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NSAA 2023 Mathematics S1

20 questions20 marksUpdated June 2026

The NSAA 2023 Mathematics S1 paper in full: all 20 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

The surface area of a solid sphere of radius R is equal to the total surface area of 10 solid closed cylinders of radius r and height 4r.

Which of the following is an expression for R in terms of r?

(The surface area of a sphere of radius R is 4πR24\pi R^2.)
  • A.R=5rR = 5r
  • B.R=12rR = 12r
  • C.R=25rR = 2\sqrt{5}r
  • D.R=1210rR = \frac{1}{2}\sqrt{10} r
  • E.R=10rR = \sqrt{10} r
  • F.R=3210rR = \frac{3}{2}\sqrt{10} r
  • G.R=15rR = \sqrt{15}r

Answer: A

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Question 2

Which of the following statements is/are correct?

1\. 4793÷3371=7193÷3347\frac{47}{93} \div \frac{33}{71} = \frac{71}{93} \div \frac{33}{47}

2\. 53% of 84 = 84% of 53

3\. (23)2=(32)2(2\sqrt{3})^2 = (3\sqrt{2})^2
  • A.none of them
  • B.1 only
  • C.2 only
  • D.3 only
  • E.1 and 2 only
  • F.1 and 3 only
  • G.2 and 3 only
  • H.1, 2 and 3

Answer: E

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Question 3

Which of the following is a correct rearrangement of

y=p−q−rs−xy = p - \frac{q-r}{s-x}

to make x the subject?
  • A.x=s−q−rp+yx = s - \frac{q-r}{p+y}
  • B.x=q−rp+y−sx = \frac{q-r}{p+y} - s
  • C.x=s−q−rp−yx = s - \frac{q-r}{p-y}
  • D.x=q−rp−y−sx = \frac{q-r}{p-y} - s
  • E.x=s−q−ry−px = s - \frac{q-r}{y-p}
  • F.x=q−ry−p−sx = \frac{q-r}{y-p} - s

Answer: C

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Question 4

Six numbers in increasing order are:

2, x, x, y, 23.5, 27.5

The mean of the six numbers is 12.

The median of the six numbers is 7.25.

What is the mode of the six numbers?
  • A.4.5
  • B.4.75
  • C.5
  • D.9.625
  • E.10
  • F.11.75

Answer: A

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Question 5

WXYZ is a square of side length 1.

WM:MX=1:2WM: MX = 1:2

XN:NY=3:1XN:NY=3:1

YP:PZ=4:1YP: PZ = 4:1

What is the area of triangle MNP?

[diagram not to scale]
Exam diagram
  • A.13\frac{1}{3}
  • B.25\frac{2}{5}
  • C.920\frac{9}{20}
  • D.130\frac{1}{30}
  • E.1960\frac{19}{60}
  • F.2360\frac{23}{60}

Answer: F

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Question 6

Which of the following is a simplification of

4x−8x2+4x−12÷x−6x3−36x\frac{4x-8}{x^2+4x-12} \div \frac{x-6}{x^3-36x}
  • A.14\frac{1}{4}
  • B.14x\frac{1}{4x}
  • C.4
  • D.4x
  • E.4x(x+6)(x−6)\frac{4x(x+6)}{(x-6)}
  • F.4x(x−6)(x+6)\frac{4x(x-6)}{(x+6)}
  • G.4x(x+6)2\frac{4}{x(x+6)^2}
  • H.4x(x−6)2\frac{4}{x(x-6)^2}

Answer: D

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Question 7

Given that

272(x−2)9(2x−3)=(81)32\frac{27^{2(x-2)}}{9^{(2x-3)}} = (81)^{\frac{3}{2}}

what is the value of x?
  • A.0
  • B.2.5
  • C.3
  • D.6
  • E.7.5
  • F.9
  • G.10.5
  • H.12

Answer: D

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Question 8

The solutions of the equation 3x2−6x−2=03x^2 - 6x - 2 = 0 are p and q, where p>qp>q.

What is the value of 3p+2q3p + 2q?
  • A.−10−2315-10 - \frac{2}{3}\sqrt{15}
  • B.−10−233-10 - \frac{2}{3}\sqrt{3}
  • C.−5−1315-5 - \frac{1}{3}\sqrt{15}
  • D.−5−133-5 - \frac{1}{3}\sqrt{3}
  • E.5+13155 + \frac{1}{3}\sqrt{15}
  • F.5+1335 + \frac{1}{3}\sqrt{3}
  • G.10+231510 + \frac{2}{3}\sqrt{15}
  • H.10+23310 + \frac{2}{3}\sqrt{3}

Answer: E

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Question 9

Last year, the salary of the coach of a football club was 80% of the salary of the star player.

At the start of the new year, the coach received a 15% increase in salary and the star player received a 38% increase in salary.

What percentage of the star player's new salary is the coach's new salary?
  • A.46%
  • B.57%
  • C.6123%61\frac{2}{3}\%
  • D.6623%66\frac{2}{3}\%
  • E.77%
  • F.8313%83\frac{1}{3}\%

Answer: D

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Question 10

Two of the angles of a triangle are 90∘90^\circ and θ\theta, where

tan⁡θ=2\tan \theta = \sqrt{2}

The length of the hypotenuse of the triangle is 363\sqrt{6} cm.

