1. hobbies and interests
2. inventors and patents

Sample PAPER 1
Paper 1 contains 15 questions, and each question is marked out of 6, for a total of 90 marks. You have
1 hour 30 minutes for this paper, which means 1 mark a minute.
1. Jupiter is the largest of all the planets in our solar system. It has a radius of 71 492 kilometres at the equator and a
radius of 66 854 kilometres at the two poles.
(a) Round the radius of Jupiter at the poles to
(i) 3 significant figures
[1 mark]
(ii) the nearest 1000 km
[1 mark]
(b) Use the radius of Jupiter at the equator to calculate an estimate of the volume of the planet in km3.
Write your answer in the form a × 10k where 1 ≤ a < 10 and k ∈ .
[4 marks]
2. A curve has the equation y = x3 − x2 − 4x + 7.
(a) Find
dy
in terms of x.
dx
[2 marks]
Two points, A and B, lie on the curve. The gradient of the curve at both points is −3.
(b) Show that the x-coordinates of A and B satisfy the equation 3x2 − 2x − 1 = 0.
[2 marks]
(c) Hence find the x-coordinates of the two points.
[2 marks]
3. Monique has written a wish list for her 18th birthday. Her list includes a laptop computer, an iPad and a
mobile phone.
The propositions p, q and r are defined as follows.
p: Dad buys her a laptop
q: Dad buys her an iPad
r: Mum buys her a mobile phone
(a) Write the following logic statements in words.
(i) ¬q ⇒ p
[1 mark]
(ii) ¬ p ⇒ (r ∧ q)
[2 marks]
(b) Complete the truth table below for the propositions given.
p
q
¬p
¬q
T
T
F
F
T
F
F
T
F
T
T
F
F
F
T
T
p ⇒ ¬q ¬p ⇒ q (p ⇒ ¬q) ∨ (¬p ⇒ q)
[3 marks]
4. The following list shows the number of downloads (in thousands) of a free application for mobile phones in ten
consecutive weeks.
Week
1
2
3
4
5
6
7
8
9
10
Downloads
(thousands)
31
31
32
33
34
35
37
46
49
64
Cambridge Mathematical Studies for the IB Diploma Standard Level (c) Cambridge University Press, 2013
Sample PAPER 1
1
(a) Find
Fi the mean number of downloads in a week.
[2 marks]
(b) Fi
Find the median and the quartiles of the data and calculate the interquartile range.
[3 marks]
(c) What is the standard deviation of the weekly downloads?
[1 mark]
5. The straight line l1 has equation y = 7x − 10 and the straight line l2 has equation 5x + 4y − 26 = 0. The two lines
intersect at the point P.
(a) Find the coordinates of P.
[3 marks]
The line l3 passes through P and is perpendicular to line l1.
(b) Find the equation of line l3. Give your answer in the form ax + by + c = 0 where a, b and c
are integers.
[3 marks]
6. The diagram below shows a cumulative frequency curve of the average daily temperatures in 50 tourist cities.
50
Cumulative frequency
40
30
20
10
0
10
20
30
40
50
60
Temperature (°F)
70
80
90
(a) Write down the median daily temperature.
[1 mark]
(b) Calculate the interquartile range of the daily temperatures.
[2 marks]
(c) Estimate how many of the cities had an average daily temperature between 50°F and 70°F.
[2 marks]
(d) Use the diagram to estimate the minimum daily temperature of the warmest 10% of the cities.
[1 mark]
7. A mathematics revision handout was uploaded onto the internet in January 2010. The number of downloads of the
handout can be modelled by the polynomial
N = 15172 + 702t − 9t2
where N is the total number of downloads in the tth month after January 2010.
(a) How many times had the file been downloaded in the sixth month after it was uploaded?
[2 marks]
(b) Calculate the number of times the file had been downloaded in January 2011.
[2 marks]
(c) In which month was the file downloaded 25 540 times?
[2 marks]
8. Penny has a collection of old music CDs and DVDs, which she decides to sell at a car boot sale. The total number of
CDs and DVDs is 53.
(a) Assuming Penny had c CDs and v DVDs, write an equation in terms of c and v for
Penny’s collection.
Penny sells each CD for 40 pence and each DVD for 70 pence. She sold all her items and earned a total
of £25.70.
2
SSample PAPER 1
Cambridge Mathematical Studies for the IB Diploma Standard Level (c) Cambridge University Press, 2013
[2 marks]
(b) Based on Penny’s sales, write a second equation in terms of c and v.
