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TMUA Beyond Horizon Set 2 Paper 1

20 questions20 marks75Updated July 2026

The TMUA Beyond Horizon Set 2 Paper 1 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 solutions does the equation xtan⁡x=2x \tan x = 2 have in the interval −3π≤x≤3π-3\pi \le x \le 3\pi?
  • A.1
  • B.2
  • C.3
  • D.4
  • E.5
  • F.6
  • G.7

Answer: E

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Question 2

1 mark
The least possible value of the gradient of the curve y=(3x+b)(x−b)2y = (3x + b)(x - b)^2 at the point where x=2x = 2, as bb varies, is:
  • A.−814-\frac{81}{4}
  • B.−64-64
  • C.−364-\frac{36}{4}
  • D.54\frac{5}{4}
  • E.5316\frac{53}{16}

Answer: B

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Question 3

1 mark
It is given that y=(1+cos⁡x)sin⁡3xy = (1 + \cos x) \sin 3x for 0<x<π0 < x < \pi. The complete set of values of xx for which yy is negative is:
  • A.0<x<π60 < x < \frac{\pi}{6}
  • B.0<x<π30 < x < \frac{\pi}{3}
  • C.π3<x<π2\frac{\pi}{3} < x < \frac{\pi}{2}
  • D.2π3<x<π\frac{2\pi}{3} < x < \pi
  • E.0<x<π20 < x < \frac{\pi}{2}
  • F.π6<x<π2\frac{\pi}{6} < x < \frac{\pi}{2}

Answer: C

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Question 4

1 mark
The function 2−x2x23\frac{2-x}{2 \sqrt[3]{x^2}} is defined for all x≠0x \neq 0. The complete set of values of xx for which the function is decreasing is
  • A.x≤−4,x>0x \le -4, x > 0
  • B.−1≤x<0-1 \le x < 0
  • C.x≤2x \le 2
  • D.x≥2x \ge 2
  • E.−1≤x≤2-1 \le x \le 2
  • F.x≤−1,x≥2x \le -1, x \ge 2

Answer: A

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Question 5

1 mark
Find the complete set of values of pp for which the equation x2−3px+y2−4y−p2+6p+7=0x^2 - 3px + y^2 - 4y - p^2 + 6p + 7 = 0 describes a circle in the xyxy-plane.
  • A.p<−73p < -\frac{7}{3}
  • B.p<12−10513p < \frac{12-10\sqrt{5}}{13} or p>12+10513p > \frac{12+10\sqrt{5}}{13}
  • C.p<10−10313p < \frac{10-10\sqrt{3}}{13} or p>10+10313p > \frac{10+10\sqrt{3}}{13}
  • D.p<12−10313p < \frac{12-10\sqrt{3}}{13} or p>12+10313p > \frac{12+10\sqrt{3}}{13}
  • E.p<−2p < -2 or p>8p > 8
  • F.All real values of pp

Answer: D

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Question 6

1 mark
Find the finite area enclosed between the line y=0y = 0 and the curve y=x2−5∣x∣−24y = x^2 - 5|x| - 24.
  • A.5443\frac{544}{3}
  • B.2643\frac{264}{3}
  • C.10883\frac{1088}{3}
  • D.162
  • E.216
  • F.432

Answer: C

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Question 7

1 mark
For any real number xx, let ⌊x⌋\lfloor x \rfloor denote the greatest integer mm such that m≤xm \le x. Evaluate the following integral:
∫−22⌊x2−1⌋dx\int_{-2}^{2} \lfloor x^2 - 1 \rfloor dx
  • A.2(3−3−2)2(3 - \sqrt{3} - \sqrt{2})
  • B.2(5−3−2)2(5 - \sqrt{3} - \sqrt{2})
  • C.2(1−3−2)2(1 - \sqrt{3} - \sqrt{2})
  • D.3(5−3−2)3(5 - \sqrt{3} - \sqrt{2})
  • E.3(1−5−3)3(1 - \sqrt{5} - \sqrt{3})
  • F.None of the above

Answer: A

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Question 8

1 mark
This question is about pairs of functions ff and gg that satisfy
f(x)−g(x)=3sin⁡xf(x) - g(x) = 3 \sin x
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.1−31 - \sqrt{3}
  • B.−1−3-1 - \sqrt{3}
  • C.0
  • D.−1-1
  • E.−2-2
  • F.−3-3
  • G.−4-4

Answer: F

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Question 9

1 mark
This sequence of transformations is applied to the curve y=3x2y = 3x^2:
1. Translation by (2 −4)\begin{pmatrix} 2 \ -4 \end{pmatrix}
2. Reflection in the yy-axis
3. Stretch parallel to the yy-axis with scale factor 3
What is the equation of the resulting curve?
  • A.y=−3x2+8x−12y = -3x^2 + 8x - 12
  • B.y=−3x2+8x−6y = -3x^2 + 8x - 6
  • C.y=9x2−16x+12y = 9x^2 - 16x + 12
  • D.y=9x2−36x+24y = 9x^2 - 36x + 24
  • E.y=−27x2+24x−36y = -27x^2 + 24x - 36
  • F.y=−27x2+24x−18y = -27x^2 + 24x - 18

Answer: D

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Question 10

1 mark
Given that
(a3+3b3)(3a3−b3)=8\left( a^3 + \frac{3}{b^3} \right) \left( \frac{3}{a^3} - b^3 \right) = 8
where aa and bb are real numbers, what is the least value of abab?
  • A.−3-\sqrt{3}
  • B.3\sqrt{3}
  • C.−23-2\sqrt{3}
  • D.232\sqrt{3}
  • E.−913-9^{\frac{1}{3}}
  • F.9139^{\frac{1}{3}}
  • G.1
  • H.-1

