New test - April 28, 2023
[209
marks]
The following diagram shows a park bounded by a fence in the shape of a
quadrilateral ABCD. A straight path crosses through the park from B to D.
AB = 85 m, AD = 85 m, BC = 40 m, CB̂D = 41°, BĈD = 120°
1a. Write down the value of angle BDC.
BDC
[1 mark]
BD
1b. Hence use triangle
BDC to find the length of path BD.
1c. Calculate the size of angle BÂD, correct to five significant figures.
ˆ
[3 marks]
[3 marks]
The size of angle BÂD rounds to 77°, correct to the nearest degree. Use
BÂD = 77° for the rest of this question.
1d. Find the area bounded by the path BD, and fences
AB and AD.
BC
[3 marks]
CD
A landscaping firm proposes a new design for the park. Fences BC and CD are to
be replaced by a fence in the shape of a circular arc BED with center A. This is
illustrated in the following diagram.
1e. Write down the distance from
A to E.
1f. Find the perimeter of the proposed park, ABED.
[1 mark]
[3 marks]
1g. Find the area of the shaded region in the proposed park.
[3 marks]
Eddie decides to construct a path across his rectangular grass lawn using pairs of
tiles.
Each tile is 10 cm wide and 20 cm long. The following diagrams show the path
after Eddie has laid one pair and three pairs of tiles. This pattern continues until
Eddie reaches the other side of his lawn. When n pairs of tiles are laid, the path
has a width of wn centimetres and a length ln centimetres.
The following diagrams show this pattern for one pair of tiles and for three pairs of
tiles, where the white space around each diagram represents Eddie’s lawn.
The following table shows the values of
wn and ln for the first three values of n.
Find the value of
2a.
a.
[1 mark]
[1 mark]
2b. b.
Write down an expression in terms of
2c.
n for
[2 marks]
wn .
[1 mark]
2d. ln .
740 cm
Eddie’s lawn has a length 740
cm .
2e. Show that Eddie needs 144 tiles.
2f. Find the value of wn for this path.
2g. Find the total area of the tiles in Eddie’s path. Give your answer in the
form a × 10k where 1 ≤ a < 10 and k is an integer.
[2 marks]
[1 mark]
[3 marks]
The tiles cost $24. 50 per square metre and are sold in packs of five tiles.
[3 marks]
2h. Find the cost of a single pack of five tiles.
To allow for breakages Eddie wants to have at least
8% more tiles than he needs.
2i. Find the minimum number of packs of tiles Eddie will need to order.
$35
[3 marks]
There is a fixed delivery cost of $35.
[2 marks]
2j. Find the total cost for Eddie’s order.
O
4. 5 m
A sector of a circle, centre O and radius 4. 5
3a. Find the angle AÔB.
m, is shown in the following diagram.
[3 marks]
3b. Find the area of the shaded segment.
8m
[5 marks]
A square field with side 8 m has a goat tied to a post in the centre by a rope such
that the goat can reach all parts of the field up to 4. 5 m from the post.
[Source: mynamepong, n.d. Goat [image online] Available at: https://thenounproject.com/term/goat/1761571/
This file is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)
https://creativecommons.org/licenses/by-sa/3.0/deed.en [Accessed 22 April 2010] Source adapted.]
3c. Find the area of a circle with radius 4. 5
m.
[2 marks]
3d. Find the area of the field that can be reached by the goat.
[3 marks]
Let V be the volume of grass eaten by the goat, in cubic metres, and t be the
length of time, in hours, that the goat has been in the field.
The goat eats grass at the rate of ddV
t
= 0. 3 te−t .
3e. Find the value of t at which the goat is eating grass at the greatest rate. [2 marks]
A large underground tank is constructed at Mills Airport to store fuel. The tank is
in the shape of an isosceles trapezoidal prism, ABCDEFGH.
AB = 70 m , AF = 200 m, AD = 40 m, BC = 40 m and CD = 110 m. Angle
ADC = 60° and angle BCD = 60°. The tank is illustrated below.
