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NSAA 2018 Mathematics S1

18 questions18 marksUpdated June 2026

The NSAA 2018 Mathematics S1 paper in full: all 18 questions, each with its answer. NSAA is the Natural Sciences Admissions Assessment. Sit it cold under exam timing, mark it, then work back through anything you missed using the solutions below.

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Question 1

A group of drivers, consisting of 200 women and 300 men, was asked if they passed their driving test at the first attempt.

Altogether 167 of the group said they passed at the first attempt.

Of the women, 143 said they did not pass at the first attempt.

How many of the men said they passed at the first attempt?

A.10
B.24
C.33
D.57
E.110
F.133
G.157

Answer: E

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Question 2

A cuboid has sides of length

x

,

\sqrt{2}x

and

2x

, measured in cm.

The volume, in cm

^3

, of the cuboid is numerically equal to twice the total surface area, in cm

^2

, of the cuboid.

What is the value of

x

?

A.10
B. $6 + 2\sqrt{2}$
C.5
D. $3 + \sqrt{2}$
E. $\frac{5}{2}$
F. $\frac{3}{2} + \frac{1}{2}\sqrt{2}$

Answer: B

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Question 3

The line joining the points with coordinates

(p, p - 1)

and

(1 - p, 2p)

is parallel to the line with equation

2x + 3y + 1 = 0

What is the value of

p

?

A.-1
B. $-\frac{1}{7}$
C. $\frac{1}{9}$
D. $\frac{1}{8}$
E.1
F. $\frac{5}{4}$
G.2
H.5

Answer: H

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Question 4

A rectangle PQRS is drawn inside a circle, with its vertices on the circumference of the circle.

Exam diagram

[diagram not to scale]

The ratio of the length of

PQ

to the length of

QR

is

2:1

The area of the rectangle

PQRS

is 96 cm

^2

.

What is the radius, in cm, of the circle?

A. $\sqrt{6}$
B.3
C. $3\sqrt{2}$
D. $2\sqrt{15}$
E. $4\sqrt{6}$
F.12
G. $12\sqrt{2}$
H. $8\sqrt{15}$

Answer: D

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Question 5

The expected number of bottles of water sold in a day at a sports ground is directly proportional to the square of the average outside temperature, in degrees Celsius, for that day.

On a day when the average outside temperature is 16°C, 64 bottles of water, the expected number, are sold.

On a warmer day, when the average outside temperature is T°C, 256 bottles of water are sold, which is 31 bottles more than the expected number for that day.

What is the value of

T

?

A.7.5
B. $\sqrt{450}$
C.30
D.32
E. $\sqrt{1148}$
F.56.25

Answer: C

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Question 6

Exam diagram

Exam diagram

[diagram not to scale]

At a cinema, drinks are sold in regular and large sizes.

The cups for these are mathematically similar.

The ratio of the heights of the cups and the ratio of the depths of the drinks are both

4:5

The volume of drink in a regular size cup is 320 cm

^3

.

What is the volume, in cm

^3

, of drink in a large size cup?

A.384
B.400
C.500
D.576
E.625
F.640

Answer: E

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Question 7

The mean of

n

numbers is

p

The mean of two of these numbers is

q

The mean of the remaining numbers is 10

Which of the following is a correct expression for

n

in terms of

p

and

q

?

A. $\frac{2(q-10)}{(p-10)}$
B. $\frac{2(q-10)}{(10-p)}$
C. $\frac{2(q-10)}{(p+10)}$
D. $\frac{2(10-q)}{(p+10)}$
E. $\frac{2(10+q)}{(p-10)}$
F. $\frac{2(10+q)}{(10-p)}$

Answer: A

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Question 8

The straight lines

5x + 2y = 20

y = 3x - 23

x = 0

enclose a region with area

K

square units.

What is the value of

K

?

A.39
B.78
C.99
D.129
E.198
F.258

Answer: C

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Question 9

A scale model of a cylindrical pillar is to be made.

The full-sized pillar has a volume of

12\pi

m

^3

.

The model will use a length scale of

1:40

The model is to be a solid cylinder made of a plastic which has a density of

\frac{4}{3}

g cm

^{-3}

.