Answer: E

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Question 11

1 mark
A hollow right circular cone rests on a sphere. The height of the cone is 4 metres and the radius of the base is 1 metre. The volume of the sphere is the same as that of the cone. Then, the distance between the centre of the sphere and the vertex of the cone is
  • A.4 metres
  • B.17\sqrt{17} metres
  • C.15\sqrt{15} metres
  • D.5 metres
  • E.6 metres
  • F.13\sqrt{13} metres

Answer: D

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Question 12

1 mark
In the range 0≤x<2π0 \le x < 2\pi, the equation
4sin⁡2x+4cos⁡x=924 \sin^2 x + 4 \cos x = \frac{9}{2}
  • A.has no solutions
  • B.has 1 solution
  • C.has 2 solutions
  • D.has 3 solutions
  • E.has 4 solutions
  • F.has 5 solutions

Answer: E

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Question 13

1 mark
Consider a rectangular cardboard box of height 3, breadth 4, and length 10 units. There is a lizard in one corner AA of the box and an insect in the corner BB, which is farthest from AA. The length of the shortest path between the lizard and the insect along the surface of the box is
  • A.52+102\sqrt{5^2 + 10^2} units
  • B.72+102\sqrt{7^2 + 10^2} units
  • C.4+32+1024 + \sqrt{3^2 + 10^2} units
  • D.3+102+423 + \sqrt{10^2 + 4^2} units
  • E.5+102+725 + \sqrt{10^2 + 7^2} units
  • F.6+102+526 + \sqrt{10^2 + 5^2} units

Answer: B

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Question 14

1 mark
The equation in xx
8x4−16x3+8x2+k=08x^4 - 16x^3 + 8x^2 + k = 0
has four real solutions:
  • A.when −30<k<4-30 < k < 4
  • B.when 4<k<304 < k < 30
  • C.when −2.5<k<0-2.5 < k < 0
  • D.when 0<k<2.50 < k < 2.5
  • E.when 0<k<0.50 < k < 0.5
  • F.when −0.5<k<0-0.5 < k < 0

Answer: F

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Question 15

1 mark
The number of pairs of positive integers x,yx, y which solve the equation
x3+6x2y+12xy2+8y3=227x^3 + 6x^2y + 12xy^2 + 8y^3 = 2^{27}
is
  • A.0
  • B.252^5
  • C.28−12^8 - 1
  • D.29+22^9 + 2
  • E.272^7
  • F.26−12^6 - 1
  • G.28+12^8 + 1

Answer: C

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Question 16

1 mark
Evaluate the following integral
∫0100ex−⌊x⌋dx\int_0^{100} e^{x-\lfloor x \rfloor}dx
  • A.e100−1100\frac{e^{100}-1}{100}
  • B.e100−1e−1\frac{e^{100}-1}{e-1}
  • C.100(e−1)100(e - 1)
  • D.e−1100\frac{e-1}{100}
  • E.1−e100\frac{1-e}{100}
  • F.e100100\frac{e^{100}}{100}

Answer: C

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Question 17

1 mark
Consider six players P1,P2,P3,P4,P5P_1, P_2, P_3, P_4, P_5, and P6P_6. A team consists of two players (Thus, there are 15 distinct teams). Two teams play a match exactly once if there is no common player. For example, team {P1,P2}\{P_1, P_2\} cannot play with {P2,P3}\{P_2, P_3\}, but will play with {P4,P5}\{P_4, P_5\}. Then the total number of possible matches is
  • A.36
  • B.40
  • C.45
  • D.54
  • E.55
  • F.60

Answer: C

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Question 18

1 mark
If (log⁡3x)(log⁡4x)(log⁡5x)=(log⁡3x)(log⁡4x)+(log⁡4x)(log⁡5x)+(log⁡5x)(log⁡3x)(\log_3 x)(\log_4 x)(\log_5 x) = (\log_3 x)(\log_4 x) + (\log_4 x)(\log_5 x) + (\log_5 x)(\log_3 x) and x≠1x \neq 1, then xx is
  • A.10
  • B.100
  • C.50
  • D.60
  • E.80
  • F.90

Answer: D

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Question 19

1 mark
The smallest value of
I(a)=∫01(x2−2a)2dxI(a) = \int_0^1 (x^2 - 2a)^2 dx,
as aa varies, is
  • A.521\frac{5}{21}
  • B.641\frac{6}{41}
  • C.823\frac{8}{23}
  • D.1
  • E.922\frac{9}{22}
  • F.445\frac{4}{45}

Answer: F

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Question 20

1 mark
Let f(x)=(tan⁡x)3/2−3tan⁡x+tan⁡xf(x) = (\tan x)^{3/2} - 3 \tan x + \sqrt{\tan x}. Consider the three integrals:
I1=∫01f(x)dxI_1 = \int_0^1 f(x)dx, I2=∫0.31.3f(x)dxI_2 = \int_{0.3}^{1.3} f(x)dx, I3=∫0.51.5f(x)dxI_3 = \int_{0.5}^{1.5} f(x)dx
Then,
  • A.I1>I2>I3I_1 > I_2 > I_3
  • B.I2>I1>I3I_2 > I_1 > I_3
  • C.I3>I1>I2I_3 > I_1 > I_2
  • D.I1>I3>I2I_1 > I_3 > I_2
  • E.I2>I3>I1I_2 > I_3 > I_1
  • F.I3>I2>I1I_3 > I_2 > I_1

Answer: D

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