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NSAA 2020 Mathematics PART A

20 questions20 marksUpdated June 2026

The NSAA 2020 Mathematics PART A 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 admission charge to a cinema is different for adults and children.

Admission for 2 adults and 3 children costs £20.

Admission for 4 adults and 4 children costs £34.

What does admission cost for 6 adults and 2 children?
  • A.£27
  • B.£29
  • C.£33
  • D.£39
  • E.£44
  • F.£48
  • G.£72

Answer: D

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Question 2

The nthn^{th} term of a sequence is 2n−52n - 5.

Which row in the table is correct for this sequence?
  • A.term-to-term rule: subtract 5, term which has a value of 17: 11th11^{th}
  • B.term-to-term rule: subtract 5, term which has a value of 17: 29th29^{th}
  • C.term-to-term rule: subtract 2, term which has a value of 17: 11th11^{th}
  • D.term-to-term rule: subtract 2, term which has a value of 17: 29th29^{th}
  • E.term-to-term rule: add 5, term which has a value of 17: 11th11^{th}
  • F.term-to-term rule: add 5, term which has a value of 17: 29th29^{th}
  • G.term-to-term rule: add 2, term which has a value of 17: 11th11^{th}
  • H.term-to-term rule: add 2, term which has a value of 17: 29th29^{th}

Answer: G

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Question 3

A fair spinner has eight equal sections.
Each section has one number written on it, as shown.

Exam diagram


The spinner is spun twice, and the two numbers scored are added.

What is the probability that the sum of the two numbers is 5?
  • A.18\frac{1}{8}
  • B.58\frac{5}{8}
  • C.116\frac{1}{16}
  • D.316\frac{3}{16}
  • E.2564\frac{25}{64}
  • F.5564\frac{55}{64}

Answer: A

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Question 4

Exam diagram

PQRS is a square with side length xx.

MM is the midpoint of side PSPS.

A circular arc, with centre MM, is drawn inside the square from SS to PP.

Another circular arc, with centre PP, is drawn inside the square from SS to QQ.

What is the area of the shaded region in terms of xx?
  • A.18πx2\frac{1}{8}\pi x^2
  • B.316πx2\frac{3}{16}\pi x^2
  • C.14πx2\frac{1}{4}\pi x^2
  • D.516πx2\frac{5}{16}\pi x^2
  • E.38πx2\frac{3}{8}\pi x^2
  • F.716πx2\frac{7}{16}\pi x^2
  • G.12πx2\frac{1}{2}\pi x^2

Answer: A

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Question 5

A balloon contains 5000cm35000\text{cm}^3 of gas.

The gas in the balloon gradually escapes so that the volume of the balloon decreases.

60% of the volume of the balloon is lost each week.

What is the volume of the balloon, in cm3\text{cm}^3, after 3 weeks?
  • A.0
  • B.128
  • C.320
  • D.800
  • E.1080

Answer: D

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Question 6

Consider the four lines with the following equations.

1 2x+6y=32x + 6y = 3

2 9y=3x−49y = 3x - 4

3 2y=6x+32y = 6x + 3

4 4x+6y−9=04x + 6y - 9 = 0

Which two lines are perpendicular?
  • A.1 and 2
  • B.1 and 3
  • C.1 and 4
  • D.2 and 3
  • E.2 and 4
  • F.3 and 4

Answer: A

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Question 7

The equilateral triangle PQRPQR has sides of length 8 cm.

A circle, centre OO, passes through each of the vertices of the triangle.