[2 marks]
(c) Solve the two equations from parts (a) and (b) to find the number of CDs and the number of
DVDs that Penny sold.
[2 marks]
9. The diagram shows the distribution of weights of a sample of mobile phones.
Weight distribution of mobile phones
12
10
8
Frequency 6
4
2
0
70
90
110 130 150
Weight of mobile phone (g)
170
190
(a) Write down the class interval in which you would find:
(i) the modal weight
[1 mark]
(ii) the median weight
[2 marks]
(b) Calculate an estimate of the mean weight of the mobile phones.
[2 marks]
(c) Give an estimate for the standard deviation of the weights of the phones.
[1 mark]
10. The following data represents the physical activity level of a group of children between 11 and 14 years of age.
The level of activity is classified as high, medium or low. The table shows the number of boys and girls at each level
of activity.
Level of activity
High
Medium
Low
Boys
85
27
18
Girls
63
42
33
Carlos wants to determine whether the level of activity is associated with gender. He carries out a χ test at the 5%
significance level.
2
(a) Write a suitable null hypothesis for the χ test.
[1 mark]
(b) State the number of degrees of freedom.
[1 mark]
2
(c) Use your GDC to find the χ test statistic for the data.
2
[2 marks]
The critical value at the 5% level of significance is 5.99.
(d) State, with reasons, whether Carlos should accept or reject the null hypothesis.
Cambridge Mathematical Studies for the IB Diploma Standard Level (c) Cambridge University Press, 2013
[2 marks]
Sample PAPER 1
3
11. The diagram shows the graph of the function y = 2 × 3x+1 − 1.
The curve passes through the points A and B with coordinates (a, 0) and (0, b), respectively.
y
B
A
x
(a) Write down the values of a and b.
[2 marks]
(b) Write down the equation of the horizontal asymptote.
[2 marks]
The curve passes through another point C(1.5, c).
(c) Find the value of c.
[2 marks]
12. ABCDEF is a triangular prism. G is the midpoint of CF. BĈF is a right angle. AB = DC = EF = 60 cm, AD = BC =
50 cm and DE = CF = 40 cm.
E
F
G
40 cm
D
C
50 cm
A
60 cm
B
(a) Calculate the length of
(i) AC
[2 marks]
(ii) AF
[2 marks]
(b) Calculate the size of the angle between AF and AG.
[2 marks]
13. The sixth, seventh and eighth terms of an arithmetic sequence are, 7x + 5, 9x and 10x + 1, respectively.
(a) Calculate the value of x.
[2 marks]
(b) Find:
(i) the common difference
[1 mark]
(ii) the first term of the sequence.
[1 mark]
(c) Calculate the 26th term of the sequence.
4
SSample PAPER 1
Cambridge Mathematical Studies for the IB Diploma Standard Level (c) Cambridge University Press, 2013
[2 marks]
14. Seb and Gabby carried out a survey of 100 students from their sixth-form college. They asked: Do you visit the
college website regularly? If you do, how do you access the website?
All 100 students in the sample regularly visited the website. The results of the survey are shown below:
Methods of accessing website
Number of students
laptop, smartphone and desktop
5
laptop and smartphone only
7
laptop and desktop only
x
smartphone and desktop only
7
laptop only
35
desktop only
23
smartphone only
15
(a) Use the information from above to complete the given Venn diagram.
Smartphone
Laptop
Desktop
[4 marks]
(b) Find the value of x.
[2 marks]
15. Señora Mercado participates in a pension scheme. In the first year she paid in $300. Her payments
increased by 4% each year in a geometric sequence.
(a) Calculate the amount she paid in the 10th year of the scheme.
[3 marks]
Señora Mercado adjusted her payments in the 11th year. She paid in a sum of £450, and plans to reduce
her payment by 2% every year until her final payment, in the 20th year.
(b) How much will she be paying in the 20th year of the scheme?
Cambridge Mathematical Studies for the IB Diploma Standard Level (c) Cambridge University Press, 2013
[3 marks]
Sample PAPER 1
5
Sample Paper 1
1. (a) (i) 66 900 km
(ii) 67 000 km
(b) 1.53 × 1015 km3 (3 s.f.)
2. (a) 3x2 − 2x − 4
(c) − 13 and 1
3. (a) (i) If Dad does not buy her an iPad then he buys her a laptop.
(ii) If Dad does not buy her a laptop then he buys her an iPad and Mum buys her a mobile phone.
