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TMUA 2022 D513/01

20 questions20 marks75Updated July 2025

The TMUA 2022 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
How many real solutions are there to the equation 2cos⁡4θ–5cos⁡2θ+3=02 \cos^4 \theta – 5 \cos^2 \theta + 3 = 0 in the interval 0≤θ≤2π0 \leq \theta \leq 2\pi?
  • A.1
  • B.2
  • C.3
  • D.4
  • E.5
  • F.6
  • G.7
  • H.8

Answer: C

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Question 2

1 mark
Find the complete set of values of pp for which the equation x2−2px+y2–6y–p2+8p+9=0x^2 - 2px + y^2 – 6y – p^2 + 8p + 9 = 0 describes a circle in the xy-plane.
  • A.p<−94p<-\frac{9}{4}
  • B.0<p<40<p<4
  • C.−1<p<9-1<p<9
  • D.p<0p<0 or p>4p > 4
  • E.p<−1p<-1 or p>9p>9
  • F.all real values of pp

Answer: D

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Question 3

1 mark
Given the following statements about a function ff
• f′′(x)=af''(x) = a for all xx
• f(0)=1,f(1)=2f(0) = 1, f(1) = 2
• ∫01f(x)dx=1\int_{0}^{1} f(x)dx = 1
find the value of aa.
  • A.-6
  • B.-3
  • C.-2
  • D.2
  • E.3
  • F.6

Answer: F

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Question 4

1 mark
These sectors of circles are similar.
The arc length of the smaller sector is 6.
The difference between the areas of the sectors is 21.
Find the positive difference between the perimeters of the sectors.
Exam diagram
  • A.4.5
  • B.7
  • C.8
  • D.9
  • E.10.5
  • F.14
  • G.15

Answer: C

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Question 5

1 mark
The terms xnx_n of a sequence follow the rule xn+1=xn+pxn+qx_{n+1} = \frac{x_n + p}{x_n + q} where pp and qq are real numbers.
Given that x1=3,x2=5x_1 = 3, x_2 = 5, and x3=7x_3 = 7, find the value of x4x_4
  • A.-5
  • B.5
  • C.517\frac{51}{7}
  • D.152\frac{15}{2}
  • E.233\frac{23}{3}
  • F.9
  • G.11
  • H.13

Answer: H

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Question 6

1 mark
Given that ∫log⁡25log⁡220x dx=log⁡2M\int_{\log_2 5}^{\log_2 20} x \,dx = \log_2 M what is the value of MM?
  • A.4
  • B.15
  • C.16
  • D.20
  • E.25
  • F.100
  • G.10000

Answer: F

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Question 7

1 mark
Find the finite area enclosed between the line y=0y = 0 and the curve y=x2–4∣x∣–12y = x^2 – 4|x| – 12
  • A.1283\frac{128}{3}
  • B.1763\frac{176}{3}
  • C.2563\frac{256}{3}
  • D.108
  • E.144
  • F.288

Answer: E

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Question 8

1 mark
A geometric sequence has first term aa and common ratio rr, where aa and rr are positive integers and rr is greater than 1.
The sum of the first nn terms of this sequence is denoted by SnS_n
It is given that the terms of the sequence satisfy S30−S20=kS10S_{30}-S_{20} = kS_{10} for some positive integer kk.
What is the smallest possible value of kk?
  • A.2102^{10}
  • B.2202^{20}
  • C.2302^{30}
  • D.210210−1\frac{2^{10}}{2^{10}-1}
  • E.210(210−1)2^{10}(2^{10}-1)

Answer: B

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Question 9

1 mark
This question is about pairs of functions ff and gg that satisfy f(x)−g(x)=2sin⁡xf(x) - g(x) = 2\sin x and f(x)g(x)=cos⁡2xf(x) g(x) = \cos^2 x for all real numbers xx.
Across all solutions for f(x)f(x), what is the minimum value that f(x)f(x) attains for any xx?
  • A.1−21-\sqrt{2}
  • B.−1−2-1-\sqrt{2}
  • C.0
  • D.-1
  • E.-2
  • F.-3
  • G.-4

Answer: E

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Question 10

1 mark
A sequence of translations is applied to the graph of y=x3y = x^3
Which of the following graphs could be the result of this sequence of translations?
I y=x3–3x2+9x–27y = x^3 – 3x^2 + 9x – 27
II y=x3–9x2+27x−3y = x^3 – 9x^2 + 27x - 3
III y=27x3–9x2+x−3y = 27x^3 – 9x^2 + x − 3
  • A.none of them
  • B.I only
  • C.II only
  • D.III only
  • E.I and II only
  • F.I and III only
  • G.II and III only
  • H.I, II and III

Answer: C

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Question 11

1 mark
Evaluate ∑n=1100log⁡10(31−n)\sum_{n=1}^{100} \log_{10} (3^{1-n})
  • A.-4950 log⁡103\log_{10} 3
  • B.4950 log⁡103\log_{10} 3
  • C.-5050 log⁡103\log_{10} 3
  • D.5050 log⁡103\log_{10} 3
  • E.1−4950log⁡1031-4950 \log_{10} 3
  • F.1+4950log⁡1031+4950 \log_{10} 3
  • G.1−5050log⁡1031-5050 \log_{10} 3
  • H.1+5050log⁡1031+5050 \log_{10} 3