4a. Find
[2 marks]
h, the height of the tank.
3
4b. Show that the volume of the tank is 624
significant figures.
000 m3 , correct to three
[3 marks]
Once construction was complete, a fuel pump was used to pump fuel into the
empty tank. The amount of fuel pumped into the tank by this pump each hour
decreases as an arithmetic sequence with terms u1 , u2 , u3 , … , un .
Part of this sequence is shown in the table.
[1 mark]
4c. Write down the common difference, d.
13th
4d. Find the amount of fuel pumped into the tank in the 13th hour.
4e. Find the value of n such that un
= 0.
[2 marks]
[2 marks]
4f. Write down the number of hours that the pump was pumping fuel into the [1 mark]
tank.
3
At the end of the 2nd hour, the total volume of fuel in the tank was 88
200 m3 .
4g. Find the total amount of fuel pumped into the tank in the first 8 hours.
[2 marks]
4h. Show that the tank will never be completely filled using this pump.
[3 marks]
5. Complete the following table by placing ticks (✓) to show which of the
[6 marks]
number sets N, Z, Q , and R these numbers belong to. The first row has
been completed as an example.
6a. Place the numbers
Venn diagram.
3
2π, − 5, 3−1 and 2 2 in the correct position on the
[4 marks]
6b. In the table indicate which two of the given statements are true by
placing a tick (✔) in the right hand column.
[2 marks]
Consider the following sets:
The universal set U consists of all positive integers less than 15;
A is the set of all numbers which are multiples of 3;
B is the set of all even numbers.
7a. Write down the elements that belong to A ∩ B.
∩
′
[3 marks]
7b. Write down the elements that belong to A ∩ B′ .
7c. Write down
n (A ∩ B′ ).
[2 marks]
[1 mark]
Little Green island originally had no turtles. After 55 turtles were introduced to the
island, their population is modelled by
N (t) = a × 2−t + 10, t ⩾ 0,
where a is a constant and
8a. Find the value of a .
t is the time in years since the turtles were introduced.
[2 marks]
8b. Find the time, in years, for the population to decrease to 20 turtles.
[2 marks]
8c. There is a number m beyond which the turtle population will not
decrease.
[2 marks]
Find the value of m . Justify your answer.
In this question, give all answers to two decimal places.
Velina travels from New York to Copenhagen with 1200 US dollars (USD). She
exchanges her money to Danish kroner (DKK). The exchange rate is 1 USD =
7.0208 DKK.
9a. Calculate the amount that Velina receives in DKK.
[2 marks]
At the end of her trip Velina has 3450 DKK left that she exchanges to USD. The
bank charges a 5 % commission. The exchange rate is still 1 USD = 7.0208 DKK .
9b. Calculate the amount, in DKK, that will be left to exchange after
commission.
[2 marks]
9c. Hence, calculate the amount of USD she receives.
[2 marks]
John purchases a new bicycle for 880 US dollars (USD) and pays for it with a
Canadian credit card. There is a transaction fee of 4.2 % charged to John by the
credit card company to convert this purchase into Canadian dollars (CAD).
The exchange rate is 1 USD = 1.25 CAD.
10a. Calculate, in CAD, the total amount John pays for the bicycle.
[3 marks]
John insures his bicycle with a US company. The insurance company produces the
following table for the bicycle’s value during each year.
The values of the bicycle form a geometric sequence.
10b. Find the value of the bicycle during the 5th year. Give your answer to [3 marks]
two decimal places.
10c. Calculate, in years, when the bicycle value will be less than 50 USD.
[2 marks]
During the 1st year John pays 120 USD to insure his bicycle. Each year the
amount he pays to insure his bicycle is reduced by 3.50 USD.
10d. Find the total amount John has paid to insure his bicycle for the first 5
years.
[3 marks]
An archaeological site is to be made accessible for viewing by the public. To do
this, archaeologists built two straight paths from point A to point B and from point
B to point C as shown in the following diagram. The length of path AB is 185 m,
∧
the length of path BC is 250 m, and angle A B C is 125°.
