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Practice
Questions
international competitions and assessments for schools
ICAS Mathematics Practice Questions Paper I © EAA 2
1. Terry is in Station Rd and is going to a party in West St, which runs parallel to Station Rd. The angles between some of the streets are shown.
West St
Smith Rd
Oak
St
Station Rd
z°
x° w°
w°
y°
y°
(A) w = y
(B) x = w
(C) y = x
(D) z = y
West St
Station Rd
Smith Rd
NOT TO SCALE
Which of these statements must be true?
West St
Smith Rd
Oak
St
Station Rd
z°
x° w°
w°
y°
y°
(A) w = y
(B) x = w
(C) y = x
(D) z = y
West St
Station Rd
Smith Rd
NOT TO SCALE
2. A company uses this formula to predict total profit P based on the number of products n sold.
P = n2 + 60n – 4000
How many products are sold if there is zero profit?
(A) 0(B) 40(C) 100(D) 4000
3.
7.101 (3.019 – 0.798)–
What is the value of this expression correct to three significant figures?
(A) 3.19(B) 3.197(C) 3.20(D) 3.200
4. This scatter diagram shows the relationship between the air temperature T and the number of people P visiting a beachside shopping centre.
Num
ber o
f peo
ple
(P)
Air temperature (T )
(A) P = 5T 2
(B) P = – 5T
(C) P = –
(D) P = —5T
—T5
Which formula could describe the relationship between the air temperature and the number of people?
Num
ber o
f peo
ple
(P)
Air temperature (T )
(A) P = 5T 2
(B) P = – 5T
(C) P = –
(D) P = —5T
—T5
3 ICAS Mathematics Practice Questions Paper I © EAA
5. This picture is based on the style of the Dutch artist Piet Mondrian (1872–1944).
2
NOT TO SCALE
26
6
4
4
Which expression gives the total area of the three coloured rectangles in the picture?
(A) 6a2 + 48ab – 32b2(B) 20a2 + 24b2(C) 4a2 + 36ab – 24b2(D) 6a2 + 16b2
6. A ship leaves position X and travels north-west for 1000 km to position Y, as shown.
Y
X
N
1000 km
NOT TO SCALE
This ship then travels due south.
How far south, to the nearest km, does the ship have to travel before it is at a position bearing south-west from X?
(A) 2000(B) 1414(C) 1000(D) 707
7. A farmer purchases water released from a dam. There are two pricing schemes and these depend on the flow rate of water (in litres per second).
Pricing Flow rate Cost scheme (L/s) ($/1000 L)
1 6 000 $23.50
2 10 000 $35.60
What is the difference, in whole dollars per second, between pricing schemes 1 and 2?
(A) 12(B) 105(C) 215(D) 497
8. This diagram shows a cubic section of a sodium chloride crystal. Each green sphere represents a chloride ion and each yellow sphere represents a sodium ion.
The distance between the centre of a chloride ion and a sodium ion is as shown here.
What is the approximate volume, in m3, of the prism whose vertices are the centres of the eight ions that form the corners of this crystal section?
(A) 1.4 × 10−15
(B) 2.1 × 10−17
(C) 2.7 × 10−15
(D) 8.6 × 10−17
ICAS Mathematics Practice Questions Paper I © EAA 4
9. This square poster shows a circular star chart. The line PR is a tangent to the circle at Q, such that PQ = 20 cm and QR = 6 cm.
STAR
CHART
Canis Major
Canis Minor
Eridanus
Puppis
Hydra
Volans
Dorado
Sirius
Procyon
20 cm
P
RQ6
cm
A
P
B CRQ
NOT TO SCALE
r
r
rr
What is the width of this poster, in cm?
10. Mario knows that a number is divisible by nine if the sum of its digits is divisible by nine.
He has eight cards with the digits 1 to 8 written on them as shown.
Mario selects three of these cards to make a three-digit number that is divisible by nine. He then replaces these three cards and repeats this selection procedure to select different three-digit numbers divisible by nine.
How many even three-digit numbers is it possible for him to find in this way?
END OF PAPER
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5 ICAS Mathematics Practice Questions Paper I © EAA
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M THE UNIVERSITY OF NEW SOUTH WALES
International Competit ions and Assessments for Schools
PRACTICE QUESTIO
NS
*045911*
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DCBA
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International Competit ions and Assessments for SchoolsM
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ICAS Mathematics Practice Questions Paper I © EAA
QUESTION KEY SOLUTION STRAND LEVEL OF DIFFICULTY
1 A
West St
Smith Rd
Oak
St
Station Rd
z°
x° w°
w°
y°
y°
(A) w = y
(B) x = w
(C) y = x
(D) z = y
West St
Station Rd
Smith Rd
NOT TO SCALE
Station Rd and West St are parallel, while Smith Rd crosses them. This results in angle y and angle w, that are alternate, being equal.Therefore, statement A is the correct statement.
