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NSAA 2017 Mathematics PART A

18 questions18 marksUpdated October 2025

The NSAA 2017 Mathematics PART A 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

Evaluate

\frac{(\sqrt{12} + \sqrt{3})^2}{(\sqrt{12}-\sqrt{3})^2}

A.1
B.3
C. $\frac{5}{3}$
D. $\frac{7}{3}$
E. $3\sqrt{3}$
F.9

Answer: F

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Question 2

Solve fully the inequality

2x^2 \geq 15-x

A. $x \leq -3$
B. $x \geq 2.5$
C. $x \leq -1.5, x \geq 5$
D. $-1.5 \leq x \leq 5$
E. $x \leq -3, x \geq 2.5$
F. $-3 \leq x \leq 2.5$

Answer: E

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Question 3

The equation gives y in terms of x:

y=3(\frac{x}{2}-1)^2 -5

Which one of the following is a rearrangement for x in terms of y?

A. $x=2 \pm 2 \sqrt{\frac{y-5}{3}}$
B. $x=2 \pm 2 \sqrt{\frac{y+5}{3}}$
C. $x=2 \pm 3 \sqrt{\frac{y+5}{3}}$
D. $x=-2 \pm 2 \sqrt{\frac{y+5}{3}}$
E. $x=-2 \pm 3 \sqrt{\frac{y+5}{2}}$
F. $x=2+2(\frac{y+5}{3})^2$
G. $x=-2+2(\frac{y+5}{3})^2$

Answer: B

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Question 4

A fruit stall sells apples costing £x each, and pears costing £y each.

Sam bought 2 apples and 5 pears, and the total cost of these was £ P.

Lesley bought 3 apples and 2 pears, and the total cost of these was £ Q.

Which of the following is an expression for the cost, in pounds (£), of a pear?

A. $\frac{2Q-3P}{3}$
B. $\frac{2Q-3P}{11}$
C. $\frac{Q-P}{3}$
D. $\frac{Q-P}{11}$
E. $\frac{P-Q}{3}$
F. $\frac{3P-2Q}{3}$
G. $\frac{3P-2Q}{11}$

Answer: G

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Question 5

P

is directly proportional to

Q

squared.

When

P

is 2,

Q

is 4.

Q

is inversely proportional to

R

.

When

Q

is 2,

R

is 5.

What is

P

in terms of

R

?

A. $P = \frac{5}{R}$
B. $P = \frac{5}{4R}$
C. $P = \frac{1}{800R^2}$
D. $P = \frac{5}{4R^2}$
E. $P = \frac{25}{2R^2}$
F. $P = \frac{800}{R^2}$
G. $P = \frac{R^2}{50}$
H. $P = \frac{25R^2}{2}$

Answer: E

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Question 6

Two sequences are defined by the following rules:

In sequence S the

n^{th}

term is

7n+1

In sequence T the

n^{th}

term is

99-n^2

What is the smallest value of

n

for which the

n^{th}

term of sequence S is greater than the

n^{th}

term of sequence T?

A.6
B.7
C.8
D.13
E.14
F.15

Answer: C

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Question 7

Which one of the following is a simplification of

2- \frac{x^2(9x^2-4)}{x^3(2-3x)}

A. $-1 - \frac{2}{x}$
B. $-1 + \frac{2}{x}$
C. $5 - \frac{2}{x}$
D. $5 + \frac{2}{x}$
E. $5 - \frac{3}{x}$
F. $5 + \frac{3}{x}$

Answer: D

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Question 8

The parallelogram

OPQR

, labelled clockwise, is in the first quadrant (

x \geq 0, y \geq 0

) with

O

at the origin.

The point R has coordinates

\begin{pmatrix} \frac{3a}{2}, \ 0 \end{pmatrix}

and the point Q has coordinates (2a, a + 1).

The area of OPQR is 9 square units.

What are the coordinates of point P?

A. $(\frac{\sqrt{3}}{2}, 1+\sqrt{3})$
B.(1, 3)
C.(1.5, 4)
D.(2, 3)
E.(3, 4)
F. $(2\sqrt{3}, 1+\sqrt{3})$

Answer: B

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Question 9

What is the value of x that makes the following expression correct?