(b)
p
q
¬p
¬q
p ⇒ ¬q
¬p ⇒ q
(p ⇒ ¬q) ∨ (¬p ⇒ q)
T
T
F
F
F
T
T
T
F
F
T
T
T
T
F
T
T
F
T
T
T
F
F
T
T
T
F
T
4. (a) 39 200
(b) Median = 34 500, Q1 = 32 000, Q3 = 46 000, IQR = 14 000
5. (a) (2, 4)
(b) x + 7y − 30 = 0
6. (a) 56°F
(b) 11°F
(c) 25
7. (a) 19 060
(b) 21 805
(c) 19.8 (20th month)
8. (a) c + v = 53
(b) 40c + 70v = 2570
(c) 38 CDs, 15 DVDs
9. (a) (i) 130–150 g
(b) 126 g
(c) 23.7 g (3 s.f.)
10. (a)
(b)
(c)
(d)
(c) 10 157
(d) 74°F
(ii) 110–130 g
The level of activity is independent of gender, i.e. it is the same for boys and girls.
2
10.7 (3 s.f.)
2
Reject H0 since the χ statistic is higher than the critical value.
11. (a) a = −1.63 (3 s.f.), b = 5
(b) y = −1
12. (a) (i) 10 61 = 78.1 cm (3 s.f.)
(c) 30.2 (3 s.f.)
(ii) 10 77 = 87.7 cm (3 s.f.)
(b) 12.8° (3 s.f.)
13. (a) 6
(b) (i) 7
(c) 187
(ii) 12
14. (a)
Laptop
Smartphone
7
35
5
x
U
15
7
23
Desktop
(b) 8
15. (a) $427.00
(b) $375.19
Cambridge Mathematical Studies for the IB Diploma Standard Level (c) Cambridge University Press, 2013
Sample Paper 1
1
Sample PAPER 2
This paper contains 6 questions. The total number of marks is again 90, but the marks do not have to be
evenly spread among the questions as in Paper 1.
1. The diagram below shows the outline of a rounders pitch. ABCD is a square of side 12 metres. AED
E is a right angle.
F is the intersection of the diagonals BD and AC and the diagonals bisect each other. AF is parallel to ED.
C (2nd post)
12 m
12 m
F
(3rd post) D
B
(1st post)
12 m
(4th post) E
A
(a) Show by calculation that AE is 8.5 m to 1 d.p.
[2 marks]
(b) Calculate:
(i) the size of angle C F .
[2 marks]
(ii) the size of angle C E to the nearest degree.
[1 mark]
(c) What type of quadrilateral is shape AFDE?
[1 mark]
The section of the pitch formed by triangle EDC is to be re-turfed.
(3rd post) D
8.5 m
12 m
(4th post) E
C (2nd post)
(d) Work out the perimeter of EDC.
[3 marks]
(e) Calculate the area of triangle EDC.
[2 marks]
The groundsman is ordering some turf for the pitch. To make allowance for wastage he needs to order 5%
more than required. The turf costs £3.84 per square metre and is delivered in 1 m2 rolls; delivery is free.
(f) How many rolls of turf should the groundsman order?
[2 marks]
(g) Calculate the total amount to be paid for the turf.
[2 marks]
[Total 15 marks]
2. The following table shows the source of text messages Tanya has received on her new phone.
Source (sender)
Number of messages
Parents
Siblings
Friends and others
31
147
322
Cambridge Mathematical Studies for the IB Diploma Standard Level (c) Cambridge University Press, 2013
Sample PAPER 2
1
Tanya’s sister accidentally deleted one of the text messages.
Tanya
(a) Calculate
C
the probability that the deleted message:
(i) was sent by a sibling
[2 marks]
(ii) did not come from a parent
[1 mark]
(iii) did not come from a sibling given that it was not sent by a parent.
[2 marks]
At the end of the first week of using her new phone, Tanya had received 100 messages.
(b) Estimate the number of text messages that came from friends and others.
[2 marks]
Text messages that Tanya receives from her parents contain news about themselves (mum and dad) or about Tanya’s
siblings.
(c) Complete the following tree diagram.
0.4
About parents
..............
About siblings
0.8
About parents
..............
About siblings
From Mum
0.75
Messages
from parents
..............
From Dad
[2 marks]
Tanya receives a new message from her parents.
(d) Calculate the probability that the message:
(i) is from her mum and about her siblings
[2 marks]
(ii) is about her parents.
[2 marks]
(e) Given that the message is about her parents, calculate the probability that it came from her dad.
[3 marks]
[Total 16 marks]
3. The advertising director of a telecommunications company has collected data from branches of his company in nine
different countries. The data shows the sales of products (y) in millions of dollars and the corresponding amount
spent on advertising (x), also in millions of dollars.
