TMUA 2017 D513/12: Questions & Worked Solutions

Question 1

1 mark

Given that

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

, which one of the following is a correct expression for

\frac{dy}{dx}

?

A. $\frac{9}{4}x^{-\frac{1}{2}} + \frac{3}{2}x^{-\frac{3}{2}} - \frac{3}{4}x^{-\frac{5}{2}}$
B. $\frac{9}{4}x^{-\frac{1}{2}} - \frac{3}{2}x^{-\frac{3}{2}} + \frac{3}{4}x^{-\frac{5}{2}}$
C. $\frac{9}{4}x^{-\frac{1}{2}} - \frac{3}{2}x^{-\frac{3}{2}} - \frac{3}{4}x^{-\frac{5}{2}}$
D. $-\frac{9}{4}x^{-\frac{1}{2}} + \frac{3}{2}x^{-\frac{3}{2}} + \frac{3}{4}x^{-\frac{5}{2}}$
E. $-\frac{9}{4}x^{-\frac{1}{2}} + \frac{3}{2}x^{-\frac{3}{2}} - \frac{3}{4}x^{-\frac{5}{2}}$
F. $-\frac{9}{4}x^{-\frac{1}{2}} - \frac{3}{2}x^{-\frac{3}{2}} - \frac{3}{4}x^{-\frac{5}{2}}$

Answer: A

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Question 2

1 mark

PQRS

is a rectangle.
The coordinates of

P

and

Q

are

(0, 6)

and

(1, 8)

respectively.
The perpendicular to

PQ

at

Q

meets the

x

-axis at

R

.
What is the area of

PQRS

?

A. $\frac{5}{2}$
B. $4\sqrt{10}$
C.20
D. $8\sqrt{10}$
E.40

Answer: E

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Question 3

1 mark

The first term of a geometric progression is

2\sqrt{3}

and the fourth term is

\frac{9}{4}

.
What is the sum to infinity of this geometric progression?

A. $-2(2 - \sqrt{3})$
B. $4(2\sqrt{3} - 3)$
C. $\frac{16(8\sqrt{3} + 9)}{37}$
D. $\frac{4(2\sqrt{3} - 3)}{7}$
E. $\frac{4(2\sqrt{3} + 3)}{7}$
F. $2(2 + \sqrt{3})$
G. $4(2\sqrt{3} + 3)$

Answer: G

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Question 4

1 mark

The following question appeared in an examination:

Given that

\tan x = \sqrt{3}

, find the possible values of

\sin 2x

.

A student gave the following answer:

\tan x = \sqrt{3}

so

x = 60^\circ

and

2x = 120^\circ

,
therefore

\sin 2x = \frac{\sqrt{3}}{2}

.

Which one of the following statements is correct?

A. $\frac{\sqrt{3}}{2}$ is the only possible value, and this is fully supported by the reasoning given in the student's answer.
B. $\frac{\sqrt{3}}{2}$ is the only possible value, but the reasoning given should consider other possible values of $x$ for which $\tan x = \sqrt{3}$ .
C. $\frac{\sqrt{3}}{2}$ is the only possible value, but the reasoning given should consider other possible values of $x$ for which $\sin 2x = \frac{\sqrt{3}}{2}$ .
D. $\frac{\sqrt{3}}{2}$ is not the only possible value because the reasoning given should have considered other possible values of $x$ for which $\tan x = \sqrt{3}$ .
E. $\frac{\sqrt{3}}{2}$ is not the only possible value because the reasoning given should have considered other possible values of $x$ for which $\sin 2x = \frac{\sqrt{3}}{2}$ .

Answer: B

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Question 5

1 mark

Consider the following three statements:
1

10p^2 + 1

and

10p^2 – 1

are both prime when

p

is an odd prime.
2 Every prime greater than 5 is of the form

6n + 1

for some integer

n

.
3 No multiple of 7 greater than 7 is prime.

The result

91 = 7 \times 13

can be used to provide a counterexample to which of the above statements?

A.none of them
B.1 only
C.2 only
D.3 only
E.1 and 2 only
F.1 and 3 only
G.2 and 3 only
H.1, 2 and 3

Answer: B

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Question 6

1 mark

A sequence

u_0, u_1, u_2, \dots

is defined as follows:

u_0 = 1

u_n = \int_0^1 4xu_{n-1} dx \text{ for } n \ge 1

What is the value of

u_{1000}

?

