52

1 Calculate 5
2
(a) giving your answer as a fraction,
(b) giving your answer as a decimal.

2
2 Work out the exact value of 1 + 8 .
4+
16+32

3 Write down
(a) an irrational number,
(b) a prime number between 60 and 70 .
 𝑛(𝑛+1)(3𝑛−1)
4 (a) The formula for the 𝑛th term of the sequence 2,15,48,110,210, … is .
2
Find the 9th term.
(b) The 𝑛th term of the sequence 12,19,28,39,52, … is (𝑛 + 2)2 + 3. Write down the formula fo
r the 𝑛th term of the sequence 19,26,35,46,59, …

5 A rectangle has sides of length 6.1 cm and 8.1 cm correct to one decimal place.
Calculate the upper bound for the area of the rectangle as accurately as possible.

6 In 2005 there were 9 million bicycles in Beijing, correct to the nearest million.
The average distance travelled by each bicycle in one day was 6.5 km correct to one decimal
place.
Work out the upper bound for the total distance travelled by all the bicycles in one day.
 1
7 The mass of the Earth is of the mass of the planet Saturn. The mass of the Earth is 5.97
95
24
× 10 kilograms.
Calculate the mass of the planet Saturn, giving your answer in standard form, correct to 2 si
gnificant figures.

8 Maria, Carolina and Pedro receive $800 from their grandmother in the ratio
Maria: Carolina: Pedro = 7: 5: 4.
(a) Calculate how much money each receives.
2
(b) Maria spends 7 of her money and then invests the rest for two years at 5% per year sim
ple interest.
How much money does Maria have at the end of the two years?
(c) Carolina spends all of her money on a hi-fi set and two years later sells it at a loss of
20%.
How much money does Carolina have at the end of the two years?
(d) Pedro spends some of his money and at the end of the two years he has $100.
Write down and simplify the ratio of the amounts of money Maria, Carolina and Pedro have
at the end of the two years.
(e) Pedro invests his $100 for two years at a rate of 5% per year compound interest.
Calculate how much money he has at the end of these two years.
9 (a) The scale of a map is 1: 20000000.
On the map, the distance between Cairo and Addis Ababa is 12 cm.
i) Calculate the distance, in kilometres, between Cairo and Addis Ababa.
ii) On the map the area of a desert region is 13 square centimetres.
Calculate the actual area of this desert region, in square kilometres.
(b) i) The actual distance between Cairo and Khartoum is 1580 km.
On a different map this distance is represented by 31.6 cm.
Calculate, in the form 1: 𝑛, the scale of this map.
ii) A plane flies the 1580 km from Cairo to Khartoum.
It departs from Cairo at 1155 and arrives in Khartoum at 1403.
Calculate the average speed of the plane, in kilometres per hour.
 𝑛(𝑛+1)
10 1 + 2 + 3 + 4 + 5 + ⋯ + 𝑛 =
2
(a) i) Show that this formula is true for the sum of the first 8 natural numbers.
ii) Find the sum of the first 400 natural numbers.
(b) i) Show that 2 + 4 + 6 + 8 + ⋯ + 2𝑛 = 𝑛(𝑛 + 1).
ii) Find the sum of the first 200 even numbers.
iii) Find the sum of the first 200 odd numbers.
(c) i) Use the formula at the beginning of the question to find the sum of the first 2𝑛 natur
al numbers.
ii) Find a formula, in its simplest form, for
1 + 3 + 5 + 7 + 9 + ⋯ + (2𝑛 − 1).
Show your working.
11 Water flows into a tank at a rate of 3 litres per minute.
(a) If the tank fills completely in 450 seconds, what is the capacity of the tank in ml ?
A hole is drilled in the bottom of the tank. Water flows out of the hole at a rate of 60ml pe
r second.
(b) How long will the tank take to drain completely if the flow of water in continues at the
same rate?