The archaeologists plan to build two more straight paths, AD and DC. For the
∧
paths to go around the site, angle B A D is to be made equal to 85° and angle
∧
B C D is to be made equal to 70° as shown in the following diagram.
11a.
Find the size of angle
∧
C A D.
∧
[1 mark]
11b.
Find the size of angle
∧
A C D.
[2 marks]
A factory packages coconut water in cone-shaped containers with a base radius of
5.2 cm and a height of 13 cm.
12a. Find the slant height of the cone-shaped container.
[2 marks]
12b. Find the slant height of the cone-shaped container.
[2 marks]
12c. Show that the total surface area of the cone-shaped container is 314
cm2, correct to three significant figures.
[3 marks]
The factory designers are currently investigating whether a cone-shaped container
can be replaced with a cylinder-shaped container with the same radius and the
same total surface area.
12d. Find the height, h , of this cylinder-shaped container.
[4 marks]
12e. The factory director wants to increase the volume of coconut water sold [4 marks]
per container.
State whether or not they should replace the cone-shaped containers with
cylinder‑shaped containers. Justify your conclusion.
Harry travelled from the USA to Mexico and changed 700 dollars (USD) into pesos
(MXN).
The exchange rate was 1 USD = 18.86 MXN.
13a. Calculate the amount of MXN Harry received.
[2 marks]
On his return, Harry had 2400 MXN to change back into USD.
There was a 3.5 % commission to be paid on the exchange.
13b. Calculate the value of the commission, in MXN, that Harry paid.
[2 marks]
13c. The exchange rate for this exchange was 1 USD = 17.24 MXN.
[2 marks]
Calculate the amount of USD Harry received. Give your answer correct to the
nearest cent.
A solid glass paperweight consists of a hemisphere of diameter 6 cm on top of a
cuboid with a square base of length 6 cm, as shown in the diagram.
The height of the cuboid, x cm, is equal to the height of the hemisphere.
14a. Write down the value of x.
[1 mark]
14b. Calculate the volume of the paperweight.
[3 marks]
14c. 1 cm3 of glass has a mass of 2.56 grams.
[2 marks]
Calculate the mass, in grams, of the paperweight.
The marks obtained by nine Mathematical Studies SL students in their projects (x)
and their final IB examination scores ( y) were recorded. These data were used to
determine whether the project mark is a good predictor of the examination score.
The results are shown in the table.
15a. Use your graphic display calculator to write down ȳ , the mean
examination score.
[1 mark]
15b. Use your graphic display calculator to write down r , Pearson’s product– [2 marks]
moment correlation coefficient.
The equation of the regression line y on x is y = mx + c.
15c. Find the exact value of m and of c for these data.
[2 marks]
A tenth student, Jerome, obtained a project mark of 17.
15d. Use the regression line y on x to estimate Jerome’s examination score. [2 marks]
15e. Justify whether it is valid to use the regression line y on x to estimate
Jerome’s examination score.
[2 marks]
A solid right circular cone has a base radius of 21 cm and a slant height of 35 cm.
A smaller right circular cone has a height of 12 cm and a slant height of 15 cm,
and is removed from the top of the larger cone, as shown in the diagram.
16. Calculate the radius of the base of the cone which has been removed.
The following table shows four different sets of numbers:
[2 marks]
N, Z, Q and R.
17a. Complete the second column of the table by giving one example of a
number from each set.
[4 marks]
17b. Josh states: “Every integer is a natural number”.
[2 marks]
Write down whether Josh’s statement is correct. Justify your answer.
In this question, give all answers to two decimal places.
Karl invests 1000 US dollars (USD) in an account that pays a nominal annual
interest of 3.5%, compounded quarterly. He leaves the money in the account
for 5 years.
18a. Calculate the amount of money he has in the account after 5 years.
18b. Write down the amount of interest he earned after 5 years.
