Practice igcse model questions vol1 mathematics 2019 Past Questions

Question 1 View Details

Solve the inequality $|3x - 7| < 2x + 1$ and express the solution set.

A. (6/5, 8]

B. (6/5, 9)

C. (5/4, 8)

Correct D. (6/5, 8)

Correct Answer: D

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Question 2 View Details

Find all integer values of $k$ for which there exists an integer $x$ satisfying both $2x + 3 > k$ and $x - 5 < 2k$.

Correct A. k \ge -3

B. k \le -3

C. k \ge -2

D. k > -3

Correct Answer: A

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Question 3 View Details

A rectangular field is to be divided into two equal smaller rectangles by a fence parallel to its length. The total length of fencing required for the outer perimeter and the internal dividing fence is $120$ m. Determine the dimensions of the field that give the maximum possible area and state that maximum area.

A. Length = 18 m, Width = 32 m, Maximum area = 576 m^2

B. Length = 25 m, Width = 25 m, Maximum area = 625 m^2

C. Length = 15 m, Width = 35 m, Maximum area = 525 m^2

Correct D. Length = 20 m, Width = 30 m, Maximum area = 600 m^2

Correct Answer: D

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Question 4 View Details

Find the smallest positive integer that leaves a remainder of $1$ when divided by $2$, a remainder of $2$ when divided by $3$, and a remainder of $3$ when divided by $4$.

Correct A. 11

B. 23

C. 13

D. 7

Correct Answer: A

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Question 5 View Details

In triangle $ABC$ the angle at $B$ is $60^{\circ}$ and side $AC = 10$ cm. Point $D$ lies on $AC$ such that $AD:DC = 1:2$. The line $BD$ is drawn and it is given that $\sin \angle ABD = \frac{1}{2}$. Find the length of side $AB$ (in cm).

A. 10*sqrt(3)/4

B. 20*sqrt(3)/3

Correct C. 10*sqrt(3)/3

D. 5*sqrt(3)

Correct Answer: C

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Question 6 View Details

The time $T$ (in seconds) taken for a particle to travel a distance $s$ metres under a constant force $F$ (in newtons) varies directly as $s$ and inversely as $\sqrt{F}$. When $s = 20$ m and $F = 25$ N the time is $8$ s. Find the time taken when $s = 45$ m and $F = 40$ N. Give your answer correct to two decimal places.

A. 18.00

B. 16.43

C. 13.42

Correct D. 14.23

Correct Answer: D

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Question 7 View Details

In a stationery shop the price of a pen is $p$ dollars and the price of a notebook is $n$ dollars. Three pens and two notebooks cost $\$5.20$. Five pens and four notebooks cost $\$9.40$. If the price of each pen is increased by $\$0.30$, the total cost of three pens and two notebooks becomes $\$6.10$. Find the original price of a notebook. Give your answer in dollars to two decimal places.

Correct A. 1.10

B. 1.25

C. 1.40

D. 0.95

Correct Answer: A

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Question 8 View Details

A bag contains red, blue and green balls. The probability of drawing a red ball is twice the probability of drawing a blue ball, and the probability of drawing a green ball is $0.2$. Two balls are drawn successively without replacement. Find the probability that exactly one of the two drawn balls is red. Express your answer as a fraction in lowest terms.

Correct A. 8/15

B. 7/15

C. 8/13

D. 9/16

Correct Answer: A

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Question 9 View Details

The sum of the first five terms of an arithmetic sequence is $40$ and the sum of the first nine terms is $100$. Find the twelfth term of the sequence. Give your answer as an integer.

A. 26

Correct B. 22

C. 20

D. 24

Correct Answer: B

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Question 10 View Details

A number is increased by $25\%$ and then the resulting number is decreased by $10\%$ of its new value. The final result is $108$. What was the original number? Give your answer as an integer.

A. 84

B. 108

C. 100

Correct D. 96

Correct Answer: D

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Question 11 View Details

The diagram shows a piecewise linear graph that passes through the points (0,0), (2,4), (4,4) and (6,2) in that order. The graph is drawn above the x‑axis. Find the value of $x$ (to one decimal place if necessary) for which the area enclosed between the graph and the x‑axis from $x=0$ up to that $x$ is exactly $10$ square units.

Correct A. 3.5

B. 2.8

C. 4.2

D. 5.0

Correct Answer: A

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Question 12 View Details

Given real numbers $a$ and $b$ satisfy $(a-b)^{2}=9$ and $ab=8$. Find the value of $\displaystyle \frac{a^{3}-b^{3}}{a-b}$.

A. 39

Correct B. 33

C. 27

D. 24

Correct Answer: B

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Question 13 View Details

Triangle $ABC$ has vertices $A(2,1)$, $B(5,1)$ and $C(5,4)$. It is first reflected in the line $y=x$, then rotated $90^{\circ}$ clockwise about the origin, and finally translated by the vector $(3,-2)$. (i) Find the coordinates of the image of point $C$. (ii) Describe the single transformation that maps triangle $ABC$ directly onto its final position.

A. Image of C: (6, -6); Overall transformation: rotation 90 clockwise about the origin followed by translation (3, -2)

B. Image of C: (8, -4); Overall transformation: reflection in the y‑axis followed by translation (3, -2)

Correct C. Image of C: (8, -6); Overall transformation: reflection in the x‑axis followed by translation (3, -2)

D. Image of C: (8, -6); Overall transformation: reflection in the x‑axis followed by translation (2, -3)

Correct Answer: C

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Question 14 View Details

A box contains 5 red, 4 blue and 3 green balls. One ball is drawn at random, its colour noted, then it is replaced and another ball of the same colour is added to the box. A second ball is then drawn. Let $X$ be the number of red balls drawn in the two draws. (a) Determine the probability distribution of $X$. (b) Find the expected value $E(X)$. (c) Find the variance $\operatorname{Var}(X)$.

