Sec 4 E-Math O-Level style Paper 1: free PDF with answers

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Mathematics

Sec 4 E-Math · O-Level style Paper 1

All topics

Practice paper in the same style as the school and national exam papers. Not a copy of any past year paper.

2 hours 15 minutes

Name

Class

Index number

Candidates answer on the Question Paper.

Read these instructions first

Write your name, class and index number in the spaces at the top of this page.

Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs.

Do not use staples, paper clips, glue or correction fluid.

Answer all questions.

If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks.

The use of an approved scientific calculator is expected, where appropriate.

If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place.

For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π.

The number of marks is given in brackets [ ] at the end of each question or part question.

The total of the marks for this paper is 90.

For Examiner's Use

/ 90

Mathematical Formulae

Compound interestTotal amount

= P\left(1 + \dfrac{r}{100}\right)^{n}

MensurationCurved surface area of a cone

= \pi r l

Surface area of a sphere

= 4\pi r^{2}

Volume of a cone

= \dfrac{1}{3}\pi r^{2} h

Volume of a sphere

= \dfrac{4}{3}\pi r^{3}

Area of triangle

ABC = \dfrac{1}{2}ab\sin C

Arc length

= r\theta

, where

\theta

is in radiansSector area

= \dfrac{1}{2}r^{2}\theta

, where

\theta

is in radians

Trigonometry

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

a^{2} = b^{2} + c^{2} - 2bc\cos A

StatisticsMean

= \dfrac{\sum fx}{\sum f}

Standard deviation

= \sqrt{\dfrac{\sum fx^{2}}{\sum f} - \left(\dfrac{\sum fx}{\sum f}\right)^{2}}

(a)Express

\dfrac{3x - 5}{3} + \dfrac{x - 3}{5}

as a single fraction in its simplest form.

Answer (a)[2]

(b)

A = 5 - 2x^{2} - 4x

B = -4x^{2} + 1

and

C = -2 + 5x - 4x^{2}

. Subtract

A

from the sum of

B

and

C

. Give your answer in its simplest form.

Answer (b)[2]

(a)Mei Ling buys a washing machine on hire purchase. Mei Ling pays a deposit of $360, followed by 30 monthly instalments of $39.60. Find the total amount Mei Ling pays for it.

Answer (a)$[2]

(b)The price of a cup of kopi fell from $2.40 to $2.10. Find the percentage decrease in the price.

Answer (b)%[2]

(a)A car travels 14 km on each litre of petrol. Find the distance, in km, it can travel on 36 litres of petrol.

Answer (a)[2]

(b)A bicycle travels at a constant speed of 27 km/h. Each of its wheels has a diameter of 70 cm and turns without slipping. Find the number of complete turns (revolutions) each wheel makes in one minute, correct to the nearest whole number.

Answer (b)[2]

(a)The diagram shows an isosceles triangle. State whether the dashed line is a line of symmetry of the shape.

Answer (a)[1]

(b)The diagram shows a five-pointed star. Find the smallest angle through which it can be rotated about its centre to look exactly the same.

Answer (b)°[2]

(a)A school wants to show the proportion of its pupils who travel to school by bus, by MRT, by car and on foot. Which statistical diagram is the most suitable?

Answer (a)[1]

(b)The pictogram shows the favourite hawker food of a class. Find the number of pupils for Laksa.

Answer (b)[2]

(a)Make

x

the subject of

y = \dfrac{3x + 1}{2x + 1}

Answer (a)[2]

(b)Express

\dfrac{1}{x + 1} + \dfrac{2}{x + 4}

as a single fraction in its simplest form.

Answer (b)[2]

In the diagram, the lines

AD

and

BC

meet at

X

, and

AB

is parallel to

DC

. Find

DC

Answercm[3]

(a)A ladder leans against an upright wall, making an angle of

71^\circ

with level ground. It reaches

6

m up the wall. Find the length of the ladder,

l

m, correct to 3 significant figures.

Answer (a)m[2]

(b)In the diagram,

\angle ABC = 90^\circ

. Find

x

, correct to 3 significant figures.

Answer (b)cm[2]

(a)The diagram shows a ball of radius

6

cm. Find its surface area, correct to 3 significant figures.

Answer (a)cm²[2]

(b)A solid cone has base radius

r

cm and height

h

cm. Its volume is equal to the volume of a solid cylinder of radius

r

cm and height

4r

cm. Find

h

in terms of

r

. Hence show that the total surface area of the cone is

k\pi r^2

cm², and find

k

, leaving your answer in surd form.

