Practice igcse model questions vol1 mathematics 2022 Past Questions

Question 1 View Details

The diagram shows a straight line that passes through the point (A(2,5)) and meets the x‑axis at point (B(10,0)). The line also passes through point (C(8,k)). (i) Find the value of (k). (ii) Determine the y‑intercept of the line. (iii) Find the area of the triangle formed by the line and the coordinate axes.

A. k = 0.75, y‑intercept = 7.0, area = 35.0

B. k = 2.0, y‑intercept = 4.0, area = 20.0

Correct C. k = 1.25, y‑intercept = 6.25, area = 31.25

D. k = 1.5, y‑intercept = 5.5, area = 30.0

Correct Answer: C

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Question 2 View Details

Given that \((3x - 2)(x + 4) = ax^{2} + bx + c\) for all real \(x\), find the integers \(a\), \(b\) and \(c\). Then evaluate the expression \(\frac{2a - b}{c}\).

A. 5/2

B. 3/2

C. -1/2

Correct D. 1/2

Correct Answer: D

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Question 3 View Details

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) in a plane satisfy \(|\mathbf{u}| = 5\), \(|\mathbf{v}| = 7\) and \(|\mathbf{u} + \mathbf{v}| = 9\). (i) Find the angle \(\theta\) between \(\mathbf{u}\) and \(\mathbf{v}\) (give your answer to the nearest degree). (ii) Find the value of the dot product \(\mathbf{u} \cdot \mathbf{v}\).

A. θ = 30°, u·v = 20

B. θ = 60°, u·v = 12

C. θ = 90°, u·v = 0

Correct D. θ = 84°, u·v = 3.5

Correct Answer: D

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Question 4 View Details

The resistance \(R\) (in ohms) of a uniform wire varies directly as its length \(L\) (in metres) and inversely as the square of its radius \(r\) (in millimetres). A wire of length 2 m and radius 1 mm has a resistance of 8 Ω. Find the resistance of a wire of length 3 m and radius 2 mm.

A. 2 Ω

B. 6 Ω

Correct C. 3 Ω

D. 4 Ω

Correct Answer: C

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Question 5 View Details

The quadratic equation \(x^{2} - kx + 12 = 0\) has two real roots such that one root is twice the other. (i) Find the value of \(k\). (ii) Find the sum of the squares of the two roots.

A. k = 3√6, sum of squares = 28

B. k = 2√6, sum of squares = 30

Correct C. k = 3√6, sum of squares = 30

D. k = 3√5, sum of squares = 30

Correct Answer: C

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Question 6 View Details

A bag contains red, blue and green balls. The total number of balls is 10. The number of blue balls is twice the number of green balls. A ball is drawn at random, its colour noted, then it is replaced and an extra ball of the same colour is added to the bag. A second ball is then drawn at random. The probability that both drawn balls are red is \(\frac{2}{11}\). Find the number of green balls initially in the bag.

A. 4

B. 3

C. 1

Correct D. 2

Correct Answer: D

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Question 7 View Details

The decimal \(0.\overline{27}\) (the digits 27 repeat indefinitely) is expressed as a fraction in lowest terms. What is the denominator of that fraction?

A. 12

B. 13

C. 9

Correct D. 11

Correct Answer: D

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Question 8 View Details

Points \(A(0,0)\), \(B(5,k)\) and \(C(8,4)\) are the vertices of a triangle. The distance from point \(B\) to the line \(3x-4y+2=0\) is exactly \(1\) unit, and the area of triangle \(ABC\) is \(12\) square units. Find the value of \(k\).

A. 4.5

B. 6.5

Correct C. 5.5

D. 5

Correct Answer: C

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Question 9 View Details

The following table shows the number of books read in a month by a class of 30 students. \[\begin{array}{c|c} \text{Books read} & \text{Frequency} \\\hline 0 & 4 \\ 1 & a \\ 2 & 7 \\ 3 & 5 \\ 4 & b \\ 5 & 3 \\ 6 & 2 \end{array}\] The mean number of books read is \(2.57\) (to two decimal places) and the mode is \(2\) books. Find the missing frequencies \(a\) and \(b\) and state the median number of books read.

