Practice igcse model questions vol1 mathematics 2021 Past Questions

Question 1 View Details

In the triangle shown, $\angle A = 40^{\circ}$, $\angle B = 70^{\circ}$ and side $AB = 8$ cm. Point $D$ lies on side $BC$ such that $BD:DC = 1:2$. Find the length of $AD$ (give your answer correct to 2 decimal places).

A. 7.03

B. 6.84

Correct C. 7.57

D. 8.12

Correct Answer: C

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Question 2 View Details

A sequence is formed by interleaving two subsequences. The terms in odd positions (1st, 3rd, 5th, …) constitute an arithmetic progression with first term 3 and common difference 4. The terms in even positions (2nd, 4th, 6th, …) constitute a geometric progression with first term 2 and common ratio 3. Find the 7th term of the overall sequence and the sum of the first 7 terms.

Correct A. 15, 62

B. 13, 62

C. 17, 70

D. 15, 58

Correct Answer: A

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Question 3 View Details

The graph below shows a parabola opening upwards. It passes through the points $(0,5)$, $(1,2)$ and $(5,2)$. Determine the equation of the parabola in the form $y = ax^{2} + bx + c$ (give $a, b, c$ to one decimal place).

A. 0.6x^2 - 4.0x + 5.0

B. 0.5x^2 - 3.6x + 5.0

Correct C. 0.6x^2 - 3.6x + 5.0

D. 0.6x^2 - 3.6x + 4.5

Correct Answer: C

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Question 4 View Details

In triangle $ABC$ we have $BC = 15$ cm, $AC = 12$ cm and $\angle A = 40^{\circ}$. The foot $D$ of the altitude from $A$ to $BC$ divides $BC$ so that $BD = 9$ cm. Find the length of side $AB$ (to 2 decimal places) and the area of $\triangle ABC$ (to 2 decimal places).

A. 9.87 cm; 38.20 cm^2

B. 10.49 cm; 45.00 cm^2

Correct C. 10.49 cm; 40.46 cm^2

D. 11.23 cm; 42.10 cm^2

Correct Answer: C

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Question 5 View Details

The Venn diagram shows three sets $A, B, C$ within a universal set of 50 elements. The numbers displayed are: $B$ only = 8, $C$ only = 5, $A\cap B$ only = 4, $A\cap C$ only = 3, $A\cap B\cap C$ = 1, and $\text{none}$ = 10. The numbers in the regions $A$ only and $B\cap C$ only are not shown. It is also given that set $A$ contains 20 elements in total. Determine (i) the number of elements that belong to exactly two of the sets, and (ii) the number of elements that belong to at most one set.

Correct A. 14, 35

B. 13, 35

C. 15, 36

D. 14, 34

Correct Answer: A

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Question 6 View Details

Points A(2,3) and B(5,k) are endpoints of a line segment. The midpoint of AB lies on the line $y = x + 1$. Find the value of k. Then, a line through B is drawn parallel to the line $2x - y + 4 = 0$. Determine the coordinates of the point where this line meets the x‑axis.

A. k = 4; intersection point (2,0)

B. k = 6; intersection point (0,0)

Correct C. k = 6; intersection point (2,0)

D. k = 8; intersection point (4,0)

Correct Answer: C

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Question 7 View Details

A standard deck of 52 playing cards is shuffled and three cards are drawn at random without replacement. Given that at least one of the three cards drawn is a heart, find the probability that exactly two of the three cards are hearts. Express your answer as a fraction in lowest terms.

A. 233/997

B. 235/997

C. 234/996

Correct D. 234/997

Correct Answer: D

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Question 8 View Details

A game consists of rolling a fair six‑sided die. Each time a 6 is rolled the player wins £10; otherwise the player loses £2. The game is played 15 independent times. Let $Y$ be the total profit after the 15 rolls. (a) Find the probability that the player makes a profit (i.e., $Y>0$). (b) Find the expected profit and the standard deviation of $Y$. Give the probability as a decimal to three decimal places, and the expected profit and standard deviation to two decimal places.

