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TMUA Beyond Horizon Set 3 Paper 1

20 questions75Updated July 2026

The TMUA Beyond Horizon Set 3 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

Consider the inequality x+3−x−1≥1\sqrt{x + 3} - \sqrt{x - 1} \ge 1. Find the range of xx that satisfies this inequality.
  • A.[−2,3.25][-2, 3.25]
  • B.[−1,2.5][-1, 2.5]
  • C.[1,3][1, 3]
  • D.[2,3.5][2, 3.5]
  • E.[−1,2][-1, 2]
  • F.[0,3][0, 3]
  • G.[1,3.25][1, 3.25]

Answer: G

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Question 2

Let d1,d2,…,dkd_1, d_2, \dots, d_k be all the factors of a positive integer nn including 1 and nn. Suppose d1+d2+⋯+dk=72d_1 + d_2 + \dots + d_k = 72. Find the value of
1d1+1d2+⋯+1dk\frac{1}{d_1} + \frac{1}{d_2} + \dots + \frac{1}{d_k}
  • A.k272\frac{k^2}{72}
  • B.72k\frac{72}{k}
  • C.72n\frac{72}{n}
  • D.72k2\frac{72}{k^2}
  • E.72n2\frac{72}{n^2}
  • F.cannot be found from the given information

Answer: C

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Question 3

How many values of xx satisfy the equation
3cos⁡2x+4sin⁡x=33 \cos^2 x + 4 \sin x = 3\nin the range 0≤x<2π0 \le x < 2\pi?
  • A.0
  • B.1
  • C.2
  • D.3
  • E.4
  • F.5

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Question 4

Which power of xx which has the greatest coefficient in the expansion of
(1+23x)12\left(1 + \frac{2}{3}x\right)^{12}
  • A.x3x^3
  • B.x4x^4
  • C.x5x^5
  • D.x6x^6
  • E.x7x^7
  • F.x8x^8

Answer: C

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Question 5

Which of the following numbers is the largest in value?
  • A.tan⁡(5π4)\tan \left(\frac{5\pi}{4}\right)
  • B.sin⁡2(5π4)\sin^2 \left(\frac{5\pi}{4}\right)
  • C.log⁡10(5π4)\log_{10} \left(\frac{5\pi}{4}\right)
  • D.log⁡2(5π4)\log_2 \left(\frac{5\pi}{4}\right)
  • E.log⁡3(5π4)\log_3 \left(\frac{5\pi}{4}\right)
  • F.tan⁡(3π4)\tan \left(\frac{3\pi}{4}\right)

Answer: D

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Question 6

If 2log⁡(α−2β)=log⁡(α)+log⁡(β)2 \log(\alpha - 2\beta) = \log(\alpha) + \log(\beta), find the value of αβ\frac{\alpha}{\beta}.
  • A.1
  • B.2
  • C.3
  • D.4
  • E.1 and 4
  • F.2 and 3

Answer: D

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Question 7

Three boys of class I, 4 boys of class II and 5 boys of class III sit in a row. The number of ways they can sit, so that boys of the same class sit together is
  • A.3!4!5!3!4!5!
  • B.(12)!3!4!5!\frac{(12)!}{3!4!5!}
  • C.(3!)24!5!(3!)^2 4! 5!
  • D.3×4!5!3 \times 4! 5!
  • E.(4!)25!6!(4!)^2 5! 6!
  • F.3!(4!)25!3!(4!)^2 5!

Answer: C

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Question 8

The area enclosed between the line y=kxy = kx and the curve y=x3y = x^3 is 8. What is the value of kk?
  • A.2
  • B.4
  • C.2\sqrt{2}
  • D.5\sqrt{5}
  • E.424\sqrt{2}
  • F.252\sqrt{5}

Answer: k=4

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Question 9

The coefficient of x4x^4 in the expansion of (1+x−2x2)7(1 + x - 2x^2)^7 is
  • A.−81-81
  • B.−91-91
  • C.81
  • D.91
  • E.101
  • F.−101-101

Answer: B

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Question 10

The curve y=2x3−9x+5y = 2x^3 - 9x + 5 has turning points at x=αx = \alpha and x=βx = \beta, where β>α\beta > \alpha. Find
∫αβ2x3−9x+5 dx\int_{\alpha}^{\beta} 2x^3 - 9x + 5 \, dx
  • A.−162-16\sqrt{2}
  • B.−20-20
  • C.−20+122-20 + 12\sqrt{2}
  • D.0
  • E.525\sqrt{2}
  • F.565\sqrt{6}
  • G.20220\sqrt{2}

Answer: F

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Question 11

A rectangle is drawn in the region enclosed by the curves pp and qq, where
p(x)=10−3x2p(x) = 10 - 3x^2
q(x)=x2−3q(x) = x^2 - 3\nsuch that the sides of the rectangle are parallel to the xx- and yy-axes. What is the maximum possible area of the rectangle?
  • A.269\frac{26}{9}
  • B.26399\frac{26\sqrt{39}}{9}
  • C.5393\frac{5\sqrt{39}}{3}
  • D.1063\frac{10\sqrt{6}}{3}
  • E.5395\sqrt{39}
  • F.10610\sqrt{6}

