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TMUA Beyond Horizon Set 1 Paper 1

20 questions75Updated July 2026

The TMUA Beyond Horizon Set 1 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

For what values of

x

is

f(x)

a valid function?

f(x) = \frac{1}{2 \log(x^2 - 2x - 3)}

A. $(0, \infty)$
B. $(-\infty, -1) \quad x \neq 1 \pm \sqrt{5}$
C. $(-\infty, -1) \cup (3, \infty) \quad x \neq 1 \pm \sqrt{5}$
D. $(-\infty, -3) \cup (1, \infty) \quad x \neq 1 \pm \sqrt{5}$
E. $f(x)$ is never valid
F. $(-\infty, -1) \cup (3, \infty)$

Answer: C

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Question 2

For a real number

x

, let

[x]

denote the largest integer less than or equal to

x

. Find the value of

\int_{-100}^{100} [t^3] dt

A.0
B.100
C. $-100^2$
D. $-100^3$
E. $100^2$
F.-100

Answer: F

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Question 3

Two functions are defined as:

f(x) = |x^2 - 4|, x \in \mathbb{R}

g(x) = x + 2, x \in \mathbb{R}

\nWhich range of values satisfy the inequality

f(x) < g(x)

?

A. $-2 < x < 3$
B. $-2 < x < 1$
C. $1 < x < 3$
D. $x < 1$ and $x > 3$
E. $x < -2$ and $x > 3$
F. $x < -2$ and $x > 1$

Answer: C

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Question 4

What is the sum of the coefficients in

\left(\frac{2}{x} + \frac{3}{x^2} + 5x\right)^3

?

A.1000
B.750
C.125
D.343
E.100
F.729

Answer: A

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Question 5

How many solutions does this equation have in the interval

0 \le x \le 2\pi

?

\sin(\cos(\sin(x))) = \cos(\sin(\cos(x)))

A.0
B.1
C.2
D.4
E.8
F.16

Answer: D

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Question 6

How many integer pairs

(x, y)

satisfy the equation

x^2 + 6x + y^2 = 4

A.2
B.4
C.6
D.8
E.10
F.12

Answer: D

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Question 7

The number of polynomials of the form

x^3 + ax^2 + bx + c

which are divisible by

x^2 + 1

and where

a, b,

and

c

belong to

\{1, 2, \dots, 10\}

, is

A.1
B.10
C.11
D.12
E.100
F.101

Answer: B

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Question 8

There are seven greeting cards, each of a different colour, and seven envelopes of the same seven colours. The number of ways in which the cards can be put in the envelopes, so that exactly four of the cards go into the envelopes of the right colours, is

A. $\binom{7}{3}$
B. $2 \binom{7}{3}$
C. $(3!) \binom{4}{3}$
D. $(3!) \binom{7}{3} \binom{4}{3}$
E. $(3!) \binom{7}{3}$
F. $(4!) \binom{7}{3}$

Answer: B

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Question 9

Suppose that

F(n + 1) = \frac{2F(n) + 1}{2}

for

n = 1, 2, 3, \dots

and

F(1) = 2

. Then

F(101)

equals

A.50
B.52
C.54
D.56
E.58
F.None of the above

Answer: B

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Question 10

Let

p(x)

be a continuous function which is positive for all

x

and

\int_2^3 p(x) dx = c \int_0^2 p\left(\frac{x + 4}{2}\right) dx.

\nThen

A. $c = 4$
B. $c = -\frac{1}{4}$
C. $c = \frac{1}{4}$
D. $c = 2$
E. $c = -\frac{1}{2}$
F. $c = \frac{1}{2}$

Answer: F

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Question 11

How many real solutions does the equation

2 \sin^3(x) - 6 \sin(x) \cos^2(x) + \sin(x) = 0

have in the interval

0 < x < 2\pi

?

A.1
B.2
C.3
D.4
E.5
F.6

Answer: E

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Question 12

Suppose

a < b

. The maximum value of the integral

\int_a^b \left(\frac{3}{4} - x - x^2\right) dx

\nover all possible values of

a

and

b

is

A. $\frac{3}{4}$
B. $\frac{4}{3}$
C. $\frac{3}{2}$
D. $\frac{2}{3}$
E. $\frac{4}{5}$
F. $\frac{5}{4}$

Answer: B

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Question 13

The line

x = y

is tangent at

(0, 0)

to a circle of radius 1. The centre of the circle is

A. $(1, 0)$
B.either $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ or $\left(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right)$
C.either $\left(\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right)$ or $\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$
D. $(0, 1)$
E. $(0, 0)$
F.None of the Above

Answer: C

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Question 14

The

m

th term of an arithmetic progression is

x

and the

n

th term is

y

. What is the sum of the first

(m + n)

terms?

A. $\frac{m+n}{2} \left[ (x + y) + \frac{x-y}{m-n} \right]$
B. $\frac{m+n}{2} \left[ (x - y) + \frac{x+y}{m-n} \right]$
C. $\frac{1}{2} \left[ \frac{x+y}{m+n} + \frac{x-y}{m-n} \right]$
D. $\frac{1}{2} \left[ \frac{x+y}{m+n} - \frac{x-y}{m-n} \right]$
E. $\frac{m-n}{2} \left[ (x - y) + \frac{x+y}{m-n} \right]$
F. $\frac{m+n}{2} \left[ (x + y) + \frac{x+y}{m-n} \right]$

Answer: A

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Question 15

The values of

m

for which

mx^2 - 6mx + 5m + 1 > 0

for all real

x

is

A. $0 < m < \frac{1}{4}$
B. $0 \le m < \frac{1}{8}$
C. $m > 0$
D. $0 \le m < \frac{1}{4}$
E. $m < 0$
F. $m = 0$

Answer: D

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Question 16

The inequality

\sqrt{x + 6} \ge x

is satisfied for real

x

if and only if

A. $-3 \le x \le 3$
B. $-2 \le x \le 3$
C. $-6 \le x \le 3$
D. $0 \le x \le 6$
E. $-6 < x < 6$
F. $0 < x < 3$

Answer: C

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Question 17

The difference between the roots of the equation

6x^2 + \alpha x + 1 = 0

is

\frac{1}{6}

.

\alpha

is a positive number. The value of

\alpha

is

A.2
B.3
C.4
D.5
E. $\frac{8}{3}$
F.8

Answer: D

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Question 18

Let

x, y, z

be positive. The least value of the following expression is

\frac{x(1 + y) + y(1 + z) + z(1 + x)}{\sqrt{xyz}}

A. $\frac{9}{\sqrt{2}}$
B.6
C. $\frac{1}{\sqrt{6}}$
D.3
E.1
F.None of the Above

Answer: B

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Question 19

All the letters of the word PESSIMISTIC are to be arranged so that no two S's, no two I's, and S and I do not occur together. The number of such arrangements is

A.1800
B.5480
C.4800
D.1200
E.2400
F.1801

Answer: E

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Question 20

A right-circular cone

A

with base radius 3 units and height 5 units is truncated so that the radius of the top circle is 1.5 units, with the top parallel to the base. A second right-circular cone

B

with base radius 5 units and height 6 units is placed vertically inside cone

A

.
\nWhat is the volume of the cone

B

outside cone

A

plus the portion of cone

A

excluding the part inside cone

B

?

A. $\frac{1867}{40} \pi$
B. $\frac{1913}{40} \pi$
C. $\frac{2417}{40} \pi$
D. $\frac{2153}{40} \pi$
E. $\frac{1867}{20} \pi$
F. $\frac{2417}{20} \pi$

Answer: A

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