What is the mass of the model in grams?

A. $\frac{9}{640}\pi$
B. $\frac{1}{40}\pi$
C. $40\pi$
D. $\frac{1125}{8}\pi$
E. $250\pi$
F. $10000\pi$
G. $225000\pi$
H. $400000\pi$

Answer: E

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Question 10

Exam diagram

[diagram not to scale]

PQRST

is a regular pentagon.

RSU

is an equilateral triangle.

What is the size of angle

STU

?

A.48°
B.54°
C.60°
D.66°
E.84°

Answer: D

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Question 11

The original price of an item is

p

The price is increased by 125%

The increased price is then decreased by 40% to

q

The relationship between

p

and

q

can be expressed as

mp = q

What is the value of

m

?

A. $\frac{7}{20}$
B. $\frac{17}{20}$
C. $\frac{27}{20}$
D. $\frac{33}{20}$
E. $\frac{37}{20}$

Answer: C

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Question 12

80% of a number is equal to two-thirds of a second number.

The whole number ratio of the first number to the second number in its lowest terms is

x:y

What is the value of

x - y

?

A.7
B.2
C.1
D.0.2
E.-0.2
F.-1
G.-2
H.-7

Answer: F

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Question 13

Q

is 5 km away from

P

on a bearing of 065°

R

is 5 km away from

Q

on a bearing of 155°

What is the bearing of

P

from

R

?

A.070°
B.110°
C.225°
D.270°
E.290°
F.315°
G.335°

Answer: E

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Question 14

With school lunch, students can select tomato sauce, or mayonnaise, or both, or neither.

n

students selected both.

3n + 1

students selected tomato sauce.

3n - 1

students selected only mayonnaise.

There were

7n + 5

students in the group.

The probability of a student, chosen at random, selecting only mayonnaise is

\frac{1}{3}

By finding

n

, what is the probability of a student, chosen at random, selecting only tomato sauce?

A. $\frac{3}{11}$
B. $\frac{7}{26}$
C. $\frac{13}{33}$
D. $\frac{3}{8}$
E. $\frac{7}{13}$

Answer: A

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Question 15

Exam diagram

[diagram not to scale]

The line segment

RT

is a tangent at the point

S

to a circle with centre

O

Q

and

P

are points on the circumference of the circle such that

QS = QP

Angle

PST = 75°

What is the size of angle

QSO

?

A.15°
B.30°
C.37.5°
D.45°
E.52.5°
F.60°
G.67.5°
H.75°

Answer: C

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Question 16

Exam diagram

[diagram not to scale]

The vertical height

h

cm of an isosceles triangle is 3 cm longer than the base length of

b

cm.

The sloping side is of length

s

cm.

The area of the triangle is 14 cm

^2

.

There is one value of

s

which satisfies these conditions.

Within which range does this value of

s

lie?

A. $5 < s < 6$
B. $6 < s < 7$
C. $7 < s < 8$
D. $8 < s < 9$
E. $9 < s < 10$
F. $10 < s < 11$

Answer: C

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Question 17

The first five terms of a sequence in order are:

2 17 42 77 122

The

n^{th}

term of this sequence is

pn^2 + q

where

p

and

q

are integers.

What is the value of

\frac{p-q}{p+q}

?

A. $\frac{1}{4}$
B. $\frac{1}{2}$
C.1
D. $\frac{23}{17}$
E. $\frac{13}{7}$
F.2
G.4
H.14

Answer: G

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Question 18

A bag contains 6 red and 6 green sweets. The sweets are identical apart from their colour.

A child takes a sweet at random from the bag.

If the sweet is red, the child stops taking sweets.

If the sweet is green, it is not replaced and the child takes another sweet.

This continues until a red sweet is taken at which point the child stops taking sweets.

What is the probability that the child takes more green sweets than red sweets?

A. $\frac{3}{22}$
B. $\frac{5}{22}$
C. $\frac{3}{11}$
D. $\frac{1}{2}$
E. $\frac{8}{11}$
F. $\frac{17}{22}$

Answer: B

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