Find an expression for the circumference of the circle, in cm.
  • A.sin⁡60°8π\frac{\sin 60°}{8\pi}
  • B.8πsin⁡60°\frac{8\pi}{\sin 60°}
  • C.cos⁡60°8π\frac{\cos 60°}{8\pi}
  • D.8πcos⁡60°\frac{8\pi}{\cos 60°}
  • E.tan⁡60°8π\frac{\tan 60°}{8\pi}
  • F.8πtan⁡60°\frac{8\pi}{\tan 60°}

Answer: C

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Question 8

Find the sum of the solutions of
2(x4+3)2−(x4+3)−36=02\left(\frac{x}{4} + 3\right)^2 - \left(\frac{x}{4} + 3\right) - 36 = 0
  • A.2
  • B.32\frac{3}{2}
  • C.12\frac{1}{2}
  • D.-4
  • E.-13
  • F.-22
  • G.-26
  • H.-34

Answer: B

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Question 9

When the expression
(2x+3)2−(x−3)2(2x + 3)^2 - (x - 3)^2

is written in the form p(x+q)2+rp(x + q)^2 + r, where p,qp, q and rr are constants, what is the value of rr?
  • A.-27
  • B.-9
  • C.0
  • D.3
  • E.15

Answer: B

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Question 10

Which one of the following expressions is equivalent to
abc−abc\frac{a}{\frac{b}{c}} - \frac{\frac{a}{b}}{c}
  • A.0
  • B.a(b2−1)bc\frac{a(b^2 - 1)}{bc}
  • C.a(b2−c2)bc\frac{a(b^2 - c^2)}{bc}
  • D.a2b2−c2abc\frac{a^2b^2 - c^2}{abc}
  • E.a(c2−1)bc\frac{a(c^2 - 1)}{bc}
  • F.a2c2−b2abc\frac{a^2c^2 - b^2}{abc}
  • G.b2−a2abc\frac{b^2 - a^2}{abc}

Answer: F

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Question 11

The table shows statistics relating to the test marks of two groups of students.

| | number of students | mean | range |
|---|---|---|---|
| group X | 10 | 36 | 16 |
| group Y | 20 | 48 | 21 |

The results for the two groups of students are combined.

What can be deduced about the mean and range of the combined results?
  • A.mean = 40, range ≤\le 16
  • B.mean = 40, 16 < range < 21
  • C.mean = 40, range ≥\ge 21
  • D.mean = 44, range ≤\le 16
  • E.mean = 44, 16 < range < 21
  • F.mean = 44, range ≥\ge 21

Answer: F

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Question 12

The number of pairs of winter boots sold on a day is inversely proportional to the cube of the outside temperature on that day, measured in °C.

On a day when the outside temperature is 8°C, 250 pairs of boots are sold.

The next day, when the outside temperature is x °C, the number of pairs of boots sold is 700% more than on the previous day.

What is the value of x?
  • A.2
  • B.4
  • C.873\frac{8}{\sqrt[3]{7}}
  • D.8738\sqrt[3]{7}
  • E.16

Answer: E

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Question 13

In a sale, all prices are reduced by 25%.

A customer calculates the pre-sale price of a bicycle incorrectly by increasing the marked sale price by 25%.

The customer's calculated pre-sale price is incorrect by £15.

What is the correct pre-sale price of the bicycle?
  • A.£180
  • B.£195
  • C.£210
  • D.£225
  • E.£240

Answer: D

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Question 14

A paint colour is a mixture of red paint, blue paint and yellow paint.

The ratio of red paint to blue paint in the mixture is 18:5

The ratio of blue paint to yellow paint in the mixture is p:3p:3

The ratio of red paint to yellow paint in the mixture is 12:5

What is the value of pp?
  • A.2
  • B.4.5
  • C.5
  • D.7.5
  • E.12

Answer: A

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Question 15

Exam diagram

[diagram not to scale]

In the diagram, QS is perpendicular to PR.

PS=xPS = x cm
PQ=yPQ = y cm
QR=zQR = z cm

angle QRS=61°QRS = 61°

PSRPSR is a straight line.

Which one of the following is an expression for the length zz, in cm?
  • A.y2+x2sin⁡61°\sqrt{y^2 + x^2} \sin 61°
  • B.y2−x2sin⁡61°\sqrt{y^2 - x^2} \sin 61°
  • C.y2+x2cos⁡61°\sqrt{y^2 + x^2} \cos 61°
  • D.y2−x2cos⁡61°\sqrt{y^2 - x^2} \cos 61°
  • E.y2+x2sin⁡61°\frac{\sqrt{y^2 + x^2}}{\sin 61°}
  • F.y2−x2sin⁡61°\frac{\sqrt{y^2 - x^2}}{\sin 61°}
  • G.y2+x2cos⁡61°\frac{\sqrt{y^2 + x^2}}{\cos 61°}
  • H.y2−x2cos⁡61°\frac{\sqrt{y^2 - x^2}}{\cos 61°}

Answer: F

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Question 16

Two identical fair six-sided dice each have their faces numbered from 1 to 6, with one number on each face.

