Download File - Jason Murphy

1.
(a)
List the elements of the set A = {x│–4 ≤ x ≤ 2, x is an integer}.
(1)
A number is chosen at random from set A.
Write down the probability that the number chosen is
(b)
a negative integer;
(2)
(c)
a positive even integer;
(1)
(d)
an odd integer less than –1.
(2)
(Total 6 marks)
2.
A satellite travels around the Earth in a circular orbit 500 kilometres above the Earth’s surface.
The radius of the Earth is taken as 6400 kilometres.
(a)
Write down the radius of the satellite’s orbit.
(1)
(b)
Calculate the distance travelled by the satellite in one orbit of the Earth.
Give your answer correct to the nearest km.
(3)
(c)
Write down your answer to (b) in the form a × 10k, where 1 ≤ a < 10, k 
.
(2)
(Total 6 marks)
IB Questionbank Mathematical Studies 3rd edition
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3.
Given p = x–
(a)
y
z
, x = 1.775, y = 1.44 and z = 48,
calculate the value of p.
(2)
Barry first writes x, y and z correct to one significant figure and then uses these values to
estimate the value of p.
(b)
(i)
Write down x, y and z each correct to one significant figure.
(ii)
Write down Barry’s estimate of the value of p.
(2)
(c)
Calculate the percentage error in Barry’s estimate of the value of p.
(2)
(Total 6 marks)
4.
The seventh term, u7, of a geometric sequence is 108. The eighth term, u8, of the sequence is 36.
(a)
Write down the common ratio of the sequence.
(1)
(b)
Find u1.
(2)
The sum of the first k terms in the sequence is 118 096.
(c)
Find the value of k.
(3)
(Total 6 marks)
5.
U is the set of all the positive integers less than or equal to 12.
A, B and C are subsets of U.
A = {1, 2, 3, 4, 6,12}
B = {odd integers}
C = {5, 6, 8}
IB Questionbank Mathematical Studies 3rd edition
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(a)
Write down the number of elements in A  C.
(1)
(b)
List the elements of B.
(1)
(c)
Complete the following Venn diagram with all the elements of U.
(4)
(Total 6 marks)
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6.
The planet Earth takes one year to revolve around the Sun. Assume that a year is 365 days and
the path of the Earth around the Sun is the circumference of a circle of radius 150 000 000 km.
diagram not to scale
(a)
Calculate the distance travelled by the Earth in one day.
(4)
(b)
Give your answer to part (a) in the form a × 10k where 1 ≤ a < 10 and k 
.
(2)
(Total 6 marks)
7.
80 matches were played in a football tournament. The following table shows the number of
goals scored in all matches.
(a)
Number of goals
0
1
2
3
4
5
Number of matches
16
22
19
17
1
5
Find the mean number of goals scored per match.
(2)
(b)
Find the median number of goals scored per match.
(2)
IB Questionbank Mathematical Studies 3rd edition
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A local newspaper claims that the mean number of goals scored per match is two.
(c)
Calculate the percentage error in the local newspaper’s claim.
(2)
(Total 6 marks)
8.
A manufacturer in England makes 16 000 garden statues. 12 % are defective and cannot be sold.
(a)
Find the number of statues that are non-defective.
(2)
The manufacturer sells each non-defective statue for 5.25 British pounds (GBP) to an American
company. The exchange rate from GBP to US dollars (USD) is 1 GBP = 1.6407 USD.
(b)
Calculate the amount in USD paid by the American company for all the non-defective
statues. Give your answer correct to two decimal places.
(2)
The American company sells one of the statues to an Australian customer for 12 USD.
The exchange rate from Australian dollars (AUD) to USD is 1 AUD = 0.8739 USD.
(c)
Calculate the amount that the Australian customer pays, in AUD, for this statue.
Give your answer correct to two decimal places.
(2)
(Total 6 marks)
9.
Shiyun bought a car in 1999. The value of the car V, in USD, is depreciating according to the
exponential model
V = 25 000 × 1.5–0.2t, t ≥ 0,
where t is the time, in years, that Shiyun has owned the car.
(a)
Write down the value of the car when Shiyun bought it.
(1)
IB Questionbank Mathematical Studies 3rd edition
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(b)
Calculate the value of the car three years after Shiyun bought it. Give your answer correct
to two decimal places.
(2)
(c)
Calculate the time for the car to depreciate to half of its value since Shiyun bought it.
(3)
(Total 6 marks)
10.
Give all answers in this question to the nearest whole currency unit.
In January 2008 Larry had 90 000 USD to invest for his retirement in January 2011.