12 Using a table of differences if necessary, calculate the rule for the 𝑛th term of the sequence
8,24,58,116,204, ….
 13 Find the coordinates of the point of intersection of the straight lines
4𝑥 + 𝑦 = 17
3𝑥 − 2𝑦 = 10

14 The graph of 𝑦 = −4𝑥2 + 16𝑥 − 13 is drawn on the grid. [max: 7]

a) Explain why 𝑦 = 3.7 has no solution. [1]

b) The equation −2𝑥2 + 7𝑥 − 5 = 0 can be solved by adding a straight line to the grid above. Find the
equation of this line. [1]
c) By drawing this straight line, solve the equation − 2𝑥2 + 7𝑥 − 5 = 0. [3]

15 Solve the following simultaneous equations. [6]

4𝑥2 + 3𝑥𝑦 + 𝑦2 = 8
𝑥𝑦 + 4 = 0
 16 The table shows some values for 𝑦 = 2 × 0.5𝑥 − 1. [max: 11]

𝑥 −1 −0.5 0 0.5 1 1.5 2

𝑦 3 1.83 0.41 0 −0.29

a)
i) Complete the table. [2]
ii) On the grid, draw the graph of 𝑦 = 2 × 0.5𝑥 − 1 for −1 ⩽ 𝑥 ⩽ 2. [4]

a) By drawing a suitable straight line, solve the equation 2 × 0.5𝑥 + 2𝑥 − 3.5 = 0 for − 1 ⩽ 𝑥 ⩽ 2. [3]

b) There are no solutions to the equation 2 × 0.5𝑥 − 1 = 𝑘 where 𝑘 is an integer.

Complete the following statements. [2]
The highest possible value of 𝑘 is:
The equation of the asymptote to the graph of 𝑦 = 2 × 0.5𝑥 − 1 is:
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

17 The function f: 𝑥 ↦ 𝑥 2 − 4𝑥 + 𝑘 is defined for the domain 𝑥 ⩾ 𝑝, where 𝑘 and 𝑝

are constants. [max: 6]

i Express f(𝑥) in the form (𝑥 + 𝑎)2 + 𝑏 + 𝑘, where 𝑎 and 𝑏 are constants. He

nce, state the minimum value of f(x) and the corresponding value of x. [4]

ii Find the set of values of k where f(x)=0 does not have any solutions. [2]
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

18 In Figure 1, AB = 14 cm, BC = 11 cm and AK = BL = CM = DN = 𝑥 cm. If the area of

KLMN is now 97 cm2 , find 𝑥. [4]

19 Simplify or factorise the following: [6]

𝑥 2 −4
a) 𝑥 2 −5𝑥+6

b) 𝑎𝑥 + 𝑏𝑥 + 𝑘𝑎𝑦 + 𝑘𝑏𝑦
𝑎 2 𝑥 2 −𝑏2 𝑦 2
c) 𝑎 2 𝑥 2 +2𝑎𝑏𝑥𝑦+𝑏2 𝑦 2
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

20

Not to scale
A circle, centre 𝑂, touches all the sides of the regular octagon 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻 sha
ded in the diagram.
The sides of the octagon are of length 12 cm.
𝐵𝐴 and 𝐺𝐻 are extended to meet at 𝑃.
𝐻𝐺 and 𝐸𝐹 are extended to meet at 𝑄.
(a) i) Show that angle 𝐵𝐴𝐻 is 135∘ .
ii) Show that angle 𝐴𝑃𝐻 is 90∘ .
(b) Calculate
i) the length of 𝑃𝐻,
ii) the length of 𝑃𝑄,
iii) the area of triangle 𝐴𝑃𝐻,
iv) the area of the octagon.
(c) Calculate
i) the radius of the circle,
ii) the area of the circle as a percentage of the area of the octagon.
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

21

The largest possible circle is drawn inside a semi-circle, as shown in the diagr
am.
The distance 𝐴𝐵 is 12 centimetres.
(a) Find the shaded area.
(b) Find the perimeter of the shaded area

22 𝐴𝐵𝐶𝐷𝐸 is a regular pentagon.

𝐷𝐸𝐹 is a straight line.
Calculate
(a) angle 𝐴𝐸𝐹,
(b) angle DAE.
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

23 Two similar vases have heights which are in the ratio 3: 2.

(a) The volume of the larger vase is 1080 cm3.
Calculate the volume of the smaller vase.
(b) The surface area of the smaller vase is 252 cm2.
Calculate the surface area of the larger vase.

