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TMUA Beyond Horizon Set 4 Paper 1

20 questions20 marks75Updated July 2026

The TMUA Beyond Horizon Set 4 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

Find the area bounded by the straight lines x=−1x = -1 and x=1x = 1 and the graphs of f(x)f(x) and g(x)g(x), where f(x)=x3f(x) = x^3 and g(x)={x5if −1≤x≤0, xif 0≤x≤1,g(x) = \begin{cases} x^5 & \text{if } -1 \le x \le 0, \ x & \text{if } 0 \le x \le 1, \end{cases}
  • A.13\frac{1}{3}
  • B.18\frac{1}{8}
  • C.12\frac{1}{2}
  • D.14\frac{1}{4}
  • E.15\frac{1}{5}
  • F.16\frac{1}{6}

Answer: A

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Question 2

The largest volume of a cube that can be enclosed in a sphere of diameter 2 cm, is, in cm3^3,
  • A.1
  • B.222\sqrt{2}
  • C.π\pi
  • D.833\frac{8}{3\sqrt{3}}
  • E.89\frac{8}{9}
  • F.223\frac{2\sqrt{2}}{3}

Answer: D

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Question 3

If α,β\alpha, \beta are the roots of ax2+bx+c=0ax^2 + bx + c = 0, then the equation whose roots are α2,β2\alpha^2, \beta^2 is
  • A.a2x2+(b2−2ac)x+c2=0a^2x^2 + (b^2 - 2ac)x + c^2 = 0
  • B.a2x2−(b2+2ac)x+c2=0a^2x^2 - (b^2 + 2ac)x + c^2 = 0
  • C.a2x2−(b2−2ac)x+c2=0a^2x^2 - (b^2 - 2ac)x + c^2 = 0
  • D.a2x2−(a2−2bc)x+c2=0a^2x^2 - (a^2 - 2bc)x + c^2 = 0
  • E.b2x2−(b2−2ac)x+c2=0b^2x^2 - (b^2 - 2ac)x + c^2 = 0
  • F.none of the above equations

Answer: C

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Question 4

Evaluate the following sum
(1⋅1!)+(2⋅2!)+(3⋅3!)+⋯+(50⋅50!)(1 \cdot 1!) + (2 \cdot 2!) + (3 \cdot 3!) + \dots + (50 \cdot 50!)
  • A.51!51!
  • B.2.5!2.5!
  • C.51!−151! - 1
  • D.51!+151! + 1
  • E.50!+150! + 1
  • F.50!+150! + 1

Answer: C

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Question 5

The number of distinct points common to the curves x2+4y2=1x^2 + 4y^2 = 1 and 4x2+y2=44x^2 + y^2 = 4 is
  • A.0
  • B.1
  • C.2
  • D.3
  • E.4
  • F.5

Answer: C

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Question 6

In a parallelogram PQRSPQRS, PP is the point (−1,−1)(-1, -1), QQ is (8,0)(8, 0) and RR is (7,5)(7, 5). Then SS is the point
  • A.(−1,4)(-1, 4)
  • B.(−2,4)(-2, 4)
  • C.(−2,72)(-2, \frac{7}{2})
  • D.(−32,4)(-\frac{3}{2}, 4)
  • E.(−2,5)(-2, 5)
  • F.(−1,5)(-1, 5)

Answer: B

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Question 7

How many real solutions are there to the equation
2tan⁡4x−5tan⁡2x+3=02 \tan^4 x - 5 \tan^2 x + 3 = 0
in the interval 0≤x<2π0 \le x < 2\pi?
  • A.0
  • B.1
  • C.2
  • D.3
  • E.4
  • F.5
  • G.6
  • H.7
  • I.8

Answer: 8

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Question 8

Given that dydx=4x2−3−2xx3,x≠0\frac{dy}{dx} = 4x^2 - \frac{3 - 2x}{x^3}, x \neq 0 and y=6y = 6 when x=1x = 1, find yy in terms of xx.
  • A.y=43x3+32x−2−23x−1+316y = \frac{4}{3}x^3 + \frac{3}{2}x^{-2} - \frac{2}{3}x^{-1} + \frac{31}{6}
  • B.y=2x3+x−2−x−1+5y = 2x^3 + x^{-2} - x^{-1} + 5
  • C.y=43x3+32x−2−2x−3+236y = \frac{4}{3}x^3 + \frac{3}{2}x^{-2} - 2x^{-3} + \frac{23}{6}
  • D.y=43x3+32x−2−2x−1+316y = \frac{4}{3}x^3 + \frac{3}{2}x^{-2} - 2x^{-1} + \frac{31}{6}
  • E.y=43x3+32x−2−23x−3+4y = \frac{4}{3}x^3 + \frac{3}{2}x^{-2} - \frac{2}{3}x^{-3} + 4
  • F.y=3x3+x−2−x−1+3y = 3x^3 + x^{-2} - x^{-1} + 3