A. $2^{1000}$
B. $4^{1000}$
C. $\frac{4}{1000!}$
D. $\frac{4}{1001!}$
E. $\frac{2^{1000}}{1000!}$
F. $\frac{4^{1000}}{1000!}$
G. $\frac{2^{1000}}{1001!}$
H. $\frac{4^{1000}}{1001!}$

Answer: A

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Question 7

1 mark

The graphs of two functions are shown here:
•

y = a^x

is shown with a solid line, where

a

is a positive real number
•

y = f(x)

is shown with a dashed line

Which of the following statements (1, 2, 3, 4) could be true?
1

f(x) = b^x

for some

b > a

2

f(x) = b^x

for some

b < a

3

f(x) = a^{kx}

for some

k > 1

4

f(x) = a^{kx}

for some

k < 1

A.1 only
B.2 only
C.3 only
D.4 only
E.1 and 3 only
F.1 and 4 only
G.2 and 3 only
H.2 and 4 only

Answer: E

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Question 8

1 mark

Which one of the following numbers is smallest in value?

A. $\log_2 7$
B. $(2^{-3} + 2^{-2})^{-1}$
C. $2^{(\pi/3)}$
D. $\frac{1}{4(\sqrt{2}-1)^3}$
E. $4 \sin^2(\frac{\pi}{4})$

Answer: E

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Question 9

1 mark

Consider the following attempt to prove this true theorem:

Theorem:

a^3 + b^3 = c^3

has no solutions with

a, b

and

c

positive integers.

Attempted proof:
Suppose that there are positive integers

a, b

and

c

such that

a^3 + b^3 = c^3

.
I We have

a^3 = c^3 – b^3

.
II Hence

a^3 = (c - b)(c^2 + cb + b^2)

.
III It follows that

a = c − b

and

a^2 = c^2 + cb + b^2

, since

a < a^2

and

c - b < c^2 + cb + b^2

.
IV Eliminating

a

, we have

(c - b)^2 = c^2 + cb + b^2

.
V Multiplying out, we have

c^2 – 2cb + b^2 = c^2 + cb + b^2

.
VI Hence

3cb = 0

so one of

b

and

c

is zero.
But this is a contradiction to the original assumption that all of

a, b

and

c

are positive. It follows that the equation has no solutions.

Comment on this proof by choosing one of the following options:

A.The proof is correct
B.The proof is incorrect and the first mistake occurs on line I.
C.The proof is incorrect and the first mistake occurs on line II.
D.The proof is incorrect and the first mistake occurs on line III.
E.The proof is incorrect and the first mistake occurs on line IV.
F.The proof is incorrect and the first mistake occurs on line V.
G.The proof is incorrect and the first mistake occurs on line VI.

Answer: D

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Question 10

1 mark

f(x)

is a function defined for all real values of

x

.
Which one of the following is a sufficient condition for

\int_1^3 f(x) dx = 0

?

A. $f(2) = 0$
B. $f(1) = f(3) = 0$
C. $f(-x) = -f(x)$ for all $x$
D. $f(x + 2) = -f(2 – x)$ for all $x$
E. $f(x - 2) = -f(2 – x)$ for all $x$

Answer: D

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Question 11

1 mark

The function

f(x)

is increasing and

f(0) = 0

.
The positive constants

a

and

b

are such that

a < b

.
The area of the region enclosed by the curve

y = f(x)

, the

x

-axis and the lines

x = a

and

x = b

is denoted by

R

.
The function

g(x)

is defined by

g(x) = f(x) + 2f(b)

.
Which of the following is an expression for the area enclosed by the curve

y = g(x)

, the

x

-axis and the lines

x = a

and

x = b

?

A. $R + (b - a)f(b)$
B. $R + 2(b − a)f(b)$
C. $R + 2f(b) − f(a)$
D. $R + 2f(b)$
E. $R + (f(b))^2$
F. $R + (f(b))^2 – (f(a))^2$
G. $R + 2(f(b) − f(a))f(b)$

Answer: B

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Question 12

1 mark

The diagram shows the graphs of

y = \sin 2x

and

y = \cos 2x

for

-\frac{\pi}{2} < x < \frac{\pi}{2}

.

Which one of the following is not true?

A. $\cos 2x < \sin 2x < \tan x$ for some real number $x$ with $-\frac{\pi}{2} < x < \frac{\pi}{2}$
B. $\cos 2x < \tan x < \sin 2x$ for some real number $x$ with $-\frac{\pi}{2} < x < \frac{\pi}{2}$
C. $\sin 2x < \cos 2x < \tan x$ for some real number $x$ with $-\frac{\pi}{2} < x < \frac{\pi}{2}$
D. $\sin 2x < \tan x < \cos 2x$ for some real number $x$ with $-\frac{\pi}{2} < x < \frac{\pi}{2}$
E. $\tan x < \sin 2x < \cos 2x$ for some real number $x$ with $-\frac{\pi}{2} < x < \frac{\pi}{2}$
F. $\tan x < \cos 2x < \sin 2x$ for some real number $x$ with $-\frac{\pi}{2} < x < \frac{\pi}{2}$

Answer: C

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Question 13

1 mark

The positive real numbers

a \times 10^{-3}

,

b \times 10^{-2}

and

c \times 10^{-1}

are each in standard form, and

(a \times 10^{-3}) + (b \times 10^{-2}) = (c \times 10^{-1}).