24 A conical beaker has a base radius of 8 cm and a height of 15 cm.

A conical tank full of acid is a similar shape to the beaker.
The beaker can be filled with acid from the tank exactly 1728 times.
Work out the base radius and height of the tank.
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

25

𝐴, 𝐵, 𝐶 and 𝐷 lie on a circle, centre 𝑂.

𝐵𝐷 is a diameter of the circle and 𝑃𝐴𝑇 is the tangent to the circle at 𝐴. Angl
e 𝐴𝐵𝐷 = 62∘ and angle 𝐶𝐷𝐵 = 28∘ .
Find
(a) angle 𝐴𝐶𝐷,
(b) angle 𝐴𝐷𝐵,
(c) angle 𝐷𝐴𝑇,
(d) angle 𝐶𝐴𝑂.

26
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

27

𝐴𝐵𝐶𝐷 is a cyclic quadrilateral. 𝑃𝑄 is a tangent to the circle at 𝐴.

Given that 𝑃𝐴̂𝐷 = 56∘ , 𝐷𝐴̂𝐵 = 80∘ and 𝐶𝐷𝐵
̂ = 36∘ , prove that 𝐴𝐷 and 𝐶𝐵 are pa
rallel.

28 A spray can is used to paint a wall.

The thickness of the paint on the wall is 𝑡. The distance of the spray can fro
m the wall is 𝑑.
𝑡 is inversely proportional to the square of 𝑑.
𝑡 = 0.2 when 𝑑 = 8.
Find 𝑡 when 𝑑 = 10.
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

1 1 𝑐−𝑑
29 Write + − as a single fraction in its simplest form.
𝑐 𝑑 𝑐𝑑

3𝑥−4 𝑥+3
30 Solve the inequality 7
< 4
.
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

31 Answer the whole of this question on a sheet of graph paper.

Tiago does some work during the school holidays.
In one week he spends 𝑥 hours cleaning cars and 𝑦 hours repairing cycles.
The time he spends repairing cycles is at least equal to the time he spends cle
aning cars.
This can be written as 𝑦 ⩾ 𝑥.
He spends no more than 12 hours working.
He spends at least 4 hours cleaning cars.
(a) Write down two more inequalities in 𝑥 and/or 𝑦 to show this information.
(b) Draw 𝑥 and 𝑦 axes from 0 to 12 , using a scale of 1 cm to represent 1 un
it on each axis.
(c) Draw three lines to show the three inequalities. Shade the unwanted region
s.
(d) Tiago receives $3 each hour for cleaning cars and $1.50 each hour for repa
iring cycles.
i) What is the least amount he could receive?
ii) What is the largest amount he could receive?
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

32

The quadrilateral 𝑃𝑄𝑅𝑆 shows the boundary of a forest. A straight 15 kilometr

e road goes due East from 𝑃 to 𝑅.
(a) The bearing of 𝑆 from 𝑃 is 030∘ and 𝑃𝑆 = 7 km.
i) Write down the size of angle SPR.
ii) Calculate the length of 𝑅𝑆.
(b) Angle 𝑅𝑃𝑄 = 55∘ and 𝑄𝑅 = 14 km.
i) Write down the bearing of 𝑄 from 𝑃.
ii) Calculate the acute angle 𝑃𝑄𝑅.
iii) Calculate the length of 𝑃𝑄.
(c) Calculate the area of the forest, correct to the nearest square kilometre.
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

33

The diagram shows a pyramid on a rectangular base 𝐴𝐵𝐶𝐷, with 𝐴𝐵 = 6 cm an

d 𝐴𝐷 = 5 cm.
The diagonals 𝐴𝐶 and 𝐵𝐷 intersect at 𝐹.
The vertical height 𝐹𝑃 = 3 cm.
(a) How many planes of symmetry does the pyramid have?
(b) Calculate the volume of the pyramid.
1
[The volume of a pyramid is 3 × area of base × height.]
(c) The midpoint of 𝐵𝐶 is 𝑀.
Calculate the angle between 𝑃𝑀 and the base.
(d) Calculate the angle between 𝑃𝐵 and the base.
(e) Calculate the length of 𝑃𝐵.
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

