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← All past papers

Mapped to the TMUA specification

ENGAA 2023

40 questions40 marks60Updated August 2025

Engineering Admissions Assessment 2023 is one of the closest available past papers for the Test of Mathematics for University Admission. Below, all 40 questions are mapped to the TMUA topics they test, each with its answer. Sit it cold under exam timing, then work back through anything you missed using the solutions.

What this paper covers on the TMUA

Every question mapped to the Test of Mathematics for University Admission specification, so you know exactly which areas it tests.

Algebra (21)
Geometry (9)
Number (7)
Ratio and proportion (6)
Units (4)

Algebra and functions (10)
Exponentials and logarithms (4)
Trigonometry (4)
Integration (3)
Sequences and series (3)
Coordinate geometry in the (x,y)-plane (2)
Differentiation (2)

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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

.

Which of the following is an expression for

R

in terms of

r

?

(The surface area of a sphere of radius

R

is

4\pi R^2

.)

A. $R = 5r$
B. $R = 12r$
C. $R = 2\sqrt{5}r$
D. $R = \frac{1}{2}\sqrt{10} r$
E. $R = \sqrt{10} r$
F. $R = \frac{3}{2}\sqrt{10} r$
G. $R = \sqrt{15}r$

Answer: A

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Question 2

1 mark

A spaceship of mass 10000 kg is moving at

2.0\,ms^{-1}

relative to a space station.

The spaceship is captured by a robotic arm attached to the space station and brought to rest by a force of 1000 N.

How far will the spaceship move in its initial direction relative to the space station while the force is being applied?

(Assume that the acceleration of the space station is negligible.)

A.0.050 m
B.0.10 m
C.0.20 m
D.5.0 m
E.10 m
F.20 m

Answer: F

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Question 3

1 mark

Which of the following is a correct rearrangement of

y = p - \frac{q-r}{s-x}

to make

x

the subject?

A. $x = s - \frac{q-r}{p+y}$
B. $x = \frac{q-r}{p+y} - s$
C. $x = s - \frac{q-r}{p-y}$
D. $x = \frac{q-r}{p-y} - s$
E. $x = s - \frac{q-r}{y-p}$
F. $x = \frac{q-r}{y-p} - s$

Answer: C

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Question 4

1 mark

A circuit is set up as shown. All three resistors are identical.

When the switch is open, the reading on the ammeter is 1.0 A and the power transferred from the battery is 1.0 W.

Exam diagram

The switch is now closed.

What is the new reading on the ammeter and what is the new power transferred from the battery?

A.ammeter reading / A: 0.67, power transferred / W: 0.67
B.ammeter reading / A: 0.67, power transferred / W: 1.3
C.ammeter reading / A: 0.67, power transferred / W: 1.5
D.ammeter reading / A: 0.67, power transferred / W: 2.0
E.ammeter reading / A: 1.0, power transferred / W: 1.0
F.ammeter reading / A: 1.0, power transferred / W: 1.5
G.ammeter reading / A: 1.0, power transferred / W: 2.0
H.ammeter reading / A: 1.0, power transferred / W: 3.0

Answer: F

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Question 5

1 mark

Exam diagram

[diagram not to scale]

WXYZ is a square of side length 1.

WM: MX = 1:2

XN:NY=3:1

YP: PZ = 4:1

What is the area of triangle MNP?

A. $\frac{1}{3}$
B. $\frac{2}{5}$
C. $\frac{9}{20}$
D. $\frac{1}{30}$
E. $\frac{19}{60}$
F. $\frac{23}{60}$

Answer: F

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Question 6

1 mark

A spring is initially unstretched. A force

F

is used to stretch the spring. The extension

x

and the energy

E

stored in the stretched spring are measured for different values of

F

.

The graph shows how the energy

E

, in J, varies with the extension squared,

x^2

, in cm

^2

.

Exam diagram

What is the magnitude of

F

when the spring stores 0.015 J of energy?