Advertising cost (x)
Sales (y)
3.4
2.2
1.3
3.2
3
2.6
2.6
2.2
1.7
25.2
19.9
18.7
24.7
21.9
23.1
22.8
20.1
19.9
(a) Use your GDC to find:
(i) x, the mean advertising cost
[2 marks]
(ii) y, the mean sales figure.
[2 marks]
(b) Draw a scatter diagram to illustrate the data. Use a scale of 2 cm to represent $1 million on the x-axis and 1 cm to
represent $5 million on the y-axis.
[4 marks]
2
(c) Plot the point P(x y ) on your scatter diagram.
[1 mark]
(d) Describe the correlation between sales and advertising costs.
[1 mark]
SSample PAPER 2
Cambridge Mathematical Studies for the IB Diploma Standard Level (c) Cambridge University Press, 2013
(e) Use your GDC to find:
(i) the product moment correlation coefficient, r
[1 mark]
(ii) the equation of the regression line of y on x.
[2 marks]
(f) Use your equation of the regression line to estimate:
(i) the estimated sales figures when the advertising costs amount to $2 million
[2 marks]
(ii) the advertising cost when the projected value of sales is $21 million.
[3 marks]
[Total 18 marks]
4. The average attendance at basketball league matches was found to be normally distributed with a mean of 17 500 and
a standard deviation of 1600.
(a) With the aid of sketches, calculate the probability that one of the matches would have an attendance:
(i) higher than 19 500
[2 marks]
(ii) not more than 15 000
[2 marks]
(iii) between 16 000 and 18 000
[2 marks]
(b) Given that there were 40 matches played one weekend, find the number of matches with an estimated
attendance above 19 500.
[1 mark]
(c) A match is graded as ‘well attended’ if the attendance is ranked in the top 15%. Work out the minimum
attendance of the ‘well attended’ matches.
[2 marks]
[Total 9 marks]
5. The number of overseas students admitted to a university in the year 2001 was 300. In 2011, admissions of overseas
students had risen to 740.
Assume that the number of admissions of overseas students at the university increased every year, forming an
arithmetic sequence such that u1 = 300 and u11 = 740.
(a) Find the yearly increase in admissions of overseas students.
[3 marks]
(b) Work out the total number of overseas students admitted to the university between 2001 and 2011
(inclusive).
[3 marks]
Mr Fonseca, the director of admissions, believes that the number of admissions of overseas students formed a
geometric rather than an arithmetic sequence. Answer the following questions using the same information from
above but assuming that the sequence is geometric.
(c) Find the common ratio.
[4 marks]
(d) Work out the number of overseas students admitted in 2005.
[2 marks]
(e) Calculate the total number of admissions from 2005 to 2011 inclusive.
[3 marks]
[Total 15 marks]
Cambridge Mathematical Studies for the IB Diploma Standard Level (c) Cambridge University Press, 2013
Sample PAPER 2
3
Sample Paper 2
1. (b) (i) 45°
(c)
(d)
(e)
(f)
(g)
(ii) 135°
Square
39.5 m
36 m2
38
£145.92
147
500
2. (a) (i)
(ii)
469
500
(iii)
322
469
(b) 64
(c)
0.4
About parents
0.6
About siblings
0.8
About parents
0.2
About siblings
From Mum
0.75
0.25
From Dad
(d) (i) 0.45
(e) 0.4
(ii) 0.5
3. (a) (i) £2.47 million (3 s.f.)
(b), (c)
30
(ii) £21.8 million (3 s.f.)
y
25
Sales ($m)
20
15
10
5
x
0
1
2
Advertising cost ($m)
(d) Positive correlation
(e) (i) 0.919 (3 s.f.)
(f) (i) $20.4 million (3 s.f.)
4. (a) (i) 0.106 (3 s.f.)
(b) 4 (4.23) matches
(c) 19 158
5. (a) 44
6. (b) 12x2
(c) 1.094
(e) 48 x − 15
x2
4
(ii) y = 3.06x + 14.3
(ii) $2.20 million (3 s.f.)
(ii) 0.0591 (3 s.f.)
(b) 5720
3
(iii) 0.448 (3 s.f.)
(d) 430 (3 s.f.)
(f) 0.679 (3 s.f.)
(e) 4016 (3 s.f.)
(g) 33.2 m2 (3 s.f.)
Cambridge Mathematical Studies for the IB Diploma Standard Level (c) Cambridge University Press, 2013
Sample Paper 1
1