Both dice are thrown, and the number on each of the dice is recorded.

They are then both thrown again, and the number on each of the dice is recorded.

What is the probability that at least one of the four recorded numbers is even?
  • A.14\frac{1}{4}
  • B.12\frac{1}{2}
  • C.916\frac{9}{16}
  • D.34\frac{3}{4}
  • E.1516\frac{15}{16}

Answer: E

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Question 17

The quadratic equation 2x2−px−4=02x^2 - px - 4 = 0, where pp is a positive constant, has two solutions that differ by 6.

What is the value of pp?
  • A.2
  • B.474\sqrt{7}
  • C.12
  • D.4114\sqrt{11}
  • E.4344\sqrt{34}
  • F.6306\sqrt{30}

Answer: B

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Question 18

Two vertices of a square are at (1, 1) and (3, 5).

What is the difference between the perimeters of the largest and smallest possible squares that can be drawn with these points as two of their vertices?
  • A.0
  • B.43(2−2)4\sqrt{3}(2-\sqrt{2})
  • C.43(2−1)4\sqrt{3}(\sqrt{2}-1)
  • D.45(2−2)4\sqrt{5}(2-\sqrt{2})
  • E.45(2−1)4\sqrt{5}(\sqrt{2}-1)
  • F.413(2−2)4\sqrt{13}(2-\sqrt{2})
  • G.413(2−1)4\sqrt{13}(\sqrt{2}-1)
  • H.435(2−2)4\sqrt{3}\sqrt{5}(2-\sqrt{2})

Answer: D

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Question 19

The point MM is (2, 5) and the point NN is (-3,-1).

The line segment MNMN is transformed to the line segment TUTU by two transformations:

MNMN is rotated 90° clockwise about the origin to give the line segment RSRS.

RSRS is then translated by the vector (p q)\begin{pmatrix} p \ q \end{pmatrix} to give the line segment TUTU.

The coordinates of the midpoint of TUTU are (7, -2.5).

Find the vector (p q)\begin{pmatrix} p \ q \end{pmatrix}.
  • A.(2 0.5)\begin{pmatrix} 2 \ 0.5 \end{pmatrix}
  • B.(0.5 2)\begin{pmatrix} 0.5 \ 2 \end{pmatrix}
  • C.(5 −3)\begin{pmatrix} 5 \ -3 \end{pmatrix}
  • D.(−3 5)\begin{pmatrix} -3 \ 5 \end{pmatrix}
  • E.(9 −2)\begin{pmatrix} 9 \ -2 \end{pmatrix}
  • F.(−2 9)\begin{pmatrix} -2 \ 9 \end{pmatrix}

Answer: C

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Question 20

A solid cone has a base radius xx cm.

The ratio of the perpendicular height of the cone to the radius of the cone is 5:2

A solid hemisphere of radius y2\frac{y}{2} cm is made from the same material as the cone.

Which one of the following is a correct expression for
volume of the conevolume of the hemisphere\frac{\text{volume of the cone}}{\text{volume of the hemisphere}}


(Volume of a cone = 13πr2h\frac{1}{3}\pi r^2 h where rr is the radius and hh is the perpendicular height.)

(Volume of a sphere = 43πr3\frac{4}{3}\pi r^3 where rr is the radius.)
  • A.5x3y3\frac{5x^3}{y^3}
  • B.5x34y3\frac{5x^3}{4y^3}
  • C.8x35y3\frac{8x^3}{5y^3}
  • D.10x3y3\frac{10x^3}{y^3}
  • E.14x3y3\frac{14x^3}{y^3}

Answer: D

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