A.0.30 N
B.0.60 N
C.1.2 N
D.1.5 N
E.2.4 N
F.3.0 N
G.30 N
H.60 N

Answer: B

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Question 7

1 mark

Given that

\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

1 mark

A solid, cylindrical metal bar has a uniform cross-sectional area of

12\,cm^2

and a volume of

180\,cm^3

.

The bar rests on a horizontal surface on one of its circular faces.

The pressure on the surface due to the bar is

0.45\,N\,cm^{-2}

.

What is the density of the metal, in

g\,cm^{-3}

?

(gravitational field strength =

10\,N\,kg^{-1}

)

A. $2.5\,g\,cm^{-3}$
B. $3.0\,g\,cm^{-3}$
C. $3.75\,g\,cm^{-3}$
D. $7.5\,g\,cm^{-3}$
E. $15\,g\,cm^{-3}$
F. $33\,g\,cm^{-3}$

Answer: B

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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.

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. $61\frac{3}{5}\%$
D. $66\frac{2}{3}\%$
E.77%
F. $83\frac{1}{3}\%$

Answer: D

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Question 10

1 mark

Two samples of pure radioactive isotopes X and Y decay with half-lives of 2 days and 3 days, respectively.

Both X and Y decay in a single step into different stable isotopes.

Initially the number of atoms of X is twice the number of atoms of Y.

After how many days are the expected numbers of atoms of X and Y equal to each other?

A.The expected numbers of atoms of X and Y are never equal.
B.2 days
C.3 days
D.4 days
E.6 days
F.12 days

Answer: E

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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}

2. Jog 200 m at an average speed of

10\,km\,h^{-1}

3. Run 100 m at an average speed of

20\,km\,h^{-1}

What is the athlete's average speed for the complete training session, in

km\,h^{-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

A large, flat, metal plate is coated on one side with a layer of thermally insulating material of the same thickness

a

as the metal plate.

The uninsulated top surface of the metal plate is maintained at a constant temperature

T_1

.

The bottom surface of the insulating material is maintained at a constant, lower temperature

T_2

.

The system is in equilibrium.

The diagram shows this arrangement.

Exam diagram

Which graph could show how the temperature varies with distance from the top surface of the metal plate to the bottom surface of the insulating material?

Exam diagram

A.Graph showing temperature dropping linearly with a shallow slope for distance a, then dropping linearly with a steeper slope for the next distance a.
B.Graph showing temperature dropping linearly with a steep slope for distance a, then dropping linearly with a shallower slope for the next distance a.
C.Graph showing temperature dropping linearly from T1 to T2 over distance a, then continuing to drop.
D.Graph showing temperature dropping linearly from T1 to a point, then dropping vertically down.
E.Graph showing a step function, temperature is T1 for distance a, then drops to T2 for the next distance a.

Answer: A

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Question 13

1 mark

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

7\sqrt{2}\,cm^3

more than the volume of object X.

What is the volume of object X, in cm³?

A. $14 - 7\sqrt{2}$
B. $14 + 7\sqrt{2}$
C. $\frac{42 - 7\sqrt{2}}{17}$
D. $\frac{42 + 7\sqrt{2}}{17}$
E. $\frac{7\sqrt{2}}{3}$
F. $7\sqrt{2}$
G. $4 - \sqrt{2}$
H. $4 + \sqrt{2}$

Answer: H

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Question 14

1 mark

The voltage output of a power station is stepped up using a transformer before the power is transmitted to a distant town. The primary coil of this transformer has 300 turns and the secondary coil has 1500 turns.

In the town, a step-down transformer reduces the voltage supplied by the transmission cables to 33 000 V for distribution within the town. The step-down transformer supplies a current of 1500 A.

The current in the transmission cables is 450 A and both transformers are ideal and 100% efficient.

What is the voltage output of the power station?

(Assume that the resistance of the transmission cables is negligible.)