34 Sketch the graph of 𝑦 = cos⁡ 𝑥 for 0∘ ⩽ 𝑥 ⩽ 360∘ .

Hence solve the equation
3cos⁡ 𝑥 + 1 = 0
for 0∘ ⩽ 𝑥 ⩽ 360∘ . Give your answers to one decimal place.
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

35 (a) A train completed a journey of 850 kilometres with an average speed of 8

0 kilometres per hour.
Calculate, giving exact answers, the time taken for this journey in
i) hours,
ii) hours, minutes and seconds.
(b) Another train took 10 hours 48 minutes to complete the same 850 km jour
ney.
i) It departed at 19:20.
At what time, on the next day, did this train complete the journey?
ii) Calculate the average speed, in kilometres per hour, for the journey.
(c)

The solid line 𝑂𝐴𝐵𝐶𝐷 on the grid shows the first 10 seconds of a car journey.
i) Describe briefly what happens to the speed of the car between 𝐵 and 𝐶.
ii) Describe briefly what happens to the acceleration of the car between 𝐵 and
𝐶.
iii) Calculate the acceleration between 𝐴 and 𝐵.
iv) Using the broken straight line 𝑂𝐶, estimate the total distance travelled by t
he car in the whole 10 seconds.
v) Explain briefly why, in this case, using the broken line makes the answer t
o part (iv) a good estimate of the distance travelled.
[1]
vi) Calculate the average speed of the car during the 10 seconds.
Give your answer in kilometres per hour.
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

36

The diagram shows the accurate graph of 𝑦 = f(𝑥).

(a) Use the graph to find
i) f(0),
ii) 𝑓(8).
(b) Use the graph to solve
i) f(𝑥) = 0,
ii) 𝑓(𝑥) = 5.
(c) 𝑘 is an integer for which the equation f(𝑥) = 𝑘 has exactly two solutions.
Use the graph to find the two values of 𝑘.
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

37 A curve 𝐶 has equation 𝑦 = 2𝑥 3 − 4𝑥 2 + 5𝑥.

d𝑦
(a) Find d𝑥.
(b) Find the value of the gradient of the curve at the point where 𝑥 = 0.5.
(c) Explain, with a reason, whether curve 𝐶 has any turning points.

38 𝐴 and 𝐵 are sets.

Write the following sets in their simplest form.
(a) 𝐴 ∩ 𝐴′ .
(b) 𝐴 ∪ 𝐴′ .
(c) (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐵)′ .
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

39

𝑂𝐴𝐵𝐶𝐷𝐸 is a regular hexagon.

With 𝑂 as origin the position vector of 𝐶 is 𝐜 and the position vector of 𝐷 is
𝐝.
(a) Find, in terms of 𝐜 and 𝐝,
⃗⃗⃗⃗⃗ ,
i) 𝐷𝐶
ii) ⃗⃗⃗⃗⃗
𝑂𝐸 ,
iii) the position vector of 𝐵.

3
40 𝑓(𝑥) = 2𝑥 − 1, 𝑔(𝑥) = 𝑥 + 1, ℎ(𝑥) = 2𝑥
(a) Find the value of fg⁡(6).
(b) Write, as single fraction, gf⁡(𝑥) in terms of 𝑥.
(c) Find g −1 (𝑥).
(d) Find hh(3).
24
(e) Find 𝑥 when h(𝑥) = g (− 7 )
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

41 Answer the whole of this question on one sheet of graph paper.

(a) Draw and label 𝑥 - and 𝑦-axes from −8 to +8, using a scale of 1 cm to 1
unit on each axis.
(b) Draw and label triangle 𝐴𝐵𝐶 with 𝐴(2,2), 𝐵(5,2) and 𝐶(5,4).
(c) On your grid:
3
i) translate triangle 𝐴𝐵𝐶 by the vector ( ) and label this image 𝐴1 𝐵1 𝐶1 ;
−9
ii) reflect triangle 𝐴𝐵𝐶 in the line 𝑥 = −1 and label this image 𝐴2 𝐵2 𝐶2 ;
iii) rotate triangle 𝐴𝐵𝐶 by 180∘ about (0,0) and label this image 𝐴3 𝐵3 𝐶3 .