2^{3+2x} \times 8^{-x} = 4\sqrt{2}

A.-2.25
B.-1.75
C.-1.5
D.-0.5
E.-0.25

Answer: D

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Question 10

There are 100 students in Year 10.

Each student studies exactly one of French, German, and Spanish.

X

girls study French and there are

3X

girls in total.

2Y

boys study German.

There are 35 students studying Spanish of which

Y

are boys.

Which of the following is an expression for the total number of students studying German?

A.X+2Y
B.X+Y+35
C.X+3Y-35
D.2X+2Y
E.2X + Y - 35
F.2X+3Y-35
G.2X + Y + 35

Answer: F

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Question 11

An exterior angle of a regular polygon with

n

sides is

4^{\circ}

larger than an exterior angle of a regular polygon with (

n+3

) sides.

What is the value of

n

?

A.10
B.12
C.15
D.18
E.21
F.24
G.27

Answer: C

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Question 12

The bearing of a ship

R

from a lighthouse

L

is

220^{\circ}

A canoe

C

is due North of

R

.

C

is the same distance from the ship and the lighthouse.

What is the bearing of

L

from

C

?

A. $070^{\circ}$
B. $080^{\circ}$
C. $090^{\circ}$
D. $100^{\circ}$
E. $140^{\circ}$

Answer: B

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Question 13

The hands of a 12-hour analogue clock move continuously. When the time on the clock is 4:00, the angle between the minute hand and the hour hand is

120^{\circ}

.

What is the angle between the two hands at 4:40?

A. $80^{\circ}$
B. $100^{\circ}$
C. $110^{\circ}$
D. $120^{\circ}$
E. $140^{\circ}$

Answer: B

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Question 14

The cost of manufacturing a cake is directly proportional to the volume of the cake.

The baker makes a 70% profit when he sells a large rectangular cake.

The baker sells a large rectangular cake for £6.80

The baker decides to sell smaller rectangular cakes. The length, width, and height of the smaller cakes are all half of those of the large rectangular cake.

He sells a pack of 6 of the smaller cakes for £6.50

How much profit does he make on the pack of smaller cakes?

A.£0.50
B.£2.93
C.£3.00
D.£3.50
E.£4.97

Answer: D

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Question 15

A pet shop has 4 female rabbits and

x

male rabbits for sale.

A customer buys 2 of the rabbits, chosen at random, and each rabbit is equally likely to be chosen.

The probability that both the chosen rabbits are male is

\frac{1}{3}

.

What is the value of

x

?

A.2
B.4
C.6
D.8
E.9
F.11
G.12

Answer: C

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Question 16

The diagram shows a square with side of length

x

cm. A circle is drawn with centre O which lies at the mid-point of one of the sides of the square. This side forms part of a diameter of the circle. The circle passes through two corners of the square as shown.

Exam diagram

What is the area, in cm

^2

, of the shaded part of the semi-circle?

A. $(\pi-1)x^2$
B.( $\frac{\pi-2}{2})x^2$
C.( $\frac{3\pi-2}{2})x^2$
D.( $\frac{3\pi-4}{4})x^2$
E.( $\frac{5\pi-4}{4})x^2$
F.( $\frac{5\pi-8}{8})x^2$

Answer: F

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Question 17

A cylindrical hollow metal pipe is 16 cm long.

It has an external diameter of 10 cm and an internal diameter of 8 cm.

The density of the metal from which the pipe is made is 8 grams per cm

^3

.

[diagram not to scale]

Exam diagram

What is the mass of the pipe in grams?

A. $8\pi$
B. $16\pi$
C. $18\pi$
D. $72\pi$
E. $128\pi$
F. $512\pi$
G. $1152\pi$
H. $4608\pi$

Answer: G

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Question 18

The shortest distance between two opposite sides of a regular hexagon is 12 cm.

Exam diagram

Find the area, in cm

^2

, of the regular hexagon.

A. $36\sqrt{3}$
B.72
C. $54\sqrt{3}$
D.108
E. $72\sqrt{3}$
F.144
G. $144\sqrt{3}$
H. $288\sqrt{3}$

Answer: E

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