TMUA Beyond Horizon Set 3 Paper 2: Questions & Worked Solutions

Question 1

1 mark

Which of these numbers is the largest?

A. $\frac{2+\sqrt{5}}{5\sqrt{2}}$
B. $\frac{\sqrt{9!}}{6!}$
C. $\frac{\log_3 26}{\log_2 33}$
D. $\frac{\sqrt{7}}{5}$
E. $\frac{\pi}{5}$

Answer: B

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Question 2

1 mark

The graph of the function

y = 2^{x^2-4x+3}

can be obtained from the graph of

y = 2^{x^2}

by

A.a stretch parallel to the y-axis followed by a translation parallel to the y-axis
B.a translation parallel to the x-axis followed by a stretch parallel to the y-axis
C.a translation parallel to the x-axis followed by a stretch parallel to the x-axis
D.a translation parallel to the x-axis followed by reflection in the y-axis
E.reflection in the y-axis followed by translation parallel to the y-axis

Answer: B

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Question 3

1 mark

Find the highest power of

x

in

\left[ \left((2x^5 + 6)^4 + (4x^6 + 3)^3\right)^7 + \left((2x^4 + 3x^5)^5 - (x^7 - 6)^4\right)^5 \right]^{10}

A.1200
B.1300
C.1400
D.1500
E.1600

Answer: C

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Question 4

1 mark

Three positive numbers

a, b, c

satisfy

\log_b a = 2

,

\log_b (c - 3) = 3

,

\log_a (c + 5) = 2

. This information

A.specifies $a$ uniquely
B.is satisfied by two values of $a$
C.is satisfied by infinitely many values of $a$
D.is contradictory

Answer: A

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Question 5

1 mark

Which of these people A-E must be telling the truth if they each speak as follows?

A.I'm not a liar
B.I'm the only liar
C.If E is a liar, so is D
D.If A is a liar, so am I
E.None of us are liars

Answer: A

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Question 6

1 mark

(*) is the statement that the functions

f(x) = b^{ax}

,

g(x) = b^{x^a}

,

h(x) = b^{a^x}

are all defined and increasing for

x > 0

. Find the range of

a

and

b

for which (*) is true.

A. $a > 0$ and $b > 0$
B. $a < 0$ and $b > 1$
C. $a > 1$ and $b > 0$
D. $a > 1$ and $b > 1$
E. $a > 0$ and $0 < b < 1$
F. $a = 1$ and $b > 0$

Answer: D

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Question 7

1 mark

Each of the positive real numbers

a, b, c, d, e

is decreased by 20%. Find the resulting percentage change in the value of the following expression:

\frac{a - b}{cd} - \frac{b^2 + c^2}{d^3 + ae^2}

A.No change
B.Decrease of 20%
C.Increase of 20%
D.Decrease of 25%
E.Increase of 25%

Answer: D

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Question 8

1 mark

Let

f

be a linear function with the properties that

f(1) \le f(2)

,

f(3) \ge f(4)

, and

f(5) = 5

. Which of the following is true?

A. $f(0) < 0$
B. $f(0) = 0$
C. $f(1) < f(0) < f(-1)$
D. $f(0) = 5$
E. $f(0) > 5$

Answer: D

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Question 9

1 mark

When

n

standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of

S

. The smallest possible value of

S

is

A.333
B.335
C.337
D.339
E.341

Answer: C

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Question 10

1 mark

A hockey team consists of 1 goalkeeper, 4 defenders, 4 midfielders and 2 forwards. There are 4 substitutes: 1 goalkeeper, 1 defender, 1 midfielder and 1 forward. A substitute may only replace a player of the same category. Given that a maximum of 3 substitutes may be used and that there are still 11 players on the pitch at the end, how many different teams could finish the game?

A.110
B.118
C.121
D.125
E.132

Answer: B

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Question 11

1 mark

Consider the unit circle

x^2 + y^2 = 1

. The locus of a point

P

such that the tangents

PA, PB

at the points

A, B

respectively of the circle are so that

\angle AOB = 60^\circ

, where

O

is the origin, is

A.a circle of radius $\frac{2}{\sqrt{3}}$ with centre $O$
B.a circle of radius $\sqrt{3}$ with centre $O$
C.a circle of radius 2 with centre $O$
D.a pair of straight lines

Answer: A

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Question 12

1 mark

Let

l, m, n

be any three positive integers such that

l^2 + m^2 = n^2

. Then,

A.3 always divides $mn$
B.3 always divides $lm$
C.3 always divides $ln$
D.3 does not divide $lmn$

Answer: B

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Question 13

1 mark

Suppose

n \ge 2

is a fixed positive integer and

f(x) = x^n|x|, x \in \mathbb{R}

. Then

A. $f$ is differentiable everywhere only when $n$ is even
B. $f$ is differentiable everywhere except at 0 if $n$ is odd
C. $f$ is differentiable everywhere
D.none of the above is true

Answer: C

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Question 14

1 mark

There are 100 people in a queue waiting to enter a hall. The hall has exactly 100 seats numbered from 1 to 100. The first person in the queue enters the hall, chooses any seat and sits there. The

n

-th person in the queue, where

n

can be

2, \dots, 100

, enters the hall after the

(n-1)

-th person is seated. He sits in seat number

n

if he finds it vacant; otherwise he takes any unoccupied seat. Find the total number of ways in which 100 seats can be filled up, provided the 100-th person occupies seat number 100.

A.100
B. $100^{100}$
C. $2^{98}$
D. $2^{99}$
E. $2^{100}$

Answer: C

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Question 15

1 mark

Find an expression for

\int_1^n (-1)^{\lfloor x \rfloor} \lfloor x \rfloor^{-1} dx

where

n \in \mathbb{N}

.

A. $\sum_{k=1}^n (-1)^k \frac{1}{k}$
B. $\sum_{k=1}^{n-1} (-1)^k \frac{1}{k}$
C. $\sum_{k=1}^{n-1} (-1)^k \frac{1}{k^2}$
D. $\sum_{k=1}^n (-1)^k \frac{1}{k^2}$

Answer: B

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Question 16

1 mark

The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is

A.11
B.12
C.13
D.14
E.15

Answer: D

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Question 17

1 mark

For all

x

such that

1 \le x \le 3

, the inequality

(x - 3a)(x - a - 3) < 0

holds for

A.no value of $a$
B.all $a$ satisfying $\frac{2}{3} < a < 1$
C.all $a$ satisfying $0 < a < \frac{1}{3}$
D.all $a$ satisfying $\frac{1}{3} < a < \frac{2}{3}$
E.all $a$ satisfying $0 < a < \frac{2}{3}$
F.all $a$ satisfying $0 < a < 1$

Answer: C

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Question 18

1 mark

A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?

A.10
B.11
C.18
D.19
E.20

Answer: C

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Question 19

1 mark

If

a

and

b

are positive numbers and

c

and

d

are real numbers, positive or negative, then

a^c \le b^d

A.if $a \le b$ and $c \le d$
B.if either $a \le b$ or $c \le d$
C.if $a \ge 1, b \ge 1, d \ge c$
D.if $a \le 2, b \le 2, d \ge 2$
E.if $a \le 0, b \le 1, d \le c$
F.is not implied by any of the above conditions

Answer: C

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Question 20

1 mark

Consider the triangular array of numbers with 0, 1, 2, 3, . . . along the sides. Let

f(n)

denote the sum of the numbers in row

n

. What is the remainder when

f(100)

is divided by 100?

A.12
B.30
C.50
D.62
E.74

Answer: E

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