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igcse model questions vol1 2024 mathematics | Objective

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Question 1 View Details

A circle passes through the points \(A(2,3)\) and \(B(8,7)\) and is tangent to the line \(3x - 4y + 5 = 0\). Find the \(y\)-coordinate of the centre of the circle.

A. 30

Correct B. 28

C. 32

D. 24

Correct Answer: B

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Question 2 View Details

The quadratic equation \(p x^{2}+q x+r=0\) has two real roots \(\alpha\) and \(\beta\). It is known that \(\alpha+2\beta=7\) and \(\alpha\beta=6\). The coefficient \(p\) is the smallest positive integer that makes both \(q\) and \(r\) integers. Find the value of \(p+q+r\).

Correct A. 2

B. 3

C. 1

D. 4

Correct Answer: A

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Question 3 View Details

A sequence is defined by \(a_{1}=2\) and \(a_{n+1}=3a_{n}+(-1)^{n}\) for \(n\ge 1\). Find the sum of the first six terms \(S_{6}=a_{1}+a_{2}+\dots +a_{6}\).

Correct A. 637

B. 639

C. 635

D. 647

Correct Answer: A

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Question 4 View Details

The line through the point \(P(4,-1)\) makes an angle of \(30^{\circ}\) with the line joining \(Q(1,2)\) and \(R(7,8)\). It meets the \(x\)-axis at a point that lies to the right of the origin. Find the coordinates of this intersection point.

A. (6+√2, 0)

B. (5+√3, 0)

Correct C. (6+√3, 0)

D. (6-√3, 0)

Correct Answer: C

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Question 5 View Details

Triangle \(ABC\) has vertices \(A(0,0)\), \(B(4,0)\) and \(C(0,3)\). It is first reflected in the line \(y=x\), then rotated \(90^{\circ}\) clockwise about the point \((2,2)\), and finally dilated with centre \((2,2)\) and a scale factor \(k\) such that the image of point \(B\) coincides with \((6,6)\). Determine the value of \(k\) and the area of the final image of triangle \(ABC\).

A. k = 2, area = 18

B. k = 3, area = 36

C. k = 1, area = 12

Correct D. k = 2, area = 24

Correct Answer: D

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Question 6 View Details

In triangle ABC, side BC = \(10\) cm, side AB = \(7\) cm. Angle B is twice angle A. Find (i) the measure of angle C (to the nearest 0.1°) and (ii) the length of the altitude from C to side AB (to the nearest 0.1 cm).

A. C = 34.2°, altitude ≈ 9.4 cm

B. C = 31.8°, altitude ≈ 8.7 cm

C. C = 28.5°, altitude ≈ 10.2 cm

Correct D. C = 31.8°, altitude ≈ 9.9 cm

Correct Answer: D

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Question 7 View Details

A class of 30 students sat two exams. For exam 1 the mean score was \(68\) and the standard deviation \(8\). For exam 2 the mean was \(72\) and the standard deviation \(6\). The correlation coefficient between the two exam scores was \(0.6\). One student scored \(85\) on exam 1 and \(90\) on exam 2. (i) If this student's scores are removed, what is the new mean of exam 2 (to two decimal places)? (ii) Using the remaining 29 students, write the equation of the regression line that predicts exam‑2 score from exam‑1 score, in the form \(\hat{Y}=a+bX\) (give \(a\) and \(b\) to two decimal places).

Correct A. New mean = 71.38; regression line: Ŷ = 48.91 + 0.33X

B. New mean = 70.12; regression line: Ŷ = 49.50 + 0.30X

C. New mean = 71.38; regression line: Ŷ = 47.80 + 0.35X

D. New mean = 72.05; regression line: Ŷ = 48.91 + 0.28X

Correct Answer: A

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Question 8 View Details

Find all real numbers \(x\) that satisfy both \(\displaystyle\frac{2x-5}{x+3}\le 1\) and \(\displaystyle\frac{x-4}{x-2}>-2\). Express the solution set in interval notation.

A. (-3,2) ∪ (8\/3,9]

Correct B. (-3,2) ∪ (8/3,8]

C. (-3,2] ∪ (8\/3,8)

D. (-3,2) ∪ [8\/3,8]

Correct Answer: B

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Question 9 View Details

A garden consists of a rectangle of length \(20\) m and width \(w\) m, together with a semicircular lawn attached to one of the rectangle's shorter sides. The diameter of the semicircle equals the width \(w\). The total perimeter of the garden (including the curved edge of the semicircle but not the side of the rectangle that is shared with the semicircle) is \(70\) m. (i) Determine the width \(w\) (to two decimal places). (ii) Find the total area of the garden (to one decimal place) in square metres.

A. w = 9.80 m; total area = 225.0 m²

B. w = 12.89 m; total area = 312.5 m²

C. w = 10.45 m; total area = 260.3 m²

Correct D. w = 11.67 m; total area = 286.9 m²

Correct Answer: D

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Question 10 View Details

Simplify \(\displaystyle\frac{2^{3}\sqrt{50}}{5\cdot2^{1/2}}\) and give your answer in simplest integer form.

Correct A. 8

B. 16

C. 4

D. 10

Correct Answer: A

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Question 11 View Details

A company produces two types of gadget, A and B. The profit from each A is \(p\) pounds and from each B is \(q\) pounds. It is known that the profit per unit of A is £5 less than twice the profit per unit of B. When the company sells 5 A and 3 B it makes a profit of £105, and when it sells 2 A and 6 B it makes a profit of £90. Find the profit per unit of B (in pounds).