A.1980 V
B.6600 V
C.22000 V
D.110000 V
E.550000 V

Answer: C

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Question 15

1 mark

The equation

\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. $2\sqrt{5}$
C. $4\sqrt{5}$
D. $20\sqrt{5}$
E. $40\sqrt{5}$
F. $200\sqrt{5}$

Answer: B

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Question 16

1 mark

A transverse wave with an amplitude of 3.0 cm travels along a stretched string. The wave has a frequency of 12 Hz and a wavelength of 0.25 m.

What is the average speed of a particle in the string as the string oscillates during a time of 2.0 s?

A. $36\,cm\,s^{-1}$
B. $72\,cm\,s^{-1}$
C. $125\,cm\,s^{-1}$
D. $144\,cm\,s^{-1}$
E. $300\,cm\,s^{-1}$

Answer: D

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Question 17

1 mark

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

\vec{XP} = <4,-3>

.

A point Q is such that

\vec{QY} =<7,6>

.

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

1 mark

A battery and two resistors X and Y are connected in series.

Exam diagram

The power transferred by the battery is 6 W.

The resistance of X is

10\,\Omega

.

The voltage across Y is 4 V.

What is the current in the circuit?

A. $\frac{2}{5}\,A$
B. $\frac{3}{5}\,A$
C. $\frac{3}{4}\,A$
D. $1\,A$
E. $\sqrt{\frac{3}{10}}\,A$
F. $\sqrt{\frac{3}{5}}\,A$

Answer: B

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Question 19

1 mark

Find the maximum value of

2^{\sin x} \times 3^{-\sin x}

where

0^\circ \le x \le 360^\circ

.

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

Answer: C

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Question 20

1 mark

A diver at the bottom of a lake of depth

d

fills a syringe with an ideal gas and seals the nozzle. The piston remains free to move. The volume of the gas in the syringe at the bottom of the lake is

90\,cm^3

.

As the diver returns to the surface, the temperature of the gas does not change. At the surface of the lake the gas in the syringe is at atmospheric pressure and the volume of the gas is

720\,cm^3

.

What is the volume of the gas in the syringe at a depth

\frac{d}{2}

?

A. $160\,cm^3$
B. $180\,cm^3$
C. $206\,cm^3$
D. $225\,cm^3$
E. $288\,cm^3$
F. $315\,cm^3$
G. $360\,cm^3$
H. $405\,cm^3$

Answer: A

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Question 21

1 mark

x^2-x-6

is a factor of

x^3 + ax^2 + 2x + b

, where

a

and

b

are real constants.

What is the value of

a + b

?

A.-39
B.-21
C.-3
D. $-\frac{3}{5}$
E. $\frac{3}{5}$
F.3
G.21
H.39

Answer: H

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Question 22

1 mark

[Question crossed out in paper]
Two pipes contain air at the same temperature and pressure.

A stationary sound wave is formed in the first pipe, which is closed at one end and open at the other end. The lowest frequency of stationary sound wave that can be formed in this pipe is 4000 Hz.

The second pipe has the same length as the first pipe, but is open at both ends.

What is the lowest frequency of stationary sound wave that can be formed in the second pipe?

A.1000 Hz
B.2000 Hz
C.4000 Hz
D.8000 Hz
E.16000 Hz

Answer: D

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Question 23

1 mark

An arithmetic progression has first term

a

and common difference

d

.

The sum of the first 9 terms plus the sum of the first 10 terms is equal to the sum of the first 11 terms.

Which of the following is a correct expression for

a

in terms of

d

?

A. $a = -\frac{13}{4}d$
B. $a = -\frac{15}{4}d$
C. $a = -\frac{16}{3}d$
D. $a = -\frac{19}{3}d$
E. $a = -7d$
F. $a = -8d$

Answer: A

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Question 24

1 mark

[Question crossed out in paper]
A cannon ball of mass 2.6 kg is fired horizontally from a cannon on the top of a cliff at a speed of

90\,m\,s^{-1}

.

The height of the cliff above the horizontal ground below is 45 m.

What is the magnitude of the impulse that acts on the ball between leaving the cannon and reaching the ground?

