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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

1 mark
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.
\nWhich 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

1 mark
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

1 mark
Which of the following is a correct rearrangement of

y=p−q−rs−xy = p - \frac{q-r}{s-x}
\nto 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

1 mark
Six numbers in increasing order are:

2, x, x, y, 23.5, 27.5
\nThe mean of the six numbers is 12.
\nThe median of the six numbers is 7.25.
\nWhat 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

1 mark
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

1 mark
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

1 mark
Given that

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

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

1 mark
Last year, the salary of the coach of a football club was 80% of the salary of the star player.
\nAt the start of the new year, the coach received a 15% increase in salary and the star player received a 38% increase in salary.
\nWhat 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

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

tan⁡θ=2\tan \theta = \sqrt{2}
\nThe length of the hypotenuse of the triangle is 363\sqrt{6} cm.
\nWhat 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

1 mark
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}
\nWhat 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

1 mark
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
\nWhat 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

1 mark
Two objects X and Y are similar.
\nThe surface area of object Y is double the surface area of object X.
\nThe volume of object Y is 727\sqrt{2} cm3^3 more than the volume of object X.
\nWhat 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

1 mark
x, y and z are positive variables.
\ny is inversely proportional to the square of x.
\nWhen y = 20, x = 3
\nz is directly proportional to x.
\nWhen z = 1.2, x = 6
\nWhat 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

1 mark
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
\nhas two solutions for a.
\nWhat 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

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

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

1 mark
X and Y are the end-points of a line segment.
\nPoint P has coordinates (-8,5).
\nP 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}.
\nA point Q is such that QY⃗=(7 6)\vec{QY} = \begin{pmatrix} 7 \ 6 \end{pmatrix}.
\nWhat 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

1 mark
The 2nd term of a linear (arithmetic) sequence is equal to twice the 8th term.
\nThe 5th term is 94\frac{9}{4}.
\nWhat 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

1 mark
Find the maximum value of

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

1 mark
Two bags contain counters which are identical except for colour.
\nThe first bag contains 3 blue and 2 yellow counters.
\nThe second bag contains x red, 1 blue and 2 yellow counters.
\nA counter is picked at random from the first bag and placed in the second bag.
\nA 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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