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igcse model questions vol1 2018 mathematics | Objective

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Question 1 View Details

Simplify $ \sqrt{2^{6} \times 3^{4}} \div (2^{2} \times \sqrt{3^{2}}) $ and give your answer as an integer.

A. 4

B. 12

C. 8

Correct D. 6

Correct Answer: D

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Question 2 View Details

A shop sells pens and notebooks. The total cost of 5 pens and 3 notebooks is $23. If the cost of a pen is $2 more than twice the cost of a notebook, find the cost of one notebook in dollars.

A. 2

B. 4

C. 3

Correct D. 1

Correct Answer: D

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Question 3 View Details

In triangle $ABC$, side $AB = 8$ cm and side $AC = 6$ cm. The angle at $B$ is twice the angle at $A$. Find the length of side $BC$ correct to two decimal places.

A. 2.81

B. 3.12

C. 4.21

Correct D. 3.21

Correct Answer: D

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Question 4 View Details

Points $A(1,2)$ and $B(5,6)$ are the endpoints of a diameter of a circle. Point $C$ lies on the circle and has integer coordinates. Determine the possible coordinates of $C$.

A. (3,3) or (4,4)

Correct B. (5,2) or (1,6)

C. (5,1) or (1,5)

D. (2,6) or (6,2)

Correct Answer: B

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Question 5 View Details

Solve the inequality $\frac{2x-5}{x+1} > 1$ (with $x \neq -1$) and express the solution set in interval notation.

A. (-∞, -1) ∪ [6, ∞)

Correct B. (-∞, -1) ∪ (6, ∞)

C. (-∞, -1) ∪ (5, ∞)

D. (-∞, -2) ∪ (6, ∞)

Correct Answer: B

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Question 6 View Details

The quadratic equation (x^{2}-kx+12=0\) has two real roots whose sum is twice their difference. Find the value of $k$ and then compute the product of the roots of the equation $2x^{2}-(k+1)x+3=0$.

A. 5\/2

Correct B. 3/2

C. 3\/4

D. 2\/3

Correct Answer: B

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Question 7 View Details

Solve the inequality $|3x+2|>x^{2}-5x+6$ and express the solution set in interval notation.

A. (4-√3, 4+√3)

B. (4-2√5, 4+2√5)

Correct C. (4-2√3, 4+2√3)

D. (3-2√3, 5+2√3)

Correct Answer: C

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Question 8 View Details

Simplify $\displaystyle \frac{2^{3}\cdot5^{2}}{10^{2}}\times\sqrt{50}$ and give your answer in simplest surd form.

A. 5√2

B. 20√2

Correct C. 10√2

D. 10√5

Correct Answer: C

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Question 9 View Details

A shop sells two types of pens. Type A costs $\$p$ per pen and Type B costs $\$(p+2)$ per pen. In one day the shop sold a total of 120 pens and collected $\$660$. If the number of Type B pens sold was three times the number of Type A pens sold, find the price $p$ of a Type A pen.

A. 6

B. 5

C. 3

Correct D. 4

Correct Answer: D

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Question 10 View Details

In triangle $ABC$ we have $AB=7$ cm, $AC=9$ cm and $\angle BAC=60^{\circ}$. The internal angle bisector of $\angle BAC$ meets side $BC$ at $D$. Find the length of $BD$.

A. 7√67/12

B. 5√67/16

Correct C. 7√67/16

D. 7√61/16

Correct Answer: C

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Question 11 View Details

In a box there are only red and blue balls. The probability of drawing a red ball on a single draw is $0.4$. If two balls are drawn without replacement, the probability that both are red is $\frac{2}{15}$. Find the total number of balls in the box.

A. 15

Correct B. 10

C. 8

D. 12

Correct Answer: B

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Question 12 View Details

A certain disease affects 2% of the population. A first diagnostic test has a sensitivity of 80% and a specificity of 85%. Anyone who tests positive on the first test is given a second, different test which has a sensitivity of 90% and a specificity of 95%. Assuming the tests are independent, what is the probability that a randomly selected person who tests positive on both tests actually has the disease? Give your answer as a fraction in lowest terms.

Correct A. 96/145

B. 100/145

C. 84/145

D. 96/150

Correct Answer: A

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Question 13 View Details

The diagram shows a smooth upward‑curving quadratic curve passing through the points $(1,2)$, $(2,6)$ and $(3,12)$ on a Cartesian coordinate system. The curve represents a function $f(x)$ of the form $f(x)=ax^{2}+bx+c$. Determine all real values of $x$ for which $f(x)=20$.

A. x = 2 or x = -10

B. x = 5 or x = -4

Correct C. x = 4 or x = -5

D. x = 4 or x = 5

Correct Answer: C

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Question 14 View Details

In a survey of 150 students, they were asked whether they study Mathematics (M), Physics (P) and Chemistry (C). The results were: 80 study M, 70 study P, 60 study C; 40 study both M and P, 30 study both M and C, 25 study both P and C; 15 students study none of the three subjects. Find the number of students who study exactly two of the subjects.