(gravitational field strength =

10\,N\,kg^{-1}

; assume that air resistance is negligible)

A.8.7 Ns
B.13 Ns
C.52 Ns
D.78 Ns
E.117 Ns
F.234 Ns
G.900 Ns

Answer: D

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Question 25

1 mark

The first three terms of a convergent geometric progression are:

2p,\, p-3,\, p-7

What is the sum to infinity of this progression?

A.-54
B.-27
C. $-13\frac{1}{2}$
D.-2
E.2
F. $13\frac{1}{2}$
G.27
H.54

Answer: G

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Question 26

1 mark

[Question crossed out in paper]
A child on a sledge is travelling straight down a snow-covered slope at a constant speed of

15\,m\,s^{-1}

.

The mass of the child and sledge together is 60 kg.

The angle of the slope to the horizontal is

30^\circ

.

What is the rate at which thermal energy is being produced due to friction forces?

(gravitational field strength =

10\,N\,kg^{-1}

)

A.450 W
B.900 W
C.4500 W
D. $4500\sqrt{3}$ W
E.6750 W
F.9000 W
G.13500 W
H. $9000\sqrt{3}$ W

Answer: C

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Question 27

1 mark

The diagram shows a circle with centre

O

.

Exam diagram

[diagram not to scale]

What is the value of

x

?

A. $40^\circ$
B. $50^\circ$
C. $55^\circ$
D. $70^\circ$
E. $75^\circ$
F. $80^\circ$

Answer: E

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Question 28

1 mark

[Question crossed out in paper]
The diagram shows a battery with no internal resistance and four identical resistors P, Q, R and S.

Resistor Q dissipates 4.0 W of power.

Exam diagram

What is the total power supplied by the battery?

A.10 W
B.16 W
C.24 W
D.32 W
E.40 W

Answer: E

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Question 29

1 mark

The function

f(x) = \sqrt{2}x^2 - 6x + 4

can be written in the form

f(x) = p(x + q)^2 + r

where

p

,

q

and

r

are constants.

What is the value of

p(r – q)

?

A.2
B.7
C. $3 - \frac{\sqrt{2}}{2}$
D. $3 - 7\sqrt{2}$
E. $4 - 3\sqrt{2}$
F. $4\sqrt{2} - 6$
G. $4\sqrt{2} - 12$
H. $7\sqrt{2} - 18$

Answer: F

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Question 30

1 mark

[Question crossed out in paper]
Two identical, uniform, thin planks, each of mass 40 kg, are propped up against one another in equilibrium on a rough surface as shown. Each plank is at an angle of

30^\circ

to the vertical.

Exam diagram

[diagram not to scale]

What is the magnitude of the friction force at point P?

(gravitational field strength =

10\,N\,kg^{-1}

)

A.100 N
B. $\frac{200}{\sqrt{3}}\,N$
C.200 N
D. $200\sqrt{3}\,N$
E. $\frac{400}{\sqrt{3}}\,N$
F.400 N
G. $400\sqrt{3}\,N$

Answer: B

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Question 31

1 mark

Which of the following is equal to

\int_{-(3-\sqrt{5})}^{3-\sqrt{5}} \frac{x^2}{3-\sqrt{5}} dx

A.0
B. $\frac{2}{3}$
C. $\frac{8}{3}$
D.4
E. $\frac{28-12\sqrt{5}}{3}$
F. $28-12\sqrt{5}$

Answer: E

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Question 32

1 mark

[Question crossed out in paper]
Three blocks P, Q and R are in contact on a horizontal surface as shown.

Exam diagram

The mass of P is 4.0 kg and the masses of Q and R are each 2.0 kg.

A horizontal force of 20 N is applied to P so that all three blocks move together.

The friction force between P and the surface is 2.0 N, and the friction forces between Q and R and the surface are each 1.0 N.

What is the magnitude of the force that R exerts on Q?

