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TMUA 2019 D513/01

20 questions20 marks75Updated July 2025

The TMUA 2019 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
f(x)f(x) is a quadratic function in xx. The graph of y=f(x)y = f(x) passes through the point (1,−1)(1, -1) and has a turning point at (−1,3)(-1, 3). Find an expression for f(x)f(x).
  • A.−x2−2x+2-x^2 - 2x + 2
  • B.−x2+2x+3-x^2 + 2x + 3
  • C.x2−2xx^2 - 2x
  • D.x2+2x−4x^2 + 2x - 4
  • E.2x2+4x+12x^2 + 4x + 1
  • F.−2x2−4x+5-2x^2 - 4x + 5

Answer: A

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Question 2

1 mark
Find the complete set of values of the real constant kk for which the expression x2+kx+2x+1−2kx^2 + kx + 2x + 1 - 2k is positive for all real values of xx.
  • A.−12<k<0-12 < k < 0
  • B.k<−12k < -12 or k>0k > 0
  • C.−6−3<k<6−3-\sqrt{6} - 3 < k < \sqrt{6} - 3
  • D.k<−6−3k < -\sqrt{6} - 3 or k>6−3k > \sqrt{6} - 3
  • E.−2<k<12-2 < k < \frac{1}{2}
  • F.k<−2k < -2 or k>12k > \frac{1}{2}
  • G.0<k<40 < k < 4
  • H.k<0k < 0 or k>4k > 4

Answer: A

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Question 3

1 mark
Find the coefficient of xx in the expression: (1+x)0+(1+x)1+(1+x)2+(1+x)3+⋯+(1+x)79+(1+x)80(1 + x)^0 + (1 + x)^1 + (1 + x)^2 + (1 + x)^3 + \dots + (1+x)^{79} + (1 + x)^{80}
  • A.80
  • B.81
  • C.324
  • D.628
  • E.3240
  • F.3321
  • G.6480
  • H.6642

Answer: E

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Question 4

1 mark
The sequence xnx_n is given by: x1=10x_1 = 10, xn+1=xnx_{n+1} = \sqrt{x_n} for n≥1n \geq 1. What is the value of x100x_{100}? [Note that abca^{b^{c}} means a(bc)a^{(b^c)}]
  • A.1029910^{2^{99}}
  • B.10210010^{2^{100}}
  • C.102−9910^{2^{-99}}
  • D.102−10010^{2^{-100}}
  • E.10−29910^{-2^{99}}
  • F.10−210010^{-2^{100}}
  • G.10−2−9910^{-2^{-99}}
  • H.10−2−10010^{-2^{-100}}

Answer: C

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Question 5

1 mark
SS is a geometric sequence.
The sum of the first 6 terms of SS is equal to 9 times the sum of the first 3 terms of SS. The 7th7^{th} term of SS is 360. Find the 1st1^{st} term of SS.
  • A.4027\frac{40}{27}
  • B.409\frac{40}{9}
  • C.403\frac{40}{3}
  • D.4516\frac{45}{16}
  • E.458\frac{45}{8}
  • F.454\frac{45}{4}

Answer: E

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Question 6

1 mark
The circles with equations
(x+4)2+(y+1)2=64(x + 4)^2 + (y + 1)^2 = 64 and
(x−8)2+(y−4)2=r2(x - 8)^2 + (y - 4)^2 = r^2 where r>0r>0
have exactly one point in common. Find the difference between the two possible values of rr.
  • A.4
  • B.10
  • C.16
  • D.26
  • E.50

Answer: C

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Question 7

1 mark
A curve has equation
y=(2q−x2)(2qx+3)y = (2q - x^2)(2qx + 3)
The gradient of the curve at x=−1x = -1 is a function of qq.
Find the value of qq which minimises the gradient of the curve at x=−1x = -1.
  • A.-1
  • B.−34-\frac{3}{4}
  • C.−12-\frac{1}{2}
  • D.0
  • E.12\frac{1}{2}
  • F.34\frac{3}{4}
  • G.1

Answer: F

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Question 8

1 mark
The function ff is such that 0<f(x)<10 < f(x) < 1 for 0≤x≤10 \leq x \leq 1. The trapezium rule with nn equal intervals is used to estimate ∫01f(x) dx\int_0^1 f(x) \,dx and produces an underestimate. Using the same number of equal intervals, for which one of the following does the trapezium rule produce an overestimate?
  • A.∫01(f(x)+1) dx\int_0^1 (f(x) + 1) \,dx
  • B.∫012f(x) dx\int_0^1 2f(x) \,dx
  • C.∫−10f(x+1) dx\int_{-1}^0 f(x + 1) \,dx
  • D.∫−10f(−x) dx\int_{-1}^0 f(-x) \,dx
  • E.∫01(1−f(x)) dx\int_0^1 (1 - f(x)) \,dx

Answer: E

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Question 9

1 mark
pp is a positive constant.
Find the area enclosed between the curves y=pxy = p\sqrt{x} and x=pyx = p\sqrt{y}.
  • A.23p52−12p2\frac{2}{3}p^{\frac{5}{2}} - \frac{1}{2}p^2
  • B.43p52−p2\frac{4}{3}p^{\frac{5}{2}} - p^2
  • C.p46\frac{p^4}{6}
  • D.p43\frac{p^4}{3}
  • E.23p3−12p4\frac{2}{3}p^3 - \frac{1}{2}p^4
  • F.43p3−p4\frac{4}{3}p^3 - p^4
  • G.2p42p^4