A. 15

B. 8

C. 12

Correct D. 10

Correct Answer: D

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Question 12 View Details

A shop offers a discount of 15% on an item. After the discount the price is £68. What was the original price? Then, if a sales tax of 8% is added to the original price, what is the final price (including tax)? Give both answers in pounds, rounded to two decimal places where necessary.

A. Original price = 70; final price = 75.60

B. Original price = 78; final price = 84.24

C. Original price = 85; final price = 91.80

Correct D. Original price = 80; final price = 86.40

Correct Answer: D

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Question 13 View Details

The time \(t\) (in minutes) required for a machine to process a batch varies directly with the number of items \(n\) and inversely with the speed \(s\) (items per minute). When 120 items are processed at a speed of 30 items per minute, it takes 8 minutes. If the speed is increased by 20% and the batch size is reduced by 15 items, how long will the processing take? Give your answer to two decimal places.

A. 6.12

B. 5.45

C. 7.00

Correct D. 5.83

Correct Answer: D

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Question 14 View Details

A straight line passes through the points \((2,3)\) and \((8,k)\). The line also meets the x‑axis at \(x = 12\). Find the value of \(k\) and then determine the x‑coordinate at which the line has the value \(y = 5\).

A. k = 0.8; x = -5\/3

Correct B. k = 1.2; x = -14/3

C. k = 1.5; x = -12\/5

D. k = 2.0; x = -7\/2

Correct Answer: B

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Question 15 View Details

A rectangular garden contains a circular flower bed that is inscribed, touching all four sides of the rectangle. The diameter of the circle is 14 m. A uniform path surrounds the circle inside the rectangle. If the area of the path is 300 m² and \(\pi\) is to be taken as \(\frac{22}{7}\), find the width of the path (in metres) correct to two decimal places.

A. 0.68

Correct B. 0.72

C. 0.75

D. 0.80

Correct Answer: B

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Question 16 View Details

In a survey, set \(A\) contains 28 elements. The union of sets \(A\) and \(B\) has 50 elements. It is known that the number of elements that are in \(B\) but not in \(A\) is twice the number of elements that are in both \(A\) and \(B\). Find \(|B|\).

A. 31

B. 38

C. 35

Correct D. 33

Correct Answer: D

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Question 17 View Details

In the coordinate plane the points \(A(2,3)\), \(B(8,3)\) and \(D(5,7)\) are plotted. Point \(C\) has coordinates \((k,9)\). The line through \(A\) and \(C\) is perpendicular to the line through \(B\) and \(D\). Determine the value of \(k\).

A. 14

B. 8

Correct C. 10

D. 12

Correct Answer: C

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Question 18 View Details

The repeating decimal \(0.\overline{27}\) can be written as the fraction \(\frac{p}{q}\) in lowest terms. Find \(p+q\).

A. 15

B. 13

C. 16

Correct D. 14

Correct Answer: D

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Question 19 View Details

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

Correct A. 82.5

B. 75.0

C. 67.5

D. 90.0

Correct Answer: A

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Question 20 View Details

In a school 60% of the students study Mathematics. Among the Mathematics students, 30% also study Biology. Among the students who do not study Mathematics, 20% study Biology. If a randomly selected student is known to study Biology, what is the probability that the student also studies Mathematics? Give your answer as a fraction in lowest terms.

A. 9\/12

B. 10\/13

Correct C. 9/13

D. 7\/13

Correct Answer: C

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Question 21 View Details

The integers \(a\) and \(b\) are positive and satisfy \(\frac{a}{b} + \frac{b}{a} = \frac{25}{12}\). Find the smallest possible value of \(a+b\).

Correct A. 7

B. 10

C. 9

D. 8

Correct Answer: A

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Question 22 View Details

In triangle \(ABC\) the angles are \(\angle A = 40^{\circ}, \angle B = 70^{\circ}, \angle C = 70^{\circ}\). The side \(AB\) has length \(10\) cm. Point \(D\) lies on side \(BC\) such that \(BD:DC = 1:2\). Calculate the length of \(AD\) correct to two decimal places.

A. 8.93

Correct B. 9.47

C. 10.12

D. 11.05

Correct Answer: B

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Question 23 View Details

A shop sold 120 identical items. Three‑fifths of the items were sold at a profit of 20 % on the cost price, and the remaining items were sold at a loss of 10 % on the cost price. If the total profit for the day was ₦1 800, determine the cost price of each item (in naira, to the nearest naira).

Correct A. 188

B. 192

C. 200

D. 176

Correct Answer: A

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Question 24 View Details

Let \(f(x)=\frac{3x+5}{2x-1}\) for \(x\neq \tfrac12\). Let \(f^{-1}\) denote the inverse function of \(f\). Find all real numbers \(x\) for which \(f(x)=f^{-1}(x)\).

A. 1 \\pm \\frac{\\sqrt{12}}{2}

Correct B. 1 \pm \frac{\sqrt{14}}{2}

C. 1 \\pm \\frac{\\sqrt{16}}{2}

D. 1 \\pm \\frac{\\sqrt{14}}{3}

Correct Answer: B

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Question 25 View Details

Simplify \(\sqrt{50}\; -\; \sqrt{18}\; +\; \sqrt{8}\) and express your answer in simplest surd form.

A. 6\sqrt{2}

Correct B. 4\sqrt{2}

C. 2\sqrt{18}

D. 4\sqrt{3}

Correct Answer: B

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