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igcse model questions vol1 2020 mathematics | Objective

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Question 1 View Details

The expression (E = \frac{3x^{2} - 12x + 10}{x-2}\) is defined for integer values of (x\) with (x\neq 2\). Find the sum of all integer values of (x\) for which (E\) is also an integer.

A. 12

B. 10

C. 6

Correct D. 8

Correct Answer: D

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Question 2 View Details

A student guesses randomly on 10 multiple‑choice questions, each having 4 options. (a) Find the probability that the student gets exactly 3 questions correct. (b) Given that the student gets at least one question correct, find the probability that they got exactly 3 correct.

A. a) 262440\/1048576 \(\approx 0.2500\); b) 262440\/989527 \(\approx 0.2653\)

B. a) 262440\/1048575 \(\approx 0.2503\); b) 262440\/989527 \(\approx 0.2653\)

C. a) 262440\/1048576 \(\approx 0.2503\); b) 262440\/989527 \(\approx 0.2600\)

Correct D. a) 262440/1048576 \(\approx 0.2503\); b) 262440/989527 \(\approx 0.2653\)

Correct Answer: D

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Question 3 View Details

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

A. 7.89 cm

B. 8.53 cm

Correct C. 9.47 cm

D. 10.12 cm

Correct Answer: C

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Question 4 View Details

Class X has 20 students with a mean score of 68 and a standard deviation of 5. Class Y has 30 students with a mean score of 75 and a standard deviation of 8. Assuming the scores are independent, (a) find the mean and standard deviation of the combined 50 students, and (b) determine the \(z\)-score of a student who scored 85 in the combined set.

A. Mean = 70.8, SD ≈ 6.9; z‑score ≈ 2.10

B. Mean = 73.0, SD ≈ 8.0; z‑score ≈ 1.25

Correct C. Mean = 72.2, SD ≈ 7.72; z‑score ≈ 1.66

D. Mean = 71.5, SD ≈ 7.5; z‑score ≈ 1.73

Correct Answer: C

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Question 5 View Details

In a plane, vectors \(\mathbf{a}\) and \(\mathbf{b}\) satisfy \(|\mathbf{a}| = 5\), \(|\mathbf{b}| = 7\) and \(\mathbf{a}\cdot\mathbf{b} = 20\). Let \(\mathbf{c} = 2\mathbf{a} - 3\mathbf{b}\). Find (a) the magnitude of \(\mathbf{c}\) and (b) the angle between \(\mathbf{a}\) and \(\mathbf{c}\) (to one decimal place).

Correct A. a) \(\sqrt{301}\) \(\approx 17.35\); b) \(96.6^{\circ}\)

B. a) \(\sqrt{301}\) \(\approx 17.35\); b) \(70.5^{\circ}\)

C. a) \(\sqrt{421}\) \(\approx 20.5\); b) \(84.3^{\circ}\)

D. a) \(\sqrt{781}\) \(\approx 27.9\); b) \(110.2^{\circ}\)

Correct Answer: A

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Question 6 View Details

A rectangular garden has a length of 20 m and an unknown width \(x\) metres. A right‑angled triangular flower bed is attached to one of the 20 m sides, sharing that side as its base; the triangle's height is 12 m. The total perimeter of the combined shape (including the outer edges of the rectangle and the two outer sides of the triangle) is 100 m. Find the total area of the garden (rectangle plus triangle) in square metres, giving your answer to one decimal place.

A. 580.2

Correct B. 561.6

C. 540.0

D. 600.5

Correct Answer: B

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Question 7 View Details

The time \(T\) (in hours) taken to process \(N\) items varies directly as \(N\) and inversely as the square root of the power \(P\) (in kW) supplied. When 200 items are processed in 5 hours using 16 kW, what power is required to process 500 items in the same 5 hours? Give your answer in kilowatts.

Correct A. 100

B. 150

C. 80

D. 125

Correct Answer: A

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Question 8 View Details

A student attempts a 10‑question multiple‑choice test, each question having four options with only one correct answer. For each question the student either knows the answer (probability 0.6) and answers correctly, or does not know it (probability 0.4) and then guesses, giving a 0.25 chance of being correct. Let \(Y\) be the total number of correct answers. (a) What is the probability that \(Y = 5\)? Give your answer to three decimal places. (b) Find the expected value and the standard deviation of \(Y\), giving the standard deviation to two decimal places.

A. 0.115, 8, 1.60

B. 0.082, 6, 1.30

C. 0.095, 7, 1.35

Correct D. 0.103, 7, 1.45

Correct Answer: D

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Question 9 View Details

A shop sells \(\frac{3}{4}\) kg of nuts for ₦1,200. What is the amount payable for 2.5 kg of nuts after a 12.5 % discount is applied to the total cost? Give your answer in naira.

A. 3750

B. 3600

Correct C. 3500

D. 4000

Correct Answer: C

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Question 10 View Details

Let numbers \(a\) and \(b\) satisfy \(a + b = 6\) and \(a^{2} + b^{2} = 20\). Find the value of the expression \(E = a^{3}b + a b^{3}\).

A. 200

B. 120

C. 64

Correct D. 160

Correct Answer: D

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Question 11 View Details

The sum of the first \(n\) terms of a sequence is given by \(S_n = 3n^{2} + 5n\). Find the value of the 10th term of the sequence.

Correct A. 62

B. 58

C. 60

D. 64

Correct Answer: A

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Question 12 View Details

For what value of the parameter \(k\) does the system of equations \(\begin{cases}2x + 3y = 7\\ kx - y = 4\end{cases}\) have a solution in which \(x = y\)?