A. Probability 0.421; Expected profit £0.00; Standard deviation £15.00

B. Probability 0.489; Expected profit £2.40; Standard deviation £18.10

C. Probability 0.452; Expected profit £-1.20; Standard deviation £16.85

Correct D. Probability 0.468; Expected profit £0.00; Standard deviation £17.32

Correct Answer: D

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Question 9 View Details

Points $A(1,2)$, $B(4,6)$ and $C(7,2)$ form an isosceles triangle with $AB = BC$. A circle passes through $A$ and $C$ and its centre lies on the line $y = x + 1$. Find the equation of the circle and then determine the coordinates of the point where this circle meets the line through $B$ that is perpendicular to $AC$. Give the required point as an ordered pair.

A. (3, 5+3\sqrt{2})

B. (4, 5-3\sqrt{2})

Correct C. (4, 5+3\sqrt{2})

D. (4, 5+2\sqrt{2})

Correct Answer: C

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Question 10 View Details

Let integers $a$ and $b$ satisfy $a+b=12$ and $a^{2}+b^{2}=90$. Find the value of the expression $\dfrac{a^{3}-b^{3}}{a-b}$.

Correct A. 117

B. 144

C. 63

D. 90

Correct Answer: A

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Question 11 View Details

Two vectors $\mathbf{u}$ and $\mathbf{v}$ lie in the plane. It is known that $\mathbf{u}+\mathbf{v} = (3,4)$, the magnitude of $\mathbf{u}$ is $5$, and the x‑component of $\mathbf{u}$ is negative while $\mathbf{u}$ is perpendicular to the vector $(1,2)$. Find the magnitude of $\mathbf{v}$.

A. \sqrt{45+4\sqrt5}

B. \sqrt{50-4\sqrt5}

Correct C. \sqrt{50+4\sqrt5}

D. \sqrt{50+2\sqrt5}

Correct Answer: C

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Question 12 View Details

If the numbers $a$ and $b$ satisfy $a+b=6$ and $ab=5$, evaluate $\dfrac{a^{3}+b^{3}}{a^{2}+b^{2}}$.

A. 63/12

Correct B. 63/13

C. 65/13

D. 64/13

Correct Answer: B

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Question 13 View Details

In right‑angled triangle $ABC$ with $\angle C = 90^{\circ}$, the hypotenuse $AB$ is $13$ cm and side $AC$ is $5$ cm. The altitude from $C$ meets $AB$ at $D$. Find the length of the altitude $CD$.

A. 65/13

B. 55/13

C. 60/12

Correct D. 60/13

Correct Answer: D

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Question 14 View Details

A bag initially contains 4 red, 5 blue and 6 green balls. One ball is drawn at random, its colour noted, and then it is replaced together with two additional balls of the same colour. A second ball is then drawn. What is the probability that the two drawn balls are of different colours?

A. 147/255

B. 148/256

C. 149/255

Correct D. 148/255

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) under a constant force varies directly as $\sqrt{s}$ and inversely as the speed $v$ (m\,s$^{-1}$). If it takes 8\,s to travel 20\,m when the speed is 4\,m\,s$^{-1}$, find the time required to travel 45\,m when the speed is 6\,m\,s$^{-1}$.

A. 6

B. 12

Correct C. 8

D. 10

Correct Answer: C

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Question 16 View Details

A straight line passes through the points $ (2,3) $ and $ (8,k) $. The line meets the x‑axis at point A and the y‑axis at point B. The triangle OAB formed by the line and the coordinate axes, where O is the origin, has area $24$ square units. Find the value of $k$ and the x‑intercept of the line.