Answer: B

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Question 12

Use the trapezium rule with 3 strips to estimate
∫133log⁡10x dx\int_1^3 3 \log_{10} x \, dx
  • A.log⁡10324\log_{10} \frac{3\sqrt{2}}{4}
  • B.log⁡1012259\log_{10} \frac{1225}{9}
  • C.log⁡10278\log_{10} \frac{27}{8}
  • D.log⁡104\log_{10} 4
  • E.log⁡10122527\log_{10} \frac{1225}{27}
  • F.log⁡103174\log_{10} \frac{3\sqrt{17}}{4}

Answer: E

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Question 13

A club with xx members is organized into four committees according to the following rules:
1. Each member belongs to exactly two committees.
2. Each pair of committees has exactly one member in common.\nThen
  • A.x=4;x = 4;
  • B.x=6;x = 6;
  • C.x=8;x = 8;
  • D.xx cannot be determined from the given information.
  • E.x=10;x = 10;
  • F.x=12;x = 12;

Answer: 6

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Question 14

If α\alpha and β\beta are roots of the equation x2+5x−5=0x^2 + 5x - 5 = 0, then (1α+1)3+(1β+1)3\left(\frac{1}{\alpha + 1}\right)^3 + \left(\frac{1}{\beta + 1}\right)^3 equals
  • A.−322-322
  • B.427\frac{4}{27}
  • C.−427-\frac{4}{27}
  • D.3+53 + \sqrt{5}
  • E.4+54 + \sqrt{5}
  • F.5+25 + \sqrt{2}

Answer: B

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Question 15

The functions f,gf, g, and hh are related by
f′(x)=g(x+2)f'(x) = g(x + 2)
g′(x)=h(x−2)g'(x) = h(x - 2)\nIt follows that f′′(3x)f''(3x) equals
  • A.h(3x+2)h(3x + 2)
  • B.3h′(3x)3h'(3x)
  • C.h(3x)h(3x)
  • D.9h(3x)9h(3x)
  • E.6h′(3x)6h'(3x)
  • F.h(3x−2)h(3x - 2)

Answer: C

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Question 16

The letters of the word "MOTHER" are permuted, and all the permutations so formed are arranged in alphabetical order as in a dictionary. Then the number of permutations which come before the word "MOTHER" is
  • A.503
  • B.93
  • C.6!2−1\frac{6!}{2} - 1
  • D.308
  • E.245
  • F.182

Answer: D

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Question 17

The number of terms in the expansion of (x+y+z+w)10(x + y + z + w)^{10} is
  • A.(104)\binom{10}{4}
  • B.(133)\binom{13}{3}
  • C.(144)\binom{14}{4}
  • D.11411^4
  • E.(122)\binom{12}{2}
  • F.(153)\binom{15}{3}

Answer: B

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Question 18

Let {Fn}\{F_n\} be the sequence of numbers defined by F1=1=F2;Fn+1=Fn+Fn−1F_1 = 1 = F_2; F_{n+1} = F_n + F_{n-1} for n≥2n \ge 2. Let fnf_n be the remainder left when FnF_n is divided by 5. Then f2000f_{2000} equals
  • A.0
  • B.1
  • C.2
  • D.3
  • E.4

Answer: A

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Question 19

What is the value of the expression
1+12(n1)+13(n2)+⋯+1n+1(nn)1 + \frac{1}{2} \binom{n}{1} + \frac{1}{3} \binom{n}{2} + \dots + \frac{1}{n + 1} \binom{n}{n}
  • A.2n+1−1n+1\frac{2^{n+1}-1}{n+1}
  • B.2(2n−1)n+1\frac{2(2^n-1)}{n+1}
  • C.2n−1n\frac{2^n-1}{n}
  • D.2(2n+1−1)n+1\frac{2(2^{n+1}-1)}{n+1}
  • E.2n+2−1n+2\frac{2^{n+2}-1}{n+2}
  • F.2(2n−1)n\frac{2(2^n-1)}{n}

Answer: A

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Question 20

What is the value of this expression (q≠1q \neq 1)
(1+q)(1+q2)(1+q4)(1+q8)(1+q16)(1+q32)(1+q64),(1 + q)(1 + q^2)(1 + q^4)(1 + q^8)(1 + q^{16})(1 + q^{32})(1 + q^{64}),
  • A.1−q1281−q\frac{1 - q^{128}}{1 - q}
  • B.1−q641−q\frac{1 - q^{64}}{1 - q}
  • C.1−q21+2+⋯+61−q\frac{1 - q^{2^{1+2+\dots+6}}}{1 - q}
  • D.none of the foregoing expressions;
  • E.1−q2641−q\frac{1 - q^{2^{64}}}{1 - q}
  • F.1−q1281+q\frac{1 - q^{128}}{1 + q}

Answer: A

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