{3}{x+2}=\frac{5}{x^{2}+x-2}\) and then evaluate the expression \(\frac{x^{2}+3x+2}{x+1}\) for the solution obtained.

A. 2

B. 5

Correct C. 4

D. -1

Correct Answer: C

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Question 18 View Details

A class of 15 students is divided into Group X (12 students) and Group Y (3 students). The recorded overall mean score for the whole class is \(72\). Later it is discovered that one student's score in Group X was recorded \(5\) points higher than the actual score. Assuming all other recorded scores are correct, what is the corrected overall mean (to two decimal places)?

A. 71.57

B. 70.67

C. 72.67

Correct D. 71.67

Correct Answer: D

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Question 19 View Details

In right‑angled triangle \(ABC\) with \(\angle C = 90^{\circ}\), the hypotenuse \(AB\) is \(10\) cm and side \(AC\) is \(6\) cm. Point \(D\) lies on \(AB\) such that \(AD:DB = 1:2\). Find the length of \(CD\) (express your answer in simplest surd form).

Correct A. 4\sqrt{13}/3

B. 4\sqrt{13}/4

C. 4\sqrt{12}/3

D. 5\sqrt{13}/3

Correct Answer: A

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Question 20 View Details

The parabola \(y = x^{2} - 4x + 3\) can be written as \(y = (x-2)^{2} - 1\). A straight line \(y = kx + 1\) cuts the parabola at two distinct points. \(a)\) Determine the range of values of \(k\) for which the line meets the parabola in two distinct points. \(b\)\) For any such \(k\), find a formula for the area of the triangle formed by the two points of intersection and the vertex of the parabola. Express the area in terms of \(k\).

A. Area = |k-1| * sqrt((k+4)^2 - 8), with k > -4 + 2√2 or k < -4 - 2√2

B. Area = |k+1| * sqrt((k-4)^2 - 8), with k > -4 + 2√2 or k < -4 - 2√2

C. Area = |k+1| * sqrt((k+4)^2 + 8), with k > -4 + 2√2 or k < -4 - 2√2

Correct D. Area = |k+1| * sqrt((k+4)^2 - 8), with k > -4 + 2√2 or k < -4 - 2√2

Correct Answer: D

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Question 21 View Details

Solve the system of inequalities \(\frac{3x-4}{x+2}\le 2\) and \(x^{2}-5x+6>0\). Find the sum of all integer values of \(x\) that satisfy both inequalities.

A. 32

Correct B. 30

C. 28

D. 24

Correct Answer: B

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Question 22 View Details

Given that \(\sin\alpha+\cos\alpha=\frac{5}{4}\) and \(\alpha\) is an acute angle greater than \(45^{\circ}\), determine \(\tan\alpha\). Express your answer in simplest exact form.

Correct A. (16+5\sqrt{7})/9

B. (16+3\\sqrt{7})\/9

C. (8+5\\sqrt{7})\/9

D. (16-5\\sqrt{7})\/9

Correct Answer: A

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Question 23 View Details

A sector of a circle of radius \(r\) has an area equal to one‑third of the area of the whole circle. The sector is joined edge‑to‑edge to form a right circular cone. Find the ratio of the volume of the cone to the volume of a cylinder that has radius \(r\) and height \(r\). Give your answer as a simplified exact fraction.

Correct A. 2\sqrt{2}/81

B. \sqrt{2}\/81

C. 2\sqrt{2}\/27

D. 4\sqrt{2}\/81

Correct Answer: A

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Question 24 View Details

A straight line passes through the points \((2,3)\) and \((8,k)\). The line meets the \(x\)-axis at a point \((a,0)\) with \(2<a<8\). The total area enclosed between the line, the \(x\)-axis and the vertical lines \(x=2\) and \(x=8\) is \(30\) square units. Find the value of \(k\).

A. 5-\sqrt{46}

B. -5-\sqrt{45}

Correct C. -5-\sqrt{46}

D. -5+\sqrt{46}

Correct Answer: C

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Question 25 View Details

A right circular cone has base radius \(12\text{ cm}\) and slant height \(13\text{ cm}\). A smaller, similar cone is removed from the top so that the remaining frustum has a vertical height of \(3\text{ cm}\). Calculate the volume of the frustum. Give your answer in terms of \(\pi\) as a simplified fraction.

A. 5616/20\pi

B. 5600/25\pi

C. 5625/25\pi

Correct D. 5616/25\pi

Correct Answer: D

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