Answer (b)[3]

Marks (x)	Frequency
$24 < x \le 28$	14
$28 < x \le 32$	11
$32 < x \le 36$	10
$36 < x \le 40$	13

(a)The table shows the marks of some pupils in a test. Find the class that contains the median.

Answer (a)[2]

(b)The histogram shows the marks,

x

, of some pupils in a test. Complete the frequency table.

Answer (b)[2]

(a)Factorise

16p^{3}q - 49pq^{3}

completely.

Answer (a)[1]

(b)Expand and simplify

(4x + 6)(4x - 6)

Answer (b)[1]

(c)Factorise

5x^{2} - 18x - 8

Answer (c)[1]

Express

2x^{2} - 8x - 11

in the form

a(x + h)^2 + k

Answer[3]

Solve the inequality

\dfrac{2x + 5}{2} < \dfrac{3x + 9}{4} < 2x + 8

Answer[3]

Evaluate

\left(\dfrac{81}{256}\right)^{-\frac{1}{4}}

Answer[2]

(a)Find the length of the line segment joining

A(-1, 1)

and

B(2, -4)

, correct to 3 significant figures.

Answer (a)[2]

(b)Find the gradient of the line joining

A(-5, 3)

and

B(-2, -5)

Answer (b)[2]

(a)Given that

y = x^{3} - 3x^{2} + 2

, calculate the value of

y

when

x = -1.5

, correct to 1 decimal place.

Answer (a)[2]

(b)The diagram shows the graph of

y = x^{2} + 2x - 2

, and the straight line that must be drawn on the same axes to solve the equation

x^{2} + 3x = 0

. Use the graph to solve this equation.

Answer (b)[2]

Given that

\sin \theta = \dfrac{3}{5}

and that

\theta

is obtuse, find, without using a calculator, the value of

\sin \theta + \cos \theta

Answer[3]

(a)The diagram shows a cuboid with

AB = 8

cm and

BC = 4

cm. Find the length of the diagonal

AC

of the base, correct to 3 significant figures.

Answer (a)cm[2]

(b)From

A

B

10

km away on a bearing of

070^\circ

and

C

11

km away on a bearing of

115^\circ

. Find the distance

BC

, correct to 3 significant figures.

Answer (b)km[3]

(a)Express

112^\circ

in radians, correct to 3 significant figures.

Answer (a)rad[2]

(b)In the diagram, triangle

ABC

is congruent to triangle

RPQ

. Find

x

Answer (b)cm[2]

In the diagram, the lines

AD

and

BC

meet at

X

, and

AB

is parallel to

DC

AX = 5

cm,

XD = 8

cm and the area of triangle

XAB

15

cm². Find the area of triangle

XDC

Answercm²[3]

(a)In the diagram,

A

B

C

and

D

lie on the circle. Find

x

Answer (a)°[1]

(b)In the diagram,

PA

and

PB

are tangents to the circle with centre

O

. Find

x

Answer (b)°[2]

(a)

\xi = \{x : x \text{ is an integer and } 2 \le x \le 16\}

and

A

is the set of multiples of 5 in

\xi

. State whether

6 \in A

is true or false.

Answer (a)[1]

(b)In a group of

40

students,

12

play badminton and

17

play tennis.

4

do both. Find the number of students who play tennis only.

Answer (b)[2]

A group of

40

visitors to a zoo is made up of

15

adults and

25

children.

\dfrac{4}{5}

of the adults and

\dfrac{3}{5}

of the children bought a souvenir. One of the visitors to a zoo is chosen at random. Find the probability that the visitor is a child who did not buy a souvenir.

Answer[3]

(a)The numbers of books read by some students in two classes are shown. Class

A

25, \ 30, \ 28, \ 22, \ 35

. Class

B

5, \ 9, \ 7, \ 6, \ 8

. Which class read more books on average?

Answer (a)[1]

(b)The cumulative frequency table shows the time, in minutes, that

60

students took to finish a puzzle. Complete the frequency column.

Answer (b)[2]

(a)The Lions had 2 wins, 3 draws and 9 losses. The Tigers had 11 wins, 10 draws and 7 losses. Write down a matrix showing this information, with one row for each team and one column for each result, in the order given.

Answer (a)[2]

(b)

\overrightarrow{PQ} = 2\mathbf{a} + 5\mathbf{b}

and

\overrightarrow{RS} = 6\mathbf{a} + 15\mathbf{b}

. Find the value of

k

such that

\overrightarrow{RS} = k\overrightarrow{PQ}

Answer (b)[2]

The diagram shows the floor plan of a flat, drawn to a scale of 1 : 40. The lengths marked are measured on the plan. Find the actual floor area of the kitchen, in m².