A. 25

Correct B. 35

C. 30

D. 40

Correct Answer: B

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Question 15 View Details

The marks obtained by 25 students in two examinations are summarised as follows: for Exam A the mean is 68 and the standard deviation is 8; for Exam B the mean is 72 and the standard deviation is 6. The Pearson correlation coefficient between the two sets of marks is 0.6. Assuming a linear relationship, determine the least‑squares regression line for predicting the Exam B mark ($y$) from the Exam A mark ($x$). Then use this line to predict the Exam B mark of a student who scored 80 in Exam A. Give the predicted mark to one decimal place.

A. 75.2

B. 80.0

Correct C. 77.4

D. 78.6

Correct Answer: C

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Question 16 View Details

Vector $\mathbf{a}$ is $(2,\,-1,\,3)$ and vector $\mathbf{b}$ is $(1,\,4,\,k)$. A vector $\mathbf{c}$ is defined by $\mathbf{c}= \mathbf{a}+ \lambda \mathbf{b}$. It is known that $\mathbf{c}$ is perpendicular to $\mathbf{a}$ and that $|\mathbf{c}| = 10$. Find the possible ordered pairs $(k,\lambda)$.

Correct A. (2.859, -2.128) or (-1.212, 2.485)

B. (3.000, -1.500) or (-0.500, 3.200)

C. (2.500, -2.000) or (-1.000, 2.000)

D. (3.100, -2.300) or (-1.300, 2.600)

Correct Answer: A

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Question 17 View Details

In a school, 60% of the pupils passed Mathematics, 70% passed English and 80% passed Science. Exactly 50% of the pupils passed all three subjects. What is the minimum possible percentage of pupils who passed at least one of the three subjects?

A. 60

B. 90

Correct C. 80

D. 70

Correct Answer: C

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Question 18 View Details

The graph of $y$ against $x$ consists of a straight line segment joining $(0,2)$ and $(4,10)$ and a parabola $y = ax^{2}+bx+c$ that passes through the points $(4,10)$, $(6,22)$ and $(8,38)$. Find the value of $a$.

A. 0.6

B. 0.4

Correct C. 0.5

D. 1

Correct Answer: C

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Question 19 View Details

Apples cost $\$1.20$ per kilogram and bananas cost $\$0.80$ per kilogram. A customer buys a total of 5 kg of fruit and spends $\$5.40$. How many kilograms of apples did the customer buy?

A. 2.5

B. 3

C. 4

Correct D. 3.5

Correct Answer: D

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Question 20 View Details

In triangle XYZ, side $x = 9$ cm, side $y = 12$ cm and the included angle $Z = 60^{\circ}$. Find (i) the area of the triangle and (ii) the radius $R$ of its circumcircle. Give your answers in simplest surd form.

A. Area = 54\sqrt{3} cm^2; Circumradius = \sqrt{39} cm

B. Area = 27 cm^2; Circumradius = \sqrt{45} cm

C. Area = 27\sqrt{3} cm^2; Circumradius = 5\sqrt{3} cm

Correct D. Area = 27\sqrt{3} cm^2; Circumradius = \sqrt{39} cm

Correct Answer: D

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Question 21 View Details

Given vectors $\mathbf{a}$ and $\mathbf{b}$ with $|\mathbf{a}|=5$, $|\mathbf{b}|=8$ and $|\mathbf{a}+\mathbf{b}|=10$, determine the magnitude of the vector $2\mathbf{a}-3\mathbf{b}$.

A. \sqrt{625}

B. \sqrt{650}

C. \sqrt{580}

Correct D. \sqrt{610}

Correct Answer: D

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Question 22 View Details

The time $T$ (in seconds) taken for a particle to travel a distance $s$ under a constant force $F$ varies directly as $s$ and as the cube of its mass $m$, and varies inversely as the square root of $F$. If a particle of mass $2\,\text{kg}$ travelling $10\,\text{m}$ under a force of $4\,\text{N}$ takes $5$ seconds, how long will a particle of mass $3\,\text{kg}$ travelling $20\,\text{m}$ under a force of $9\,\text{N}$ take?

A. 20.0

B. 18.0

C. 27.0

Correct D. 22.5

Correct Answer: D

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Question 23 View Details

In a survey of 220 students, the numbers of students who play football (set $F$), basketball (set $B$) and cricket (set $C$) are shown in the Venn diagram. The diagram gives $|F|=100$, $|B|=90$, $|C|=80$, $|F\cap B|=40$, $|F\cap C|=30$, $|B\cap C|=25$, and $|F\cap B\cap C|=15$. Thirty students play none of these sports. How many students play only football?

A. 55

B. 40

Correct C. 45

D. 35

Correct Answer: C

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Question 24 View Details

A linear function is defined by $f(x)=ax+b$ for all real $x$. Its inverse function satisfies $f^{-1}(x)=2x-3$. Find the constants $a$ and $b$, and then determine the value of $x$ for which $f(x)=7$.

A. 9

B. 13

C. 15

Correct D. 11

Correct Answer: D

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Question 25 View Details

In triangle $ABC$ the side $AB$ is $8$ cm, $\angle B=50^{\circ}$ and $\angle C=60^{\circ}$. Point $D$ lies on side $BC$ such that $BD:DC=1:2$. Find the length of $AD$ correct to two decimal places.

A. 7.12

Correct B. 6.53

C. 6.01

D. 5.84

Correct Answer: B

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