Space and Geometry Easy
2 B
This is a quadratic equation. It factorises to (n + 100)(n − 40) = 0The solutions for this equation are n = − 100 and n = 40.As n is the number of products, it cannot be negative. Hence, n = 40 is the correct solution.Alternatively, substituting the options will show that n = 40 gives P = 0.
Algebra and Patterns Easy
3 CThe result of the calculation is 3.197208465.This number rounded to three significant figures is 3.20.
Number and Arithmetic Medium
4 D
The diagram shows an inverse relation between the air temperature, T, and the number of people, P. As T increases, P decreases. Note that the relation is not linear.Option A is a quadratic equation that gives a parabola when graphed, where the relation is positive (considering positive values of T). This does not describe the given data. Options B and C are both linear equations that give straight lines sloping downwards when graphed. Again, these do not describe the given data.Option D is an equation that gives a hyperbola when graphed. For small values of T, P has a large value. As the values of T increase, the values of P decrease. This correctly describes the given data.
Chance and Data Medium
ICAS Mathematics Practice Questions Paper I © EAA
5 A
The green area is 2a × a = 2a2The orange area is 4a × a = 4a2The blue area is = length × width = (6b + 2b) × (6a – 4b)= 8b × (6a – 4b)= 48ab – 32b2Therefore the total area that is coloured is = green + orange + blue= 2a2 + 4a2 + 48ab – 32b2 = 6a2 + 48ab – 32b2
Algebra and Patterns Medium
6 B
1000
1000
XW
Z
Y N
45°
45°
45°
45°
To be south-west from X and south from Y, the boat must be at position Z, as shown. Notice that Triangle XYZ has two 45 degrees angles which means it has to be an isosceles triangle. Hence, the distance from X to Z must also be 1000 km.
Using Pythagoras’ theorem in Triangle XYZ the distance that the boat must travel south is YZ = √10002 + 10002 = 1414.21356 = 1414 rounded down to the nearest km.
Space and Geometry Medium
7 C
Pricing scheme 1 has 6000 L flowing per second, at $23.50 per 1000 litres. So the cost is($23.50 ÷ 1000) × 6000 = $141 per second.Similarly, pricing scheme two is ($35.60 ÷ 1000) × 10000 = $356 per second.The difference is therefore 356 − 141 = $215 per second.
Number and Arithmetic Medium
8 A
To solve this question, a cube that is 4 × 4 × 4 units, with each unit being 2.78 × 10–6 m should be considered.
That is, the cube with side lengths
4 × 2.78 × 10–6
= 1.11200 × 10–5
Hence, the volume of the cube will be(1.11200 × 10−5)3 =1.4 × 10−15 (rounded up).
Measurement Medium/Hard
ICAS Mathematics Practice Questions Paper I © EAA
9 60
STAR
CHART
Canis Major
Canis Minor
Eridanus
Puppis
Hydra
Volans
Dorado
Sirius
Procyon
20 cm
P
RQ6
cm
A
P
B CRQ
NOT TO SCALE
r
r
rr
It is given that PQ = 20 and QR = 6AP = PQ = 20 (tangents to the circle from P are equal)Similarly QR = RC = 6.
Let r be the radius of the circle. Then,r = AP + PB = PB + 20so PB = r − 20Similarly, BR = r − 6
Using Pythagoras’ Theorem on triangle PBR,
(r − 20)2 + (r − 6)2 = 262
2r2 − 52r + 436 = 6762r2 − 52r − 240 = 0r2 − 26r − 120 = 0(r − 30)(r + 4) = 0
As r is the radius, it cannot be negative. Hence, r = 30.
Therefore, the width of the poster, the diameter, is 60 cm.
Measurement Hard
ICAS Mathematics Practice Questions Paper I © EAA
10 18
Numbers to be considered are numbers with a digit sum that is divisible by 9. So the sum of the digits must be multiples of 9: 9, 18, 27...The highest digit sum that can be obtained from the numbers 1 to 8 is 8 + 7 + 6 = 21. So only numbers whose digits sum to 9 or 18 need to be considered.
The numbers must be even, so they must be of the form: _ _ 2, _ _ 4, _ _ 6 and _ _ 8.
Take for example _ _ 2. To make this number’s digits sum to 9, the first two digits must sum to 7. We can therefore have 342, or 432. We cannot have 252 or 522 as the number 2 cannot be used twice.
This table summarises the solutions.
Possible numbers
Sum to 9
Solutions
Sum to 18
SolutionsFirst two digits sum to:
First two digits sum to:
_ _ 2 7
432
16 -342
162
612
_ _ 4 5234
14684
324 864
_ _ 6 3
126
12
486
846
216 756
576
_ _ 8 1 - 10
468
648
738
378
Therefore there are 18 possible numbers that Mario can find.Note: This is one possible method. The question can be solved using other methods.
Chance and Data Hard
ICAS Mathematics Practice Questions Paper I © EAA
Level of difficulty refers to the expected level of difficulty for the question.
Easy more than 70% of candidates will choose the correct option
Medium about 50–70% of candidates will choose the correct option
Medium/Hard about 30–50% of candidates will choose the correct option
Hard less than 30% of candidates will choose the correct option