42 Answer the whole of this question on a sheet of graph paper.

(a) Draw a label 𝑥 - and 𝑦-axes from −6 to 6 , using a scale of 1 cm to 1 uni
t.
(b) Draw triangle 𝐴𝐵𝐶 with 𝐴(2,1), 𝐵(3,3) and 𝐶(5,1).
(c) Draw the reflection of triangle 𝐴𝐵𝐶 in the line 𝑦 = 𝑥. Label this 𝐴1 𝐵1 𝐶1.
(d) Rotate triangle 𝐴1 𝐵1 𝐶1 about (0,0) through 90∘ anticlockwise. Label this
𝐴2 𝐵2 𝐶2 .
(e) Describe fully the single transformation which maps triangle 𝐴𝐵𝐶 onto tria
ngle 𝐴2 𝐵2 𝐶2 .
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

43 The speeds ( 𝑣 kilometres/hour) of 150 cars passing a 50 km/h speed limit sig
n are recorded. A cumulative frequency curve to show the results is drawn bel
ow.

(a) Use the graph to find

i) the median speed,
ii) the interquartile range of the speeds,
iii) the number of cars travelling with speeds of more than 50 km/h.

(b) A frequency table showing the speeds of the cars is

Speed (𝑣𝐤𝐦/𝐡) Frequency

30 < 𝑣 ⩽ 35 10

35 < 𝑣 ⩽ 40 17

40 < 𝑣 ⩽ 45 33

45 < 𝑣 ⩽ 50 42

50 < 𝑣 ⩽ 55 𝑛

55 < 𝑣 ⩽ 60 16

i) Find the value of 𝑛.

ii) Calculate an estimate of the mean speed.
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

(c) Answer this part of this question on a sheet of graph paper. Another frequency ta
ble for the same speeds is

Speed (𝒗 𝐤𝐦/𝐡) 30 < 𝑣 ⩽ 40 40 < 𝑣 ⩽ 55 55 < 𝑣 ⩽ 60

Frequency 27 107 16

Draw an accurate histogram to show this information. Use 2 cm to represent 5 units

on the speed axis and 1 cm to represent 1 unit on the frequency density axis (so that
1 cm2 represents 2.5 cars).
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

44
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

45 The tree diagram shows a testing procedure on calculators, taken from a large
batch.

Each time a calculator is choosen at random, the probability that it is faulty (

1
𝐹) is .
20
(a) Write down the values of 𝑝 and 𝑞.
(b) Two calculators are chosen at random. Calculate the probability that
i) both are faulty,
ii) exactly one is faulty.
(c) If exactly one out of two calculators tested is faulty, then a third calculator
is chosen at random.
Calculate the probability that exactly one of the first two calculators is faulty a
nd the third one is faulty.
(d) The whole batch of calculators is rejected either if the first two chosen are
both faulty or if a third one needs to be chosen and it is faulty.
Calculate the probability that the whole batch is rejected.
(e) In one month, 1000 batches of calculators are tested in this way.
How many batches are expected to be rejected?
Y11 0580 P2 & P4 Test Prep Bloomsbury Education

46 (a) All 24 students in a class are asked whether they like football and whether
they like basketball. Some of the results are shown in the Venn diagram belo
w.

𝜀 = { students in the class }.

𝐹 = { students who like football }.
𝐵 = { students who like basketball }.
i) How many students like both sports?
ii) How many students do not like either sport?
iii) Write down the value of 𝑛(𝐹 ∪ 𝐵).
iv) Write down the value of 𝑛(𝐹 ′ ∩ 𝐵).
v) A student from the class is selected at random.
What is the probability that this student likes basketball?
vi) A student who likes football is selected at random.
What is the probability that this student likes basketball?
(b) Two students are selected at random from a group of 10 boys and 12 girl
s.
Find the probability that
i) they are both girls,
ii) one is a boy and one is a girl.