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TMUA 2023 D513/01

20 questions20 marks75Updated July 2025

The TMUA 2023 D513/01 paper in full: all 20 questions, each with its answer. TMUA is the Test of Mathematics for University Admission. 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
Given that ∫01(ax+b) dx=1\int_{0}^{1}(ax+b)\ \mathrm{d}x = 1 and ∫01x(ax+b) dx=1\int_{0}^{1}x(ax+b)\ \mathrm{d}x = 1 find the value of a+ba+b.
  • A.-1
  • B.0
  • C.1
  • D.2
  • E.3
  • F.4
  • G.5

Answer: F

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Question 2

1 mark
The graphs of y=x2+5x+6y = x^2 + 5x + 6 and y=mx−3y=mx-3, where mm is a constant, are plotted on the same set of axes.
Given that the graphs do not meet, what is the complete range of possible values of mm?
  • A.−1<m<11-1<m< 11
  • B.m<−1,m>11m<-1, m > 11
  • C.−11<m<11-\sqrt{11} < m < \sqrt{11}
  • D.m<−11,m>11m<-\sqrt{11}, m> \sqrt{11}
  • E.−11<m<1-11<m<1
  • F.m<−11,m>1m<-11, m> 1

Answer: A

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Question 3

1 mark
For any integer n≥0n \ge 0, ∫nn+1f(x) dx=n+1\int_{n}^{n+1}f(x)\ \mathrm{d}x = n + 1
Evaluate ∫03f(x) dx+∫13f(x) dx+∫23f(x) dx+∫43f(x) dx+∫53f(x) dx\int_{0}^{3}f(x)\ \mathrm{d}x + \int_{1}^{3}f(x)\ \mathrm{d}x + \int_{2}^{3}f(x)\ \mathrm{d}x + \int_{4}^{3}f(x)\ \mathrm{d}x + \int_{5}^{3}f(x)\ \mathrm{d}x
  • A.-2
  • B.0
  • C.1
  • D.4
  • E.18
  • F.27

Answer: C

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Question 4

1 mark
Evaluate ∑n=0∞sin⁡(nπ+π3)2n\sum_{n=0}^{\infty} \frac{\sin \left(n\pi + \frac{\pi}{3}\right)}{2^n}
  • A.0
  • B.13\frac{1}{3}
  • C.33\frac{\sqrt{3}}{3}
  • D.3\sqrt{3}
  • E.3

Answer: C

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Question 5

1 mark
The following shape has two lines of reflectional symmetry.
Exam diagram

[diagram not to scale]
MNOP is a square of perimeter 4040 cm.
The vertices of rectangle RSTURSTU lie on the edge of square MNOPMNOP.
MRMR has length xx cm.
What is the largest possible value of xx such that RSTURSTU has area 2020 cm2^2?
  • A.2\sqrt{2}
  • B.10\sqrt{10}
  • C.2152\sqrt{15}
  • D.10210\sqrt{2}
  • E.5+55+\sqrt{5}
  • F.5+155+\sqrt{15}

Answer: F

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Question 6

1 mark
In the simplified expansion of (2+3x)12(2+3x)^{12}, how many of the terms have a coefficient that is divisible by 1212?
  • A.0
  • B.2
  • C.5
  • D.10
  • E.11
  • F.12
  • G.13

Answer: E

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Question 7

1 mark
P(x)P(x) and Q(x)Q(x) are defined as follows:
P(x)=2x+4P(x) = 2^x + 4
Q(x)=2(2x−2)−2(x+2)+16Q(x) = 2^{(2x-2)}-2^{(x+2)} + 16
Find the largest value of xx such that P(x)P(x) and Q(x)Q(x) are in the ratio 4:14:1, respectively.
  • A.5
  • B.12
  • C.32
  • D.log⁡23\log_2 3
  • E.log⁡25\log_2 5
  • F.log⁡212\log_2 12
  • G.log⁡220\log_2 20

Answer: F

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Question 8

1 mark
A triangle XYZXYZ is called fun if it has the following properties:
angle YXZ=30∘YXZ = 30^{\circ}
XY=3aXY = \sqrt{3} a
YZ=aYZ = a
where aa is a constant.
For a given value of aa, there are two distinct fun triangles SS and TT, where the area of SS is greater than the area of TT.
Find the ratio area of SS: area of TT
  • A.1:11:1
  • B.2:12:1
  • C.2:32:\sqrt{3}
  • D.3:1\sqrt{3}:1
  • E.3:13:1

Answer: B

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Question 9

1 mark
How many solutions are there to (1+3cos⁡3θ)2=4(1 + 3\cos3\theta)^2 = 4 in the interval 0∘≤θ≤180∘0^{\circ} \le \theta \le 180^{\circ}?
  • A.1
  • B.2
  • C.3
  • D.4
  • E.5
  • F.6

Answer: E

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Question 10

1 mark
The trapezium rule with 44 strips is used to estimate the integral:
∫−224−x2 dx\int_{-2}^{2}\sqrt{4-x^2}\ \mathrm{d}x
What is the positive difference between the estimate and the exact value of the integral?
  • A.2(π−2−23)2(\pi-2-2\sqrt{3})
  • B.2(π−1−3)2(\pi-1-\sqrt{3})
  • C.2(2π−1−3)2(2\pi-1-\sqrt{3})
  • D.4(π−1−3)4(\pi-1-\sqrt{3})
  • E.2π−332\pi-3\sqrt{3}
  • F.4π−634\pi-6\sqrt{3}