A.2.0 N
B.5.0 N
C.6.0 N
D.12 N
E.15 N
F.17 N
G.19 N

Answer: B

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Question 33

1 mark

A function

f

is defined by

f(x) = \frac{a}{x} + \frac{b}{x^2}

where

a

and

b

are constants.

It is given that

f'(1) = 2

and

f''(-1) = -2

.

What is the value of

a + b

?

A.-5
B.-4
C. $-\frac{7}{5}$
D.-1
E. $\frac{9}{5}$
F.2
G. $\frac{14}{5}$
H.3

Answer: C

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Question 34

1 mark

An object is released from rest at a height

H

above the ground and falls freely in a uniform gravitational field. At time

t

after being released, it has fallen a distance

s

and is at a height

h

above the ground, travelling at speed

v

.

Exam diagram

The graph shows two quantities plotted.

Exam diagram

Which of the rows show(s) a possible pair of quantities for the axes?

(Assume that air resistance is negligible.)

| | y-axis | x-axis |
|---|---|---|
| 1 | h | t |
| 2 | kinetic energy | h |
| 3 | gravitational potential energy | s |

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: G

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Question 35

1 mark

Find the real value of

x

that satisfies

\log_3 (x^2 + 3x + 2) = 2 + \log_3 (x^2 + 2x)

A. $\frac{1}{8}$
B. $\frac{1}{7}$
C. $\frac{1}{6}$
D. $\frac{1}{2}$
E.2
F.7

Answer: A

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Question 36

1 mark

[Question crossed out in paper]
A 20 V battery of negligible internal resistance is connected in series with a

20\,\Omega

resistor and a cylindrical conductor. The current in the circuit is 0.20 A.

The cylindrical conductor is now removed, melted and re-formed into a new cylinder. After the cylinder has cooled to its original temperature, its length is 4 times greater than that of the original cylinder.

What is the resistance of the new cylinder?

A. $5.0\,\Omega$
B. $20\,\Omega$
C. $320\,\Omega$
D. $400\,\Omega$
E. $1280\,\Omega$
F. $1600\,\Omega$

Answer: E

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Question 37

1 mark

What is the constant term in the simplified binomial expansion of

\left( \frac{2}{x} + \frac{x}{4} \right)^8

?

A. $\frac{7}{2}$
B. $\frac{35}{8}$
C. $\frac{1}{16}$
D. $\frac{7}{256}$
E. $\frac{35}{1024}$
F. $\frac{1}{2048}$

Answer: B

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Question 38

1 mark

[Question crossed out in paper]
A sound wave can travel from a source at P to a detector at R directly or by reflecting at Q.

The angle between PR and PQ is

30^\circ

and the distance from P to R is

d

as shown.

Exam diagram

There is a phase difference of

\pi

radians between the incident and reflected wave at Q.

Waves that reach R via Q are in phase with waves that reach R directly from P.

Which expression gives the greatest wavelength of sound waves for which this is true?

A. $2d(\frac{2}{\sqrt{3}} - 1)$
B. $d(\frac{2}{\sqrt{3}} - 1)$
C. $d(2 - \sqrt{3})$
D. $\frac{4d}{\sqrt{3}}$
E. $\frac{2d}{\sqrt{3}}$

Answer: A

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Question 39

1 mark

Given that

x^2 + y^2 = 1

what is the greatest possible value of

2x + 3y

?

A. $\frac{7}{2}$
B.3
C. $\frac{5\sqrt{2}}{2}$
D. $\sqrt{7}$
E. $\sqrt{10}$
F. $\frac{13\sqrt{10}}{10}$
G. $\sqrt{13}$
H. $\frac{12\sqrt{13}}{13}$

Answer: G

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Question 40

1 mark

[Question crossed out in paper]
Two springs have spring constants of

200\,N\,m^{-1}

and

600\,N\,m^{-1}

, respectively.

The springs are joined in series, end-to-end, and stretched so that their combined extension is 0.80 m.

What is the total strain energy stored in the springs?

A.48 J
B.96 J
C.112 J
D.120 J
E.240 J
F.256 J
G.512 J

Answer: A

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