[3 marks]
[1 mark]
18c. Karl decides to donate this interest to a charity in France. The charity [2 marks]
receives 170 euros (EUR). The exchange rate is 1 USD = t EUR.
Calculate the value of t.
Give your answers to parts (b), (c) and (d) to the nearest whole number.
Harinder has 14 000 US Dollars (USD) to invest for a period of five years. He has
two options of how to invest the money.
Option A: Invest the full amount, in USD, in a fixed deposit account in an
American bank.
The account pays a nominal annual interest rate of r % , compounded yearly, for
the five years. The bank manager says that this will give Harinder a return of
17 500 USD.
19a. Calculate the value of r.
[3 marks]
Option B: Invest the full amount, in Indian Rupees (INR), in a fixed deposit
account in an Indian bank. The money must be converted from USD to INR before
it is invested.
The exchange rate is 1 USD = 66.91 INR.
19b. Calculate 14 000 USD in INR.
[2 marks]
The account in the Indian bank pays a nominal annual interest rate of 5.2 %
compounded monthly.
19c. Calculate the amount of this investment, in INR, in this account after
five years.
[3 marks]
19d. Harinder chose option B. At the end of five years, Harinder converted
this investment back to USD. The exchange rate, at that time, was 1
USD = 67.16 INR.
[3 marks]
Calculate how much more money, in USD, Harinder earned by choosing option B
instead of option A.
The Tower of Pisa is well known worldwide for how it leans.
Giovanni visits the Tower and wants to investigate how much it is leaning. He
draws a diagram showing a non-right triangle, ABC.
On Giovanni’s diagram the length of AB is 56 m, the length of BC is 37 m, and
angle ACB is 60°. AX is the perpendicular height from A to BC.
20a. Use Giovanni’s diagram to show that angle ABC, the angle at which the [5 marks]
Tower is leaning relative to the
horizontal, is 85° to the nearest degree.
20b. Use Giovanni's diagram to calculate the length of AX.
[2 marks]
20c. Use Giovanni's diagram to find the length of BX, the horizontal
displacement of the Tower.
[2 marks]
Giovanni’s tourist guidebook says that the actual horizontal displacement of the
Tower, BX, is 3.9 metres.
20d. Find the percentage error on Giovanni’s diagram.
[2 marks]
20e. Giovanni adds a point D to his diagram, such that BD = 45 m, and
another triangle is formed.
Find the angle of elevation of A from D.
[3 marks]
In this question give all answers correct to two decimal places.
Javier takes 5000 US dollars (USD) on a business trip to Venezuela. He exchanges
3000 USD into Venezuelan bolívars (VEF).
The exchange rate is 1 USD = 6.3021 VEF.
21a. Calculate the amount of VEF that Javier receives.
[2 marks]
During his time in Venezuela, Javier spends 1250 USD and 12 000 VEF. On his
return home, Javier exchanges his remaining VEF into USD.
The exchange rate is 1 USD = 8.7268 VEF.
21b. Calculate the total amount, in USD, that Javier has remaining from his
5000 USD after his trip to Venezuela.
[4 marks]
A water container is made in the shape of a cylinder with internal height
internal base radius r cm.
h cm and
The water container has no top. The inner surfaces of the container are to be
coated with a water-resistant material.
22a. Write down a formula for
A, the surface area to be coated.
The volume of the water container is
22b. Express this volume in
cm3 .
[2 marks]
0.5m3 .
[1 mark]
22c. Write down, in terms of
container.
22d. Show that A
= πr2 +
r and h, an equation for the volume of this water [1 mark]
1 000 000
.
r
[2 marks]
The water container is designed so that the area to be coated is minimized.
[3 marks]
22e. Find dA .
d
r
22f. Using your answer to part (e), find the value of r which minimizes
A.
[3 marks]
[2 marks]
22g. Find the value of this minimum area.
One can of water-resistant material coats a surface area of 2000cm 2 .
22h. Find the least number of cans of water-resistant material that will coat [3 marks]
the area in part (g).
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