A. a = 5, b = 4; median = 3

Correct B. a = 5, b = 4; median = 2

C. a = 4, b = 5; median = 2

D. a = 6, b = 3; median = 2

Correct Answer: B

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Question 10 View Details

The simultaneous equations \[\begin{cases} 3x + ay = 5 \\ 2x - y = 1 \end{cases}\] have an integer solution \((x,y)\). Find the smallest positive integer value of \(a\) for which this occurs and give the corresponding values of \(x\) and \(y\).

A. a = 4, x = 1, y = 2

Correct B. a = 2, x = 1, y = 1

C. a = 1, x = 2, y = 3

D. a = 3, x = 2, y = 3

Correct Answer: B

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Question 11 View Details

A circle passes through the points \(A(1,2)\) and \(B(4,6)\). Its centre lies on the line \(3x-4y+5=0\). Find the coordinates of the centre and the radius of the circle.

Correct A. Centre (37/12, 57/16); radius 125/48

B. Centre (40\/12, 60\/16); radius 130\/48

C. Centre (3\/4, 5\/6); radius 7\/8

D. Centre (31\/12, 49\/16); radius 115\/48

Correct Answer: A

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Question 12 View Details

In triangle \(ABC\) the side lengths are \(AB=13\), \(AC=15\) and \(BC=14\). Find the length of the angle bisector from vertex \(A\) to side \(BC\).

A. 7\\sqrt{5}\/2

B. 5\\sqrt{13}\/2

C. 3\\sqrt{61}\/2

Correct D. 3\sqrt{65}/2

Correct Answer: D

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Question 13 View Details

The sum of the first \(n\) terms of an arithmetic series is given by \(S_n = 4n^{2}+6n\). Determine the first term, the common difference and the 12th term of the series.

A. First term 10, common difference 7, 12th term 89

Correct B. First term 10, common difference 8, 12th term 98

C. First term 12, common difference 8, 12th term 104

D. First term 8, common difference 9, 12th term 101

Correct Answer: B

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Question 14 View Details

In a class of 120 students, 70 study Mathematics, 65 study Physics and 50 study Chemistry. The numbers studying both Mathematics and Physics, Mathematics and Chemistry, and Physics and Chemistry are 40, 30 and 25 respectively, and 15 students study all three subjects. How many students study none of these subjects?

A. 20

B. 25

Correct C. 15

D. 10

Correct Answer: C

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Question 15 View Details

In triangle \(ABC\) we have \(\angle A = 40^{\circ}\), \(\angle B = 70^{\circ}\) and side \(AB = 10\) cm. Point \(D\) lies on side \(BC\) such that \(BD:DC = 1:2\). Find the length of \(AD\) (give your answer to 2 decimal places).

A. 8.31 cm

B. 7.92 cm

C. 10.58 cm

Correct D. 9.47 cm

Correct Answer: D

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Question 16 View Details

Simplify \( \frac{2^{3}\,\sqrt{8}}{4^{1.5}} \) and give your answer in simplest surd form.

A. 4\sqrt{2}

Correct B. 2\sqrt{2}

C. \sqrt{2}

D. 2\sqrt{3}

Correct Answer: B

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Question 17 View Details

In triangle PQR, side PQ = 7 cm, side PR = 9 cm, and \(\angle QPR = 120^{\circ}\). Point S lies on side QR such that QS = 4 cm. Find the length of PS, giving your answer correct to two decimal places.

A. 4.56 cm

Correct B. 4.32 cm

C. 3.98 cm

D. 4.12 cm

Correct Answer: B

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Question 18 View Details

Solve the inequality \( |3x - 7| < 2x + 1 \) and express the solution set in interval notation.

A. (6/5, 9)

B. (6/5, 7)

C. (5/4, 8)

Correct D. (6/5, 8)

Correct Answer: D

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Question 19 View Details

Express the repeating decimal \(0.\overline{27}\) as a fraction in lowest terms.