Answerm²[2]

END OF PAPER

Answers and worked solutions

Question 1

(a)Answer:

= \dfrac{18x - 34}{15}

[2]

\text{LCM of } 3 \text{ and } 5 =

15

\dfrac{3x - 5}{3} + \dfrac{x - 3}{5} = \dfrac{5(3x - 5) + 3(x - 3)}{15}

= \dfrac{15x - 25 + 3x - 9}{15}

= \dfrac{18x - 34}{15}

(b)Answer:

= -6x^{2} + 9x - 6

[2]

B + C - A

(-4x^{2} + 1) + (-2 + 5x - 4x^{2}) - (5 - 2x^{2} - 4x)

= -4x^{2} + 1 - 2 + 5x - 4x^{2} - 5 + 2x^{2} + 4x

= -6x^{2} + 9x - 6

Question 2

(a)Answer: The total amount paid is $1548.[2]

30 \times 39.60 = 1188

\text{Total} = 360 + 1188

\phantom{\text{Total}} = 1548

The total amount paid is $1548.

(b)Answer:

= 12.5\%

[2]

\text{Decrease} = 2.40 - 2.10

\phantom{\text{Decrease}} = 0.30

\text{Percentage decrease} = \dfrac{0.30}{2.40} \times 100\%

= 12.5\%

Question 3

(a)Answer:

504

km[2]

14 \times 36

= 504

504

(b)Answer:

\phantom{\text{Turns}} = 204.62\ldots \approx 205

[2]

\dfrac{27 \times 100\,000}{60} = 45000 \text{ cm}

\pi \times 70 = 219.91\ldots \text{ cm}

\text{Turns} = \dfrac{45000}{219.91\ldots}

\phantom{\text{Turns}} = 204.62\ldots \approx 205

Question 4

(a)Answer: NO[1]

Each corner on one side must land exactly on a corner on the other side, and each side on a side.No: the two halves do not match when folded, so the dashed line is not a line of symmetry.

(b)Answer:

\text{Angle} = 72^\circ

°[2]

It looks the same every fifth of a turn.

\text{Order} = 5

\text{Angle} = \dfrac{360^\circ}{5}

\text{Angle} = 72^\circ

Question 5

(a)Answer: PIE[1]

The diagram has to show each part as a share of a whole.Pie chart.

(b)Answer:

= 75

[2]

There are

1.5

symbols for Laksa.

1.5 \times 50

= 75

Question 6

(a)Answer:

x = \dfrac{1 - y}{2y - 3}

[2]

y(2x + 1) = 3x + 1

2xy + y = 3x + 1

2xy - 3x = 1 - y

x(2y - 3) = 1 - y

x = \dfrac{1 - y}{2y - 3}

(b)Answer:

= \dfrac{3x + 6}{(x + 1)(x + 4)}

[2]

\dfrac{1}{x + 1} + \dfrac{2}{x + 4} = \dfrac{1(x + 4) + 2(x + 1)}{(x + 1)(x + 4)}

= \dfrac{x + 4 + (2x + 2)}{(x + 1)(x + 4)}

= \dfrac{3x + 6}{(x + 1)(x + 4)}

Question 7

Answer:

DC = 9

cm[3]

\angle XAB = \angle XDC

(alternate angles)

\angle AXB = \angle DXC

(vertically opposite angles)

\dfrac{DC}{AB} = \dfrac{XD}{XA}

\mathstrut \dfrac{DC}{6} = \dfrac{6}{4}

DC = \dfrac{6}{4} \times 6

DC = 9

Question 8

(a)Answer:

= 6.35

m (3 s.f.)[2]

From the angle at the foot of the ladder: the ladder,

l

m, is the hypotenuse and the height up the wall,

6

m, is the opposite.

\sin 71^\circ = \dfrac{6}{l}

l = \dfrac{6}{\sin 71^\circ}

= 6.3457\ldots

= 6.35

m (3 s.f.)

(b)Answer:

= 8.21

cm (3 s.f.)[2]

From the 46° angle: the side

x

cm is the adjacent and the side

8.5

cm is the opposite.

\tan 46^\circ = \dfrac{8.5}{x}

x = \dfrac{8.5}{\tan 46^\circ}

= 8.2083\ldots

= 8.21

cm (3 s.f.)

Question 9

(a)Answer:

= 452

cm² (3 s.f.)[2]

Surface area

= 4\pi r^2

Surface area

= 4 \times \pi \times 6^2

= 144\pi

= 452.3893\ldots

= 452

cm² (3 s.f.)