Answer: B

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Question 11

1 mark
It is given that f(x)=x2−6xf(x) = x^2 - 6x
The curves y=f(kx)y = f(kx) and y=f(x−c)y=f(x - c) have the same minimum point, where k>0k > 0 and c>0c > 0
Which of the following is a correct expression for kk in terms of cc?
  • A.k=3−c3k = \frac{3-c}{3}
  • B.k=3c+3k = \frac{3}{c+3}
  • C.k=c−66k = \frac{c-6}{6}
  • D.k=66−ck = \frac{6}{6-c}
  • E.k=c+99k = \frac{c+9}{9}
  • F.k=9c−9k = \frac{9}{c-9}

Answer: B

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Question 12

1 mark
How many solutions are there to the equation 2tan⁡2x4sin⁡2x=1\frac{2^{\tan^2x}}{4^{\sin^2x}} = 1 in the range 0≤x≤2π0 \le x \le 2\pi?
  • A.2
  • B.3
  • C.4
  • D.5
  • E.6
  • F.7
  • G.8

Answer: F

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Question 13

1 mark
Point PP lies on the circle with equation (x−2)2+(y−1)2=16(x - 2)^2 + (y - 1)^2 = 16
Point QQ lies on the circle with equation (x−4)2+(y+5)2=16(x - 4)^2 + (y + 5)^2 = 16
What is the maximum possible length of PQPQ?
  • A.10
  • B.14
  • C.16
  • D.2342\sqrt{34}
  • E.10210\sqrt{2}
  • F.8+2108+2\sqrt{10}
  • G.16+2616+2\sqrt{6}

Answer: F

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Question 14

1 mark
The function f(x)=23x3+2mx2+nf(x) = \frac{2}{3}x^3 + 2mx^2 + n, m>0m > 0 has three distinct real roots.
What is the complete range of possible values of nn, in terms of mm?
  • A.−83m3<n<0-\frac{8}{3}m^3 < n < 0
  • B.−43m3<n<0-\frac{4}{3}m^3 < n < 0
  • C.0<n<32m20 < n < \frac{3}{2}m^2
  • D.0<n<403m30 < n < \frac{40}{3}m^3
  • E.n<−83m3n < -\frac{8}{3}m^3
  • F.n<32m2n < \frac{3}{2}m^2
  • G.n>−43m3n > -\frac{4}{3}m^3
  • H.n>403m3n > \frac{40}{3}m^3

Answer: A

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Question 15

1 mark
The difference between the maximum and minimum values of the function f(x)=acos⁡xf(x) = a^{\cos x}, where a>0a > 0 and xx is real, is 33.
Find the sum of the possible values of aa.
  • A.0
  • B.32\frac{3}{2}
  • C.52\frac{5}{2}
  • D.3
  • E.10\sqrt{10}
  • F.13\sqrt{13}

Answer: F

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Question 16

1 mark
A right-angled triangle has vertices at (2,3)(2, 3), (9,−1)(9, -1) and (5,k)(5, k).
Find the sum of all the possible values of kk.
  • A.-8
  • B.-6
  • C.0.25
  • D.2
  • E.2.25
  • F.8.25
  • G.10.25

Answer: E

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Question 17

1 mark
A circle CnC_n is defined by x2+y2=2n(x+y)x^2 + y^2 = 2n(x + y) where nn is a positive integer.
C1C_1 and C2C_2 are drawn and the area between them is shaded.
Next, C3C_3 and C4C_4 are drawn and the area between them is shaded.
This is shown in the diagram.
Exam diagram

[diagram not to scale]
This process continues until 100100 circles have been drawn.
What is the total shaded area?
  • A.100π100\pi
  • B.500π500\pi
  • C.2500π2500\pi
  • D.5050π5050\pi
  • E.10100π10100\pi
  • F.40400π40400\pi

Answer: E

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Question 18

1 mark
You are given that
S=4+8k7+16k249+32k3343+⋯+4(2k7)n+…S = 4 + \frac{8k}{7} + \frac{16k^2}{49} + \frac{32k^3}{343} + \dots + 4\left(\frac{2k}{7}\right)^n + \dots

The value for kk is chosen as an integer in the range −5≤k≤5-5 \le k \le 5
All possible values for kk are equally likely to be chosen.
What is the probability that the value of SS is a finite number greater than 33?
  • A.111\frac{1}{11}
  • B.110\frac{1}{10}
  • C.311\frac{3}{11}
  • D.310\frac{3}{10}
  • E.511\frac{5}{11}
  • F.12\frac{1}{2}
  • G.711\frac{7}{11}
  • H.710\frac{7}{10}

Answer: E

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Question 19

1 mark
The solution to the differential equation dydx=∣−6x∣\frac{\mathrm{d}y}{\mathrm{d}x} = |-6x| for all xx is y=f(x)+cy = f(x) + c, where cc is a constant.
Which one of the following is a correct expression for f(x)f(x)?
  • A.−6xx-\frac{6x}{x}
  • B.6x∣x∣\frac{6x}{|x|}
  • C.−3x∣x∣-3x|x|
  • D.3x∣x∣3x|x|
  • E.−3x2-3x^2
  • F.3x23x^2
  • G.−x3-x^3
  • H.x3x^3

Answer: D

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Question 20

1 mark
The diagram shows the graph of y=f(x)y = f(x)
The function ff attains its maximum value of 22 at x=1x = 1, and its minimum value of −2-2 at x=−1x = -1
Exam diagram

Find the difference between the maximum and minimum values of (f(x))2−f(x)(f(x))^2 - f(x)
  • A.2
  • B.94\frac{9}{4}
  • C.4
  • D.174\frac{17}{4}
  • E.6
  • F.254\frac{25}{4}
  • G.8
  • H.334\frac{33}{4}

Answer: F

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