What is the area of the triangle, in cm2^2?
  • A.922\frac{9\sqrt{2}}{2}
  • B.929\sqrt{2}
  • C.939\sqrt{3}
  • D.969\sqrt{6}
  • E.18218\sqrt{2}
  • F.18318\sqrt{3}
  • G.18618\sqrt{6}

Answer: B

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Question 11

An athlete's training session consists of several complete repetitions of a three-part programme:

1. Walk 100 m at an average speed of 6 km h−1^{-1}

2. Jog 200 m at an average speed of 10 km h−1^{-1}

3. Run 100 m at an average speed of 20 km h−1^{-1}

What is the athlete's average speed for the complete training session, in km h−1^{-1}?
  • A.7.2
  • B.9.6
  • C.11.5
  • D.12
  • E.14.4

Answer: B

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Question 12

The line joining the points with coordinates (q,2−q)(q, 2 - q) and (2q+2,q−4)(2q + 2, q - 4) is perpendicular to the line with equation 3x−4y+5=03x - 4y + 5 = 0

What is the value of q?
  • A.−25-\frac{2}{5}
  • B.−16-\frac{1}{6}
  • C.1
  • D.1811\frac{18}{11}
  • E.167\frac{16}{7}
  • F.52\frac{5}{2}
  • G.6

Answer: C

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Question 13

Two objects X and Y are similar.

The surface area of object Y is double the surface area of object X.

The volume of object Y is 727\sqrt{2} cm3^3 more than the volume of object X.

What is the volume of object X, in cm3^3?
  • A.14−7214-7\sqrt{2}
  • B.14+7214+7\sqrt{2}
  • C.42−7217\frac{42-7\sqrt{2}}{17}
  • D.42+7217\frac{42+7\sqrt{2}}{17}
  • E.723\frac{7\sqrt{2}}{3}
  • F.727\sqrt{2}
  • G.4−24-\sqrt{2}
  • H.4+24+\sqrt{2}

Answer: H

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Question 14

x, y and z are positive variables.

y is inversely proportional to the square of x.

When y = 20, x = 3

z is directly proportional to x.

When z = 1.2, x = 6

What is z when y = 80?
  • A.0.0225
  • B.0.03
  • C.0.15
  • D.0.3
  • E.0.36
  • F.0.6
  • G.1.2

Answer: D

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Question 15

The equation

(a×104+2a×1033×10−1)2=8×109\left( \frac{a \times 10^4 + 2a \times 10^3}{3 \times 10^{-1}} \right)^2 = 8 \times 10^9

has two solutions for a.

What is the positive difference between these two solutions?
  • A.0
  • B.252\sqrt{5}
  • C.454\sqrt{5}
  • D.20520\sqrt{5}
  • E.40540\sqrt{5}
  • F.2005200\sqrt{5}

Answer: B

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Question 16

For any real number x, the function V(x) is defined as:

V(x)=10x2+6x+7V(x) = 10^{x^2+ 6x +7}

What is the smallest value of V(x)?
  • A.0.001
  • B.0.01
  • C.0.1
  • D.1
  • E.10
  • F.100
  • G.1000

Answer: B

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Question 17

X and Y are the end-points of a line segment.

Point P has coordinates (-8,5).

P lies on the line segment XY such that XP: PY is 1:2 and XP⃗=(4 −3)\vec{XP} = \begin{pmatrix} 4 \ -3 \end{pmatrix}.

A point Q is such that QY⃗=(7 6)\vec{QY} = \begin{pmatrix} 7 \ 6 \end{pmatrix}.

What are the coordinates of point Q?
  • A.(7, 5)
  • B.(3, 8)
  • C.(1, -12)
  • D.(-3, -10)
  • E.(-7, -7)
  • F.(-11, -4)

Answer: E

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Question 18

The 2nd term of a linear (arithmetic) sequence is equal to twice the 8th term.

The 5th term is 94\frac{9}{4}.

What is the 20th term of the sequence?
  • A.-9
  • B.−74-\frac{7}{4}
  • C.−32-\frac{3}{2}
  • D.6
  • E.314\frac{31}{4}
  • F.272\frac{27}{2}

Answer: C

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Question 19

Find the maximum value of

2sin⁡x×3−sin⁡x2^{\sin x} \times 3^{-\sin x}

where 0∘≤x≤360∘0^\circ \le x \le 360^\circ.
  • A.23\frac{2}{3}
  • B.1
  • C.32\frac{3}{2}
  • D.2
  • E.3
  • F.6

Answer: C

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Question 20

Two bags contain counters which are identical except for colour.

The first bag contains 3 blue and 2 yellow counters.

The second bag contains x red, 1 blue and 2 yellow counters.

A counter is picked at random from the first bag and placed in the second bag.

A counter is then picked at random from the second bag. The probability that this counter is yellow is 0.2. What is the probability that this counter is red?
  • A.14\frac{1}{4}
  • B.311\frac{3}{11}
  • C.310\frac{3}{10}
  • D.13\frac{1}{3}
  • E.23\frac{2}{3}
  • F.710\frac{7}{10}
  • G.811\frac{8}{11}
  • H.34\frac{3}{4}

Answer: E

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