Which of the following statements (I, II, III, IV) must be true?
I

a > 9

II

b > 9

III

a < c

IV

b < c

A.I only
B.II only
C.I and II only
D.I and III only
E.I and IV only
F.II and III only
G.II and IV only
H.I, II, III and IV

Answer: B

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Question 14

1 mark

The diagram below shows the graph of

y = x^2 - 2bx + c

. The vertex of this graph is at the point

P

.

Which one of the following could be the graph of

y = x^2 - 2Bx + c

, where

B > b

?

A.Graph A
B.Graph B
C.Graph C
D.Graph D
E.Graph E
F.Graph F
G.Graph G
H.Graph H

Answer: F

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Question 15

1 mark

The function

f

is defined on the positive integers as follows:

f(1) = 5

, and for

n \ge 1

:

f(n + 1) = 3f(n) + 1

if

f(n)

is odd

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

if

f(n)

is even

The function

g

is defined on the positive integers as follows:

g(1) = 3

, and for

n \ge 1

:

g(n+1) = g(n) + 5

if

g(n)

is odd

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

if

g(n)

is even

What is the value of

f(1000) – g(1000)

?

A.-6
B.-5
C.1
D.2
E.4
F.8

Answer: D

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Question 16

1 mark

Consider the following statement:
(*) If

f(x)

is an integer for every integer

x

, then

f'(x)

is an integer for every integer

x

.
Which one of the following is a counterexample to (*)?

A. $f(x) = \frac{x^3 + x + 1}{4}$
B. $f(x) = \frac{x^4 + x^2 + x}{2}$
C. $f(x) = \frac{x^4 + x^3 + x^2 + x}{2}$
D. $f(x) = \frac{x^4 + 2x^3 + x^2}{4}$

Answer: C

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Question 17

1 mark

A set S of whole numbers is called *stapled* if and only if for every whole number

a

which is in S there exists a prime factor of

a

which divides at least one other number in S.
Let T be a set of whole numbers. Which of the following is true if and only if T is not stapled?

A.For every number $a$ which is in T, there is no prime factor of $a$ which divides every other number in T.
B.For every number $a$ which is in T, there is no prime factor of $a$ which divides at least one other number in T.
C.For every number $a$ which is in T, there is a prime factor of $a$ which does not divide any other number in T.
D.For every number $a$ which is in T, there is a prime factor of $a$ which does not divide at least one other number in T.
E.There exists a number $a$ which is in T such that there is no prime factor of $a$ which divides every other number in T.
F.There exists a number $a$ which is in T such that there is no prime factor of $a$ which divides at least one other number in T.
G.There exists a number $a$ which is in T such that there is a prime factor of $a$ which does not divide any other number in T.
H.There exists a number $a$ which is in T such that there is a prime factor of $a$ which does not divide at least one other number in T.

Answer: F

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Question 18

1 mark

Consider the following problem:
Solve the inequality

(\frac{1}{4})^n < (\frac{1}{32})^{10}

, where

n

is a positive integer.
A student produces the following argument:

Which step (if any) in the argument is invalid?

A.There are no invalid steps; the argument is correct
B.Only step (I) is invalid; the rest are correct
C.Only step (II) is invalid; the rest are correct
D.Only step (III) is invalid; the rest are correct
E.Only step (IV) is invalid; the rest are correct
F.Only step (V) is invalid; the rest are correct

Answer: B

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Question 19

1 mark

Which one of the following is a sufficient condition for the equation

x^3 – 3x^2 + a = 0

, where

a

is a constant, to have exactly one real root?

A. $a > 0$
B. $a \le 0$
C. $a \ge 4$
D. $a < 4$
E. $|a| > 4$
F. $|a| \le 4$
G. $a = \frac{3}{2}$
H. $|a| = \frac{3}{2}$

Answer: E

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Question 20

1 mark

I have forgotten my 5-character computer password, but I know that it consists of the letters a, b, c, d, e in some order. When I enter a potential password into the computer, it tells me exactly how many of the letters are in the correct position.

When I enter abcde, it tells me that none of the letters are in the correct position. The same happens when I enter cdbea and eadbc.

Using the best strategy, how many further attempts must I make in order to guarantee that I can deduce the correct password?

A.None: I can deduce it immediately
B.One
C.Two
D.Three
E.More than three

Answer: B

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