Answer: D

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Question 9

Evaluate
∑n=150log⁡10(21−n)\sum_{n=1}^{50} \log_{10} (2^{1-n})
  • A.−495log⁡102-495 \log_{10} 2
  • B.495log⁡102495 \log_{10} 2
  • C.−1225log⁡102-1225 \log_{10} 2
  • D.1225log⁡1021225 \log_{10} 2
  • E.1−495log⁡1021 - 495 \log_{10} 2
  • F.1+495log⁡1021 + 495 \log_{10} 2
  • G.1−1225log⁡1021 - 1225 \log_{10} 2
  • H.1+1225log⁡1021 + 1225 \log_{10} 2

Answer: C

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Question 10

In the expansion of (a+bx)4(a+bx)^4 the coefficient of x3x^3 is 4 times the coefficient of x2x^2. Given that aa and bb are non-zero positive integers, what is the smallest possible value of a+ba + b?
  • A.5
  • B.6
  • C.7
  • D.11
  • E.15
  • F.19

Answer: C

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Question 11

The functions ff and gg are given by f(x)=4x2+10x+5f(x) = 4x^2 + 10x + 5 and g(x)=x3+5x2+8x−10g(x) = x^3 + 5x^2 + 8x - 10.
What is the complete set of values of xx for which one of the functions is increasing and the other decreasing?
  • A.x≥−2x \ge -2
  • B.x≤−2x \le -2
  • C.−4≤x≤−3,x≥−2-4 \le x \le -3, x \ge -2
  • D.x≤−3,x≥−2x \le -3, x \ge -2
  • E.x≤−4,−3≤x≤−2x \le -4, -3 \le x \le -2
  • F.x≤−4,x≥−3x \le -4, x \ge -3
  • G.−3≤x≤−2-3 \le x \le -2

Answer: A

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Question 12

Find the value of
∫154−3xxxdx\int_1^5 \frac{4 - 3x}{x\sqrt{x}} dx
  • A.145−385\frac{14\sqrt{5}-38}{\sqrt{5}}
  • B.−9516-\frac{95}{16}
  • C.155−385\frac{15\sqrt{5}-38}{\sqrt{5}}
  • D.−2-2
  • E.−12-\frac{1}{2}
  • F.94\frac{9}{4}
  • G.145−383\frac{14\sqrt{5}-38}{\sqrt{3}}

Answer: A

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Question 13

Find the number of solutions of the equation
xsin⁡3x=cos⁡xx \sin 3x = \cos x
in the interval 0≤x≤3π0 \le x \le 3\pi.
  • A.1
  • B.3
  • C.5
  • D.7
  • E.9

Answer: D

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Question 14

The number of ways in which three non-negative integers n1,n2,n3n_1, n_2, n_3 can be chosen such that n1+n2+n3=10n_1 + n_2 + n_3 = 10 is
  • A.66
  • B.55
  • C.10310^3
  • D.10!3!2!1!\frac{10!}{3!2!1!}
  • E.9!3!2!1!\frac{9!}{3!2!1!}
  • F.10210^2

Answer: A

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Question 15

Find the complete set of values of the constant cc for which the cubic equation
x3−3x+c=0x^3 - 3x + c = 0
has three distinct real solutions.
  • A.0<c<20 < c < 2
  • B.−2<c<0-2 < c < 0
  • C.−2<c<2-2 < c < 2
  • D.c>−2c > -2
  • E.c<2c < 2
  • F.c<0c < 0

Answer: C

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Question 16

In the interval 0≤x≤2π0 \le x \le 2\pi, the equation
sin⁡(3cos⁡(3x)+1)=0\sin(3 \cos(3x) + 1) = 0
has exactly
  • A.2 solutions
  • B.4 solutions
  • C.6 solutions
  • D.8 solutions
  • E.10 solutions
  • F.12 solutions

Answer: f

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Question 17

An equilateral triangle has centre OO and side length 2. A straight line through OO intersects the triangle at two distinct points PP and QQ. The minimum possible length of PQPQ is
  • A.23\frac{2}{3}
  • B.12\frac{1}{2}
  • C.233\frac{2\sqrt{3}}{3}
  • D.43\frac{4}{3}
  • E.3\sqrt{3}
  • F.2\sqrt{2}

Answer: D

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Question 18

What is the area of the polygon formed by all the points (x,y)(x, y) in the plane satisfying the inequality
∣x−2∣+∣y−2∣≤8?|x - 2| + |y - 2| \le 8?
  • A.32
  • B.64
  • C.96
  • D.128
  • E.160

Answer: D

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Question 19

Let AA be a point on the circle with the equation x2+y2=4x^2 + y^2 = 4, and let BB be a point on the circle with the equation x2+y2=9x^2 + y^2 = 9. If the angle between the vectors representing the points AA and BB is 30∘30^\circ, what is the distance between the points AA and BB?
  • A.13\sqrt{13}
  • B.636\sqrt{3}
  • C.3\sqrt{3}
  • D.13−63\sqrt{13 - 6\sqrt{3}}
  • E.13+63\sqrt{13 + 6\sqrt{3}}

Answer: D

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Question 20

What is the degree of the following polynomial
d2dx2[(3x−2)4(2−x)5]+ddx[(3x+2)4(4x2−3)2]\frac{d^2}{dx^2} [(3x - 2)^4 (2 - x)^5] + \frac{d}{dx} [(3x + 2)^4 (4x^2 - 3)^2]
  • A.11
  • B.10
  • C.9
  • D.8
  • E.7
  • F.less than 7

Answer: E

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