He invested 40 000 USD in US government bonds which paid 4 % per annum simple interest.
(a)
Calculate the value of Larry’s investment in government bonds in January 2011.
(3)
Larry changed this investment into South African rand (ZAR) at an exchange rate of
1 USD = 18.624 ZAR.
(b)
Calculate the amount that Larry received in ZAR from the exchange.
(2)
He changed the remaining 50 000 USD to South African rand (ZAR) in January 2008.
The exchange rate between USD and ZAR was 1 USD = 10.608 ZAR. There was 2.5 %
commission charged on the exchange.
(c)
Calculate the value, in USD, of the commission Larry paid.
(2)
(d)
Show that the amount that Larry had to invest is 517 000 ZAR, correct to the nearest
thousand ZAR.
(3)
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In January 2008, Larry deposited this money into a bank account that paid interest at a nominal
annual rate of 12 %, compounded monthly.
(e)
Find the value of the money in Larry’s bank account in January 2011.
(3)
(Total 13 marks)
11.
Give all your numerical answers correct to two decimal places.
On 1 January 2005, Daniel invested 30 000 AUD at an annual simple interest rate in a Regular
Saver account. On 1 January 2007, Daniel had 31 650 AUD in the account.
(a)
Calculate the rate of interest.
(3)
On 1 January 2005, Rebecca invested 30 000 AUD in a Supersaver account at a nominal annual
rate of 2.5 % compounded annually.
(b)
Calculate the amount in the Supersaver account after two years.
(3)
(c)
Find the number of complete years since 1 January 2005 it will take for the amount in
Rebecca’s account to exceed the amount in Daniel’s account.
(3)
On 1 January 2007, Daniel reinvested 80 % of the money from the Regular Saver account in an
Extra Saver account at a nominal annual rate of 3 % compounded quarterly.
(d)
(i)
Calculate the amount of money reinvested by Daniel on the 1 January 2007.
(ii)
Find the number of complete years it will take for the amount in Daniel’s Extra
Saver account to exceed 30 000 AUD.
(5)
(Total 14 marks)
12.
A geometric sequence has 1024 as its first term and 128 as its fourth term.
(a)
Show that the common ratio is
1
.
2
(2)
IB Questionbank Mathematical Studies 3rd edition
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(b)
Find the value of the eleventh term.
(2)
(c)
Find the sum of the first eight terms.
(3)
(d)
Find the number of terms in the sequence for which the sum first exceeds 2047.968.
(3)
(Total 10 marks)
13.
Consider the arithmetic sequence 1, 4, 7, 10, 13, …
(a)
Find the value of the eleventh term.
(2)
(b)
The sum of the first n terms of this sequence is
n
(3n – 1).
2
(i)
Find the sum of the first 100 terms in this arithmetic sequence.
(ii)
The sum of the first n terms is 477.
(a)
Show that 3n2 – n – 954 = 0.
(b)
Using your graphic display calculator or otherwise, find the number of
terms, n.
(6)
(Total 8 marks)
14.
A shipping container is a cuboid with dimensions 16 m, 1
(a)
3
2
m and 2 m.
4
3
Calculate the exact volume of the container. Give your answer as a fraction.
(3)
IB Questionbank Mathematical Studies 3rd edition
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Jim estimates the dimensions of the container as 15 m, 2 m and 3 m and uses these to estimate
the volume of the container.
(b)
Calculate the percentage error in Jim’s estimated volume of the container.
(3)
(Total 6 marks)
15.
The sets P, Q and U are defined as
U = {Real Numbers}, P = {Positive Numbers} and Q = {Rational Numbers}.
Write down in the correct region on the Venn diagram the numbers
22
,
7
5 × 10–2
,
sin(60°)
,
0
,
3
8
,
–π
(Total 6 marks)
16.
Astrid invests 1200 euros for five years at a nominal annual interest rate of 7.2 %, compounded
monthly.
(a)
Find the interest Astrid has earned during the five years of her investment.
Give your answer correct to two decimal places.
(3)
IB Questionbank Mathematical Studies 3rd edition
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Helen invests 1200 euros in an annual simple interest scheme for five years.
She earns the same interest as Astrid.
(b)
Find the simple interest rate of this scheme.
(3)
(Total 6 marks)
17.
Daniel wants to invest $25 000 for a total of three years. There are three investment options.
Option One
pays simple interest at an annual rate of interest of 6 %.
Option Two
pays compound interest at a nominal annual rate of interest of 5 %,
compounded annually.
Option Three
pays compound interest at a nominal annual rate of interest of 4.8 %,
compounded monthly.