A. k = 18\\sqrt{2} + 24, \\; x\\text{-intercept}= 8 - 4\\sqrt{2}

B. k = -18\\sqrt{2} - 24, \\; x\\text{-intercept}= 8 + 4\\sqrt{2}

C. k = 12\\sqrt{2} - 24, \\; x\\text{-intercept}= 8 + 6\\sqrt{2}

Correct D. k = 18\sqrt{2} - 24, \; x\text{-intercept}= 8 + 4\sqrt{2}

Correct Answer: D

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Question 17 View Details

In the diagram, points A$(0,0)$ and B$(8,6)$ are shown. The perpendicular bisector of AB passes through point P$(4,3)$. The line meets the vertical line $x = 8$ at point Q. Find the coordinates of Q and the distance between Q and B.

A. Q\\left(8, -\\frac{7}{3}\\right) \\text{ and } QB = \\frac{27}{3} \\text{ units}

B. Q\\left(8, -\\frac{5}{3}\\right) \\text{ and } QB = \\frac{23}{3} \\text{ units}

C. Q\\left(8, \\frac{7}{3}\\right) \\text{ and } QB = \\frac{25}{3} \\text{ units}

Correct D. Q\left(8, -\frac{7}{3}\right) \text{ and } QB = \frac{25}{3} \text{ units}

Correct Answer: D

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Question 18 View Details

The quadratic equation $x^{2} - p x + q = 0$ has roots that are the reciprocals of the roots of the equation $2x^{2} - 5x + 3 = 0$. Find the values of $p$ and $q$.

Correct A. p = \frac{5}{3}, \; q = \frac{2}{3}

B. p = \frac{5}{3}, \; q = \frac{3}{2}

C. p = \frac{4}{3}, \; q = \frac{2}{3}

D. p = \frac{3}{5}, \; q = \frac{2}{3}

Correct Answer: A

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Question 19 View Details

Solve the inequality $\frac{2x-5}{x+1} < 3$ and express the solution set in interval notation.

A. (-\infty, -1) \cup (-8, \infty)

Correct B. (-\infty, -8) \cup (-1, \infty)

C. (-\infty, -8] \cup (-1, \infty)

D. (-\infty, -8) \cup (-1, 1)

Correct Answer: B

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Question 20 View Details

Write the decimal $0.375$ as a fraction in lowest terms.

A. \frac{3}{5}

B. \frac{5}{8}

C. \frac{2}{7}

Correct D. \frac{3}{8}

Correct Answer: D

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Question 21 View Details

Solve the inequality $|2x-5| < x+3$ and find the sum of all integer values of $x$ that satisfy it.

A. 30

Correct B. 28

C. 24

D. 26

Correct Answer: B

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Question 22 View Details

The first three terms of a sequence form an arithmetic progression and the next three terms form a geometric progression. The fourth term equals the sum of the first three terms. The sixth term is $96$ and the sum of the first six terms is $186$. Find the first term of the sequence.

A. -8

B. -10

C. -16

Correct D. -12

Correct Answer: D

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Question 23 View Details

In a class of $120$ students, $68$ study Mathematics, $54$ study Physics and $48$ study Chemistry. The numbers studying both Mathematics and Physics, Mathematics and Chemistry, and Physics and Chemistry are $30$, $25$ and $20$ respectively. If $12$ students study none of these subjects, how many students study all three subjects?

A. 14

B. 15

C. 12

Correct D. 13

Correct Answer: D

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Question 24 View Details

Find the value of $k$ for which the system of equations $\begin{cases}2x + ky = 7\\ kx - 3y = 4\end{cases}$ has a solution with $x = y$.

A. 30\/3

B. 28\/3

C. 31\/3

Correct D. 29/3

Correct Answer: D

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Question 25 View Details

The time $t$ (seconds) taken for a particle to travel a distance $s$ (metres) under a constant force $F$ (newtons) varies directly with $\sqrt{s}$ and inversely with $F$. If a force of $8$ N moves the particle $20$ m in $5$ s, find the time required to move $45$ m under a force of $12$ N.

A. 4

B. 6

C. 8

Correct D. 5

Correct Answer: D

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