(b)Answer:

\pi r \times \sqrt{145}\,r + \pi r^2 = (1 + \sqrt{145})\pi r^2

[3]

\dfrac{1}{3}\pi r^2 h = 4\pi r^3

h = 12r

l^2 = r^2 + (12r)^2

\phantom{l^2} = 145r^2

l = \sqrt{145}\,r

\pi r \times \sqrt{145}\,r + \pi r^2 = (1 + \sqrt{145})\pi r^2

Question 10

(a)Answer: 28 < x ≤ 32[2]

Total frequency

= 14 + 11 + 10 + 13

= 48

The median is the mean of the

24

th and

25

th values.Running totals:

14,\ 25,\ 35,\ 48

The running total first reaches

24

in the class

28 < x \le 32

, so the median lies in it.

(b)Answer:

70 < x \le 80

: the bar's height is

2

.[2]

50 < x \le 60

: the bar's height is

4

60 < x \le 70

: the bar's height is

8

70 < x \le 80

: the bar's height is

2

Question 11

(a)Answer:

= pq(4p + 7q)(4p - 7q)

[1]

16p^{3}q - 49pq^{3} = pq(16p^{2} - 49q^{2})

= pq(4p + 7q)(4p - 7q)

(b)Answer:

16x^2-36

[1]

16x^2-36

(c)Answer:

(5x+2)(x-4)

[1]

(5x+2)(x-4)

Question 12

Answer:

= 2(x - 2)^2 - 19

[3]

2x^{2} - 8x - 11 = 2(x^2 - 4x) -11

Half of

-4

-2

, so the bracket is

(x - 2)

(x - 2)^2 = x^{2} - 4x + 4

2x^{2} - 8x - 11 = 2\Big((x - 2)^2 - 4\Big) -11

= 2(x - 2)^2 - 8 -11

= 2(x - 2)^2 - 19

Question 13

Answer:

-\dfrac{23}{5} < x < -1

[3]

\dfrac{2x + 5}{2} < \dfrac{3x + 9}{4}

and

\dfrac{3x + 9}{4} < 2x + 8

\dfrac{2x + 5}{2} < \dfrac{3x + 9}{4}

Multiply by

4

2(2x + 5) < 3x + 9

x < -1

\dfrac{3x + 9}{4} < 2x + 8

Multiply by

4

3x + 9 < 4(2x + 8)

-5x < 23

x > -\dfrac{23}{5}

(dividing by a negative number reverses the sign)

-\dfrac{23}{5} < x < -1

Question 14

Answer:

\dfrac{4}{3}

[2]

\left(\dfrac{81}{256}\right)^{-\frac{1}{4}} = \left(\dfrac{256}{81}\right)^{\frac{1}{4}}

\sqrt[4]{\dfrac{256}{81}} = \dfrac{4}{3}

\dfrac{4}{3}

Question 15

(a)Answer:

\phantom{AB} = 5.83

units (3 s.f.)[2]

AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

AB = \sqrt{(2 - (-1))^2 + (-4 - 1)^2}

= \sqrt{34}

AB = 5.8309\ldots

\phantom{AB} = 5.83

units (3 s.f.)

(b)Answer:

\phantom{m} = -\dfrac{8}{3}

[2]

m = \dfrac{y_2 - y_1}{x_2 - x_1}

m = \dfrac{-5 - 3}{-2 - (-5)}

m = \dfrac{-8}{3}

\phantom{m} = -\dfrac{8}{3}

Question 16

(a)Answer:

y = -8.1

[2]

y = (-1.5)^{3} - 3(-1.5)^{2} + 2

\phantom{y} = -8.125

y = -8.1

(b)Answer:

x = -3

x = 0

[2]

x^{2} + 3x = 0

x^{2} + 2x - 2 = -x - 2

The line drawn is

y = -x - 2

.Number of points where they meet:

2

x = -3

x = 0

Question 17

Answer:

\sin \theta + \cos \theta = -\dfrac{1}{5}

[3]

Opposite

= 3

, hypotenuse

= 5

: adjacent

= \sqrt{5^2 - 3^2}

Adjacent

= \sqrt{16}

= 4

(Pythagoras' theorem)

\cos \theta = -\cos \alpha

(

\theta

obtuse)

\cos \theta

= -\dfrac{4}{5}

\sin \theta + \cos \theta = \dfrac{3}{5} - \dfrac{4}{5}

\phantom{\sin \theta + \cos \theta} = \dfrac{3 - 4}{5}

\sin \theta + \cos \theta = -\dfrac{1}{5}

Question 18

(a)Answer:

\phantom{AC} = 8.94

cm (3 s.f.)[2]

AC^2 = 8^2 + 4^2

AC^2 = 80

AC = \sqrt{80}

AC = 8.9442\ldots

\phantom{AC} = 8.94

cm (3 s.f.)