(a)
Calculate the value of his investment at the end of the third year for each investment
option, correct to two decimal places.
(8)
(b)
Determine Daniel’s best investment option.
(1)
(Total 9 marks)
18.
Give all answers in this question to the nearest whole currency unit.
Ying and Ruby each have 5000 USD to invest.
Ying invests his 5000 USD in a bank account that pays a nominal annual interest rate of 4.2 %
compounded yearly. Ruby invests her 5000 USD in an account that offers a fixed interest of
230 USD each year.
(a)
Find the amount of money that Ruby will have in the bank after 3 years.
(2)
(b)
Show that Ying will have 7545 USD in the bank at the end of 10 years.
(3)
IB Questionbank Mathematical Studies 3rd edition
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(c)
Find the number of complete years it will take for Ying’s investment to first exceed 6500
USD.
(3)
(d)
Find the number of complete years it will take for Ying’s investment to exceed Ruby’s
investment.
(3)
Ruby moves from the USA to Italy. She transfers 6610 USD into an Italian bank which has an
exchange rate of 1 USD = 0.735 euros. The bank charges 1.8 % commission.
(e)
Calculate the amount of money Ruby will invest in the Italian bank after commission.
(4)
Ruby returns to the USA for a short holiday. She converts 800 euros at a bank in Chicago and
receives 1006.20 USD. The bank advertises an exchange rate of 1 euro = 1.29 USD.
(f)
Calculate the percentage commission Ruby is charged by the bank.
(5)
(Total 20 marks)
19.
The first term of an arithmetic sequence is 3 and the sum of the first two terms is 11.
(a)
Write down the second term of this sequence.
(1)
(b)
Write down the common difference of this sequence.
(1)
(c)
Write down the fourth term of this sequence.
(1)
(d)
The nth term is the first term in this sequence greater than 1000.
Find the value of n.
(3)
(Total 6 marks)
IB Questionbank Mathematical Studies 3rd edition
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20.
A store sells bread and milk. On Tuesday, 8 loaves of bread and 5 litres of milk were sold for
$21.40. On Thursday, 6 loaves of bread and 9 litres of milk were sold for $23.40.
If b = the price of a loaf of bread and m = the price of one litre of milk, Tuesday’s sales can be
written as 8b + 5m = 21.40.
(a)
Using simplest terms, write an equation in b and m for Thursday’s sales.
(b)
Find b and m.
(c)
Draw a sketch, in the space provided, to show how the prices can be found graphically.
5
4
m
3
2
1
0
0
1
2
3
4
b
(Total 6 marks)
21.
Jacques can buy six CDs and three video cassettes for $163.17
or he can buy nine CDs and two video cassettes for $200.53.
(a)
Express the above information using two equations relating the price of CDs and the price
of video cassettes.
(b)
Find the price of one video cassette.
(c)
If Jacques has $180 to spend, find the exact amount of change he will receive if he buys
nine CDs.
(Total 6 marks)
IB Questionbank Mathematical Studies 3rd edition
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22.
A swimming pool is to be built in the shape of a letter L. The shape is formed from two squares
with side dimensions x and
x as shown.
x
x
Diagram not to scale
x
x
(a)
Write down an expression for the area A of the swimming pool surface.
(b)
The area A is to be 30 m2. Write a quadratic equation that expresses this information.
(c)
Find both the solutions of your equation in part (b).
(d)
Which of the solutions in part (c) is the correct value of x for the pool? State briefly why
you made this choice.
(Total 8 marks)
23.
A small manufacturing company makes and sells x machines each month. The monthly cost C,
in dollars, of making x machines is given by
C(x) = 2600 + 0.4x2.
The monthly income I, in dollars, obtained by selling x machines is given by
I(x) = 150x – 0.6x2.
(a)
Show that the company’s monthly profit can be calculated using the quadratic function
P(x) = – x2 + 150x – 2600.
(2)
IB Questionbank Mathematical Studies 3rd edition
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(b)
The maximum profit occurs at the vertex of the function P(x). How many machines
should be made and sold each month for a maximum profit?
(2)
(c)
If the company does maximize profit, what is the selling price of each machine?
(4)
(d)
Given that P(x) = (x – 20) (130 – x), find the smallest number of machines the company
must make and sell each month in order to make positive profit.
(4)
(Total 12 marks)
24.
(a)
Factorize the expression x2 − 25.
(b)
Factorize the expression x2 – 3x – 4.
(c)
Using your answer to part (b), or otherwise, solve the equation x2 – 3x – 4 = 0.
(Total 8 marks)
IB Questionbank Mathematical Studies 3rd edition
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