Correct A. 3/11

B. 4/11

C. 2/11

D. 3/10

Correct Answer: A

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Question 20 View Details

If \(a = 2^{3/2}\) and \(b = 8^{1/3}\), evaluate \(\frac{a}{b}\) and give your answer in simplest surd form.

A. \sqrt{8}

Correct B. \sqrt{2}

C. \frac{1}{\sqrt{2}}

D. \sqrt{3}

Correct Answer: B

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Question 21 View Details

Two independent tests are taken. In the first test a student answers 5 multiple‑choice questions, each with a probability of 0.6 of being correct. In the second test the student answers 4 similar questions, each with a probability of 0.5 of being correct. Let \(X\) be the total number of correct answers in the two tests. Find the probability that \(X\ge 6\). Give your answer as a decimal correct to three decimal places.

Correct A. 0.374

B. 0.424

C. 0.324

D. 0.274

Correct Answer: A

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Question 22 View Details

A point \(A(7,0)\) is transformed by the following sequence: (i) reflection in the line \(y=x\); (ii) a rotation of \(180^{\circ}\) about the point \((1,2)\); (iii) a dilation with centre at the origin and scale factor \(k\). The image of \(A\) after the three transformations is the point \(B(6,-9)\). (a) Determine the scale factor \(k\). (b) Using the same three transformations, find the image of the point \(C(-1,5)\). Give your answer as an ordered pair.

Correct A. k = 3; image of C = (-9,15)

B. k = 3; image of C = (-9,14)

C. k = 2; image of C = (-6,10)

D. k = 4; image of C = (-12,20)

Correct Answer: A

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Question 23 View Details

In triangle \(PQR\) the side opposite \(R\) has length \(PQ=12\) cm, the side opposite \(Q\) has length \(PR=10\) cm and \(\angle Q =30^{\circ}\). (i) Find the two possible measures of \(\angle R\) (to one decimal place). (ii) For each possible \(\angle R\) determine the length of side \(QR\). (iii) Calculate the area of the triangle for the case in which \(\angle R\) is obtuse. Give the area to two decimal places.

A. Angle R: 38.5° or 141.5°; QR: 18.0 cm (acute) or 2.50 cm (obtuse); area (obtuse case): 7.00 cm²

Correct B. Angle R: 36.9° or 143.1°; QR: 18.4 cm (acute) or 2.39 cm (obtuse); area (obtuse case): 7.18 cm²

C. Angle R: 40.2° or 139.8°; QR: 17.6 cm (acute) or 2.75 cm (obtuse); area (obtuse case): 6.95 cm²

D. Angle R: 35.0° or 145.0°; QR: 19.0 cm (acute) or 2.20 cm (obtuse); area (obtuse case): 7.45 cm²

Correct Answer: B

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Question 24 View Details

Real numbers \(x\) and \(y\) satisfy \[x+\frac{1}{y}=3\qquad\text{and}\qquad y+\frac{1}{x}=4.\] Find the exact value of \(x^{2}+y^{2}\). Express your answer in simplest surd form.

A. \frac{125}{12}+\frac{20}{6}\sqrt{6}

B. \frac{125}{12}+\frac{25}{6}\sqrt{5}

Correct C. \frac{125}{12}+\frac{25}{6}\sqrt{6}

D. \frac{115}{12}+\frac{25}{6}\sqrt{6}

Correct Answer: C

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Question 25 View Details

The following frequency distribution gives the ages (in years) of participants in a survey. The class mid‑points are shown in parentheses. | Age interval (mid‑point) | Frequency | | 0-9 (4.5) | 5 | | 10-19 (14.5) | \(a\) | | 20-29 (24.5) | 12 | | 30-39 (34.5) | 9 | | 40-49 (44.5) | 4 | The mean age of the whole group is \(24.5\) years. (i) Determine the value of the unknown frequency \(a\). (ii) Using this value, calculate the standard deviation of the distribution, giving your answer to one decimal place.

A. a = 7; standard deviation ≈ 13.2

Correct B. a = 7; standard deviation ≈ 11.9

C. a = 6; standard deviation ≈ 12.3

D. a = 8; standard deviation ≈ 10.5

Correct Answer: B

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