(b)Answer:

\phantom{x} = 8.09

km (3 s.f.)[3]

\angle BAC = 115^\circ - 070^\circ

= 45^\circ

\text{Cosine rule}

x^2 = 10^2 + 11^2 - 2(10)(11) \cos 45^\circ

x^2 = 65.4365\ldots

x = 8.0892\ldots

\phantom{x} = 8.09

km (3 s.f.)

Question 19

(a)Answer:

112^\circ = 1.95

rad (3 s.f.)[2]

To change degrees to radians, multiply by

\dfrac{\pi}{180}

\mathstrut 112^\circ = 112 \times \dfrac{\pi}{180}

112^\circ = 1.9547\ldots

112^\circ = 1.95

rad (3 s.f.)

(b)Answer:

x = 7

cm[2]

\triangle ABC \equiv \triangle RPQ

, so

BC

matches

PQ

PQ = BC

x = 7

Question 20

Answer:

\text{Area of } \triangle XDC = 38.4

cm²[3]

\angle XAB = \angle XDC

(alternate angles)

\angle AXB = \angle DXC

(vertically opposite angles)

\dfrac{\text{Area of } \triangle XDC}{\text{Area of } \triangle XAB} = \left(\dfrac{XD}{XA}\right)^2

\mathstrut \dfrac{\text{Area of } \triangle XDC}{15} = \left(\dfrac{8}{5}\right)^2

\text{Area of } \triangle XDC = \left(\dfrac{8}{5}\right)^2 \times 15

\text{Area of } \triangle XDC = 38.4

cm²

Question 21

(a)Answer:

x = 106

°[1]

\mathstrut x = 180^\circ - 74^\circ

(∠s in opp. segments)

x = 106

(b)Answer:

x = 108

°[2]

\angle OAP = 90^\circ

(tan. ⊥ rad.)

\angle OBP = 90^\circ

(tan. ⊥ rad.)

\mathstrut x = 360^\circ - 90^\circ - 90^\circ - 72^\circ

(∠ sum of quad.)

x = 108

Question 22

(a)Answer: FALSE[1]

A = \{5, 10, 15\}

6 \notin A

So the statement is false.

(b)Answer:

\text{play tennis only} = 13

[2]

\mathstrut \text{play badminton only} = 12 - 4

\text{play badminton only} = 8

\mathstrut \text{play tennis only} = 17 - 4

\text{play tennis only} = 13

Question 23

Answer:

P = \dfrac{1}{4}

[3]

adults:

\dfrac{4}{5} \times 15 = 12

children:

\dfrac{3}{5} \times 25 = 15

children who did not buy a souvenir:

25 - 15 = 10

10

of the 40 visitors to a zoo fit.

\mathstrut P = \dfrac{10}{40}

P = \dfrac{1}{4}

Question 24

(a)Answer: A[1]

Better on average means the higher mean.

A: 28

B: 7

Class A's is higher.Class A.

(b)Answer: Frequencies:

2, \ 6, \ 14, \ 22, \ 11, \ 5

[2]

\text{1st} = 2

\mathstrut \text{2nd} = 8 - 2

\text{2nd} = 6

\text{Total} = 60

Frequencies:

2, \ 6, \ 14, \ 22, \ 11, \ 5

Question 25

(a)Answer:

\begin{pmatrix} 2 & 3 & 9 \\ 11 & 10 & 7 \end{pmatrix}

[2]

2 \quad 3 \quad 9

11 \quad 10 \quad 7

\begin{pmatrix} 2 & 3 & 9 \\ 11 & 10 & 7 \end{pmatrix}

(b)Answer:

3 \times 5 = 15

[2]

k \times 2 = 6

k = 3

3 \times 5 = 15

Question 26

Answer:

4.2 \times 3 = 12.6 \text{ m}^2

m²[2]

10.5 \times 40 = 420 \text{ cm}

7.5 \times 40 = 300 \text{ cm}

420 \text{ cm} = 4.2 \text{ m}

300 \text{ cm} = 3 \text{ m}

4.2 \times 3 = 12.6 \text{ m}^2

The kitchen's floor area is 12.6 m².

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