Answer: D

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Question 10

1 mark
Evaluate ∫−13∣x∣(1−x) dx\int_{-1}^3 |x|(1-x)\,dx
  • A.173\frac{17}{3}
  • B.−173-\frac{17}{3}
  • C.163\frac{16}{3}
  • D.−163-\frac{16}{3}
  • E.113\frac{11}{3}
  • F.−113-\frac{11}{3}

Answer: F

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Question 11

1 mark
Find the sum of the real values of xx that satisfy the simultaneous equations:
log⁡3(xy2)=1\log_3(xy^2) = 1 and (log⁡3x)(log⁡3y)=−3(\log_3 x)(\log_3 y) = -3
  • A.13\frac{1}{3}
  • B.1
  • C.3
  • D.3193\frac{1}{9}
  • E.91279\frac{1}{27}
  • F.9139\frac{1}{3}
  • G.27
  • H.271927\frac{1}{9}

Answer: H

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Question 12

1 mark
It is given that dVdt=24π(t−1)(1+t)\frac{dV}{dt} = \frac{24\pi(t - 1)}{(1+\sqrt{t})} for t≥1t \geq 1 and V=7V = 7 when t=1t = 1. Find the value of VV when t=9t = 9.
  • A.208π+7208\pi + 7
  • B.216π+7216\pi + 7
  • C.224π+7224\pi + 7
  • D.416π+7416\pi + 7
  • E.608π+7608\pi + 7
  • F.744π+7744\pi + 7

Answer: C

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Question 13

1 mark
Find the maximum value of 4sin⁡x−4×2sin⁡x+1744^{\sin x} - 4 \times 2^{\sin x} + \frac{17}{4} for real xx.
  • A.14\frac{1}{4}
  • B.52\frac{5}{2}
  • C.132\frac{13}{2}
  • D.212\frac{21}{2}
  • E.654\frac{65}{4}
  • F.There is no maximum value.

Answer: B

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Question 14

1 mark
xx satisfies the simultaneous equations
sin⁡2x+3cos⁡2x=−1\sin 2x + \sqrt{3} \cos 2x = -1
and
3sin⁡2x−cos⁡2x=3\sqrt{3} \sin 2x - \cos 2x = \sqrt{3}
where 0∘≤x≤360∘0^{\circ} \leq x \leq 360^{\circ}.
Find the sum of the possible values of xx.
  • A.210∘210^{\circ}
  • B.330∘330^{\circ}
  • C.390∘390^{\circ}
  • D.660∘660^{\circ}
  • E.780∘780^{\circ}
  • F.930∘930^{\circ}

Answer: B

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Question 15

1 mark
Find the real non-zero solution to the equation 2(9x)8(3x)=14\frac{2^{(9^x)}}{8^{(3^x)}} = \frac{1}{4}
  • A.log⁡32\log_3 2
  • B.2log⁡322\log_3 2
  • C.1
  • D.2
  • E.log⁡23\log_2 3
  • F.2log⁡232\log_2 3

Answer: A

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Question 16

1 mark
Given that
2∫01f(x) dx+5∫12f(x) dx=142 \int_0^1 f(x) \,dx + 5 \int_1^2 f(x) \,dx = 14 and ∫01f(x+1) dx=6\int_0^1 f(x + 1) \,dx = 6 find the value of, ∫02f(x) dx\int_0^2 f(x) \,dx.
  • A.-8
  • B.-4
  • C.-2
  • D.2
  • E.4
  • F.295\frac{29}{5}
  • G.325\frac{32}{5}
  • H.14

Answer: C

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Question 17

1 mark
Find the fraction of the interval 0≤θ≤π0 \leq \theta \leq \pi for which the inequality (sin⁡(2θ)−12)(sin⁡θ−cos⁡θ)≥0(\sin(2\theta) - \frac{1}{2})(\sin \theta - \cos \theta) \geq 0 is satisfied.
  • A.112\frac{1}{12}
  • B.16\frac{1}{6}
  • C.14\frac{1}{4}
  • D.512\frac{5}{12}
  • E.712\frac{7}{12}
  • F.34\frac{3}{4}
  • G.56\frac{5}{6}
  • H.1112\frac{11}{12}

Answer: C

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Question 18

1 mark
Find the shortest distance between the curve y=x2+4y = x^2 + 4 and the line y=2x−2y = 2x - 2.
  • A.2
  • B.5\sqrt{5}
  • C.655\frac{6\sqrt{5}}{5}
  • D.3
  • E.553\frac{5\sqrt{5}}{3}
  • F.5
  • G.6

Answer: B

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Question 19

1 mark
Find the value of ∑k=090sin⁡(10+90k)∘\sum_{k=0}^{90} \sin(10 + 90k)^{\circ}
  • A.0
  • B.sin⁡10∘\sin 10^{\circ}
  • C.sin⁡100∘\sin 100^{\circ}
  • D.sin⁡190∘\sin 190^{\circ}
  • E.sin⁡280∘\sin 280^{\circ}
  • F.1

Answer: C

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Question 20

1 mark
What is the complete range of values of kk for which the curves with equations y=x3−12xy = x^3 - 12x and y=k−(x−2)2y = k - (x - 2)^2 intersect at three distinct points, of which exactly two have positive xx-coordinates?
  • A.−4<k<0-4 < k < 0
  • B.−4<k<4-4 < k < 4
  • C.−4<k<16-4 < k < 16
  • D.−16<k<0-16 < k < 0
  • E.−16<k<4-16 < k < 4
  • F.−16<k<16-16 < k < 16

Answer: E

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