Answer: A

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Question 12

1 mark
A family of quadratic curves is given by yk=2(x−k2)2+k22+4k+3y_k = 2\left(x - \frac{k}{2}\right)^2 + \frac{k^2}{2} + 4k + 3 where kk is any real number and yky_k is a function of xx.
All these curves are sketched, and the point with the lowest y-coordinate among all the curves yky_k is (a,b)(a,b).
Find the value of a+ba + b
  • A.-1
  • B.-3
  • C.-5
  • D.-7
  • E.-9

Answer: D

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Question 13

1 mark
Given that (a3+2b3)(2a3−b3)=2\left(a^3 + \frac{2}{b^3}\right)\left(\frac{2}{a^3} - b^3\right) = \sqrt{2} where aa and bb are real numbers, what is the least value of abab?
  • A.−2-\sqrt{2}
  • B.2\sqrt{2}
  • C.−22-2\sqrt{2}
  • D.222\sqrt{2}
  • E.−22-\frac{\sqrt{2}}{2}
  • F.22\frac{\sqrt{2}}{2}
  • G.−216-2^{\frac{1}{6}}
  • H.2162^{\frac{1}{6}}

Answer: A

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Question 14

1 mark
A circle has centre OO and radius 6.
P,QP, Q and RR are points on the circumference with angle POQ≥π2POQ \geq \frac{\pi}{2}
The area of the triangle POQPOQ is 939\sqrt{3}
What is the greatest possible area of triangle PRQPRQ?
  • A.18+9318+9\sqrt{3}
  • B.18318\sqrt{3}
  • C.27+9327+9\sqrt{3}
  • D.27327\sqrt{3}
  • E.36+9336+9\sqrt{3}
  • F.36336\sqrt{3}

Answer: D

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Question 15

1 mark
A rectangle is drawn in the region enclosed by the curves pp and qq, where p(x)=8−2x2p(x) = 8 - 2x^2 and q(x)=x2−2q(x) = x^2 - 2 such that the sides of the rectangle are parallel to the x- and y-axes.
What is the maximum possible area of the rectangle?
  • A.269\frac{26}{9}
  • B.529\frac{52}{9}
  • C.463\frac{4\sqrt{6}}{3}
  • D.863\frac{8\sqrt{6}}{3}
  • E.424\sqrt{2}
  • F.828\sqrt{2}
  • G.20109\frac{20\sqrt{10}}{9}
  • H.40109\frac{40\sqrt{10}}{9}

Answer: H

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Question 16

1 mark
The solutions to 7x4−6x2+1=07x^4 - 6x^2 + 1 = 0 are ±cos⁡θ\pm\cos\theta and ±cos⁡β\pm\cos\beta.
Which one of the following equations has solutions ±sin⁡θ\pm\sin\theta and ±sin⁡β\pm\sin\beta?
  • A.7x4−8x2−5=07x^4-8x^2- 5 = 0
  • B.7x4−8x2+2=07x^4-8x^2 + 2 = 0
  • C.7x4−6x2−2=07x^4-6x^2-2 = 0
  • D.7x4−6x2+1=07x^4-6x^2 + 1 = 0
  • E.7x4+6x2–1=07x^4 + 6x^2 – 1 = 0
  • F.7x4+6x2+5=07x^4 + 6x^2 + 5 = 0

Answer: B

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Question 17

1 mark
Find the complete set of values of xx for which there are two non-congruent triangles with the side lengths and angle as shown in the diagram.
Exam diagram
  • A.1<x<31<x<3
  • B.1<x<41<x<4
  • C.1<x<51<x<5
  • D.3<x<43<x<4
  • E.3<x<53<x<5
  • F.4<x<54<x<5

Answer: D

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Question 18

1 mark
It is given that f(x)=x2(x–1)2(x–2)f(x) = x^2(x – 1)^2(x – 2) and g(x)=−p(x–q)2(x–r)2g(x) = -p(x – q)^2(x – r)^2 where p,qp, q and rr are positive and q<rq<r.
Find the set of values of qq and rr that guarantees the greatest number of distinct real solutions of the equation f(x)=g(x)f(x) = g(x) for all pp.
  • A.q<1q<1 and r<1r<1
  • B.q<1q<1 and 1<r<21<r<2
  • C.q<1q<1 and r>2r>2
  • D.1<q<21<q<2 and 1<r<21<r<2
  • E.1<q<21<q<2 and r>2r>2
  • F.q>2q>2 and r>2r>2

Answer: B

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Question 19

1 mark
Circle C1C_1 is defined as x2+y2=25x^2 + y^2 = 25
A second circle C2C_2 has radius 4 and centre (a,b)(a, b) where −2≤a≤2-2\leq a \leq 2 and −3≤b≤3-3\leq b \leq 3
If the centre of C2C_2 is equally likely to be located anywhere within the given range, what is the probability that C2C_2 intersects C1C_1?
  • A.125\frac{1}{25}
  • B.925\frac{9}{25}
  • C.1625\frac{16}{25}
  • D.6−π6\frac{6-\pi}{6}
  • E.16–π24\frac{16 – \pi}{24}
  • F.24–π24\frac{24 – \pi}{24}

Answer: F

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Question 20

1 mark
nn is the number of points of intersection of the graphs y=∣x2−a2∣y = |x^2 - a^2| and y=a2∣x−1∣y = a^2 |x − 1| where aa is a real number.
What is the smallest value of nn that is not possible?
  • A.n=1n=1
  • B.n=2n=2
  • C.n=3n=3
  • D.n=4n=4
  • E.n=5n=5

Answer: B

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