A. 30/7

B. 27/5

C. 24/7

Correct D. 27/7

Correct Answer: D

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Question 13 View Details

A straight line passes through the points \(A(2,3)\) and \(B(5,k)\) and also cuts the x‑axis at the point \(C(1,0)\). Determine the value of \(k\).

A. 9

B. 6

Correct C. 12

D. 15

Correct Answer: C

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Question 14 View Details

Find the sum of all integers \(n\) satisfying \(\frac{2n-5}{\,n+3\,}>1\) with \(-10\le n\le 10\).

A. -36

B. -18

C. -24

Correct D. -30

Correct Answer: D

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Question 15 View Details

The time \(t\) (seconds) taken for a particle to travel a distance \(s\) (metres) varies directly with \(s\) and inversely with \(\sqrt{F}\), where \(F\) (newtons) is a constant force. When \(s=20\) m and \(F=4\) N the time is \(t=5\) s. Find the time required to travel \(s=45\) m when the force is increased to \(F=9\) N.

A. 11.3

B. 9.0

C. 6.0

Correct D. 7.5

Correct Answer: D

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Question 16 View Details

Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) lie in the same plane with \(|\mathbf{a}| = 5\) and \(|\mathbf{b}| = 8\). The vector \(\mathbf{c} = 2\mathbf{a} - 3\mathbf{b}\) has magnitude \(15\). Find the angle between \(\mathbf{a}\) and \(\mathbf{b}\) (to the nearest degree).

Correct A. 20

B. 10

C. 30

D. 45

Correct Answer: A

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Question 17 View Details

A point \(P(3,-1)\) is reflected in the line \(y=-x\), then dilated from the origin with scale factor \(k\) (where \(k>0\)). The resulting point is then translated by the vector \((t,2t)\) to give the point \(Q(5,7)\). Find the values of \(k\) and \(t\), and then determine the image of the point \(R(-4,2)\) after the same sequence of transformations.

Correct A. (16/5, 56/5)

B. (16/5, 48/5)

C. (8/5, 28/5)

D. (12/5, 56/5)

Correct Answer: A

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Question 18 View Details

A positive integer \(N\) leaves a remainder of \(5\) when divided by \(7\), a remainder of \(8\) when divided by \(9\), and a remainder of \(10\) when divided by \(11\). Find the smallest possible value of \(N\).

Correct A. 691

B. 682

C. 695

D. 703

Correct Answer: A

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Question 19 View Details

The diagram shows a parabola passing through the points \((1,3)\), \((3,7)\) and \((5,15)\). Assuming the parabola has the equation \(y = ax^{2} + bx + c\), determine the values of \(a, b, c\) and then find the value of \(y\) when \(x = 7\).

A. 33

B. 30

C. 25

Correct D. 27

Correct Answer: D

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Question 20 View Details

In a school, 60% of the students study mathematics and 40% study biology. Among the mathematics students, 25% also study physics. Among the biology students, an unknown proportion study physics, but overall 20% of all students study physics. If a randomly selected student is known to study physics, what is the probability that they also study mathematics? Express your answer as a fraction in lowest terms.

A. 2/3

B. 4/5

C. 1/2

Correct D. 3/4

Correct Answer: D

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Question 21 View Details

In triangle \(ABC\), \(AB = 12\) cm, \(AC = 15\) cm and \(\angle BAC = 60^{\circ}\). The internal bisector of \(\angle BAC\) meets side \(BC\) at \(D\). Find the length of \(BD\).

Correct A. \dfrac{4}{3}\sqrt{21}

B. \dfrac{4}{3}\sqrt{20}

C. \dfrac{5}{3}\sqrt{21}

D. \dfrac{8}{3}\sqrt{21}

Correct Answer: A

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Question 22 View Details

The function \(f\) is defined for all real numbers \(x \neq -m\) by \(f(x)=\dfrac{mx+2}{x+m}\). It is known that \(f\) is its own inverse. (a) Determine the value of \(m\). (b) Using this value, solve the equation \(f(x)=3\) for \(x\).

A. m = 1; x = \dfrac{2}{3}

B. m = 0; x = \dfrac{3}{2}

C. m = 0; x = \dfrac{1}{3}

Correct D. m = 0; x = \dfrac{2}{3}

Correct Answer: D

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Question 23 View Details

What is the remainder when \(7^{2021}\) is divided by \(9\)?

Correct A. 4

B. 1

C. 0

D. 7

Correct Answer: A

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Question 24 View Details

A price is first increased by \(12\%\) and then decreased by \(12\%\) of the new price. State the overall percentage change in the price (give a negative sign for a decrease).

Correct A. -1.44\%

B. -0.96\%

C. -2.00\%

D. -1.20\%

Correct Answer: A

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Question 25 View Details

The lines \(2x - y + 3 = 0\) and \(x + 2y - 5 = 0\) intersect at point \(P\). The line through \(P\) is required to be perpendicular to the line joining \(A(1,2)\) and \(B(4,6)\). Find the equation of this perpendicular line in the form \(ax + by + c = 0\) with integer coefficients, and then determine the distance from the origin to this line.

A. 15x+20y+49=0; distance = \dfrac{49}{25}

Correct B. 15x+20y-49=0; distance = \dfrac{49}{25}

C. 12x+16y-49=0; distance = \dfrac{49}{20}

D. 15x+20y-45=0; distance = \dfrac{45}{25}

Correct Answer: B

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