Correct A. P(X=0)=14/39, P(X=1)=35/78, P(X=2)=5/26; E(X)=5/6; Var(X)=245/468

B. P(X=0)=15\/39, P(X=1)=35\/78, P(X=2)=4\/26; E(X)=5\/7; Var(X)=240\/468

C. P(X=0)=13\/39, P(X=1)=36\/78, P(X=2)=5\/26; E(X)=4\/6; Var(X)=250\/468

D. P(X=0)=14\/39, P(X=1)=34\/78, P(X=2)=6\/26; E(X)=5\/5; Var(X)=245\/460

Correct Answer: A

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Question 15 View Details

A sequence is defined by $a_{1}=2$ and $a_{n+1}=3a_{n}+4$ for $n\ge 1$. (i) Find an explicit formula for $a_{n}$. (ii) Using this formula, calculate the sum of the first five terms of the sequence.

A. a_n = 3·3^{n-1} - 2; Sum of first 5 terms = 353

Correct B. a_n = 4·3^{n-1} - 2; Sum of first 5 terms = 474

C. a_n = 4·3^{n-1} + 2; Sum of first 5 terms = 494

D. a_n = 4·3^{n-1} - 1; Sum of first 5 terms = 479

Correct Answer: B

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Question 16 View Details

A contractor charges a daily labour rate of $d$ dollars and a material cost that depends on the length of the job. For a 5‑day job the total charge is $2000, and for a 3‑day job the total charge is $1100. The material cost for the 5‑day job is twice the material cost for the 3‑day job. Find the daily labour rate $d$ in dollars.

A. 250

Correct B. 200

C. 300

D. 150

Correct Answer: B

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Question 17 View Details

The time $t$ (in minutes) taken to fill a tank varies directly as the volume $V$ (in litres) and inversely as the square of the pump speed $s$ (in litres per minute). If it takes 12 minutes to fill a 180‑litre tank with a pump speed of 6 L min$^{-1}$, how long will it take to fill a 300‑litre tank with a pump speed of 9 L min$^{-1}$? Give your answer to one decimal place.

A. 7.2

Correct B. 8.9

C. 9.5

D. 10.0

Correct Answer: B

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Question 18 View Details

The quadratic equation $x^{2} - 7x + 10 = 0$ has roots $r_{1}$ and $r_{2}$. Find the value of $r_{1}^{2} + r_{2}^{2}$.

Correct A. 29

B. 49

C. 21

D. 39

Correct Answer: A

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Question 19 View Details

A shop sells apples at $2.40 per kilogram. A customer buys $ \frac{3}{4}$ kg of apples at this price and then buys $1\frac{1}{2}$ kg at a reduced price of $1.80 per kilogram. What is the total amount paid, in dollars? Give your answer to two decimal places.

Correct A. 4.50

B. 5.10

C. 4.20

D. 3.90

Correct Answer: A

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Question 20 View Details

In a certain population, 2 % of people have a particular disease. A first screening test has a sensitivity of 90 % and a specificity of 85 %. Individuals who test positive are given a second, more accurate test with a sensitivity of 95 % and a specificity of 98 %. What is the probability that a person who tests positive on both tests actually has the disease? Express your answer as a fraction in lowest terms.

A. 300/334

Correct B. 285/334

C. 260/334

D. 295/334

Correct Answer: B

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Question 21 View Details

The Venn diagram shows the numbers of students studying three subjects $A$, $B$ and $C$. The total number of students is 120. In the diagram the following numbers are shown: only $A$: 30, only $C$: 20, only $B$: ?, $A\cap B$ only: ?, $A\cap C$ only: 15, $B\cap C$ only: ?, and $A\cap B\cap C$: 10. It is also given that the number of students who study exactly one subject is twice the number who study exactly two subjects. Find the number of students who study exactly two subjects.

A. 20

Correct B. 30

C. 35

D. 25

Correct Answer: B

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Question 22 View Details

The quadratic equation $x^{2}-(k+2)x+(k-3)=0$ has two real roots that differ by 4. Find the product of the roots.

A. -2

Correct B. -3

C. -4

D. 0

Correct Answer: B

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Question 23 View Details

In the diagram, the line through points $P(6,2)$ and $Q(4,5)$ meets the x‑axis at $A$ and the y‑axis at $B$. Find the area of triangle $OAB$, where $O$ is the origin.

A. 121/4

B. 119/3

Correct C. 121/3

D. 124/3

Correct Answer: C

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Question 24 View Details

In triangle $ABC$ the side $AB = 12$ cm, $\angle A = 30^{\circ}$ and $\angle B = 45^{\circ}$. The altitude from $B$ to side $AC$ is 6 cm. Find the length of side $AC$.

A. 12\sqrt{3}

B. 12(2-\sqrt{3})

C. 12(\sqrt{3}+1)

Correct D. 12(\sqrt{3}-1)

Correct Answer: D

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Question 25 View Details

Given that $\cos x = \frac{3}{5}$ with $x$ acute, find the exact value of $\tan (2x + 30^{\circ})$.

A. (7-24\\sqrt{3})\/(24-7\\sqrt{3})

B. (7+24\\sqrt{3})\/(24+7\\sqrt{3})

Correct C. (7-24\sqrt{3})/(24+7\sqrt{3})

D. (24-7\\sqrt{3})\/(7+24\\sqrt{3})

Correct Answer: C

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