TMUA 2018 D513/02: Questions & Worked Solutions

Question 1

1 mark

The function f is given, for

x > 0

, by

f(x) = \frac{x^3 - 4x}{2\sqrt{x}}

Find the value of

f'(4)

.

A.3
B.9
C.9.5
D.12
E.39.5
F.88

Answer: C

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Question 2

1 mark

Find the value of the constant term in the expansion of

\left(x^6 - \frac{1}{x^2}\right)^{12}

A.-495
B.-220
C.-66
D.66
E.220
F.495

Answer: B

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Question 3

1 mark

Consider the following statement:
A car journey consists of two parts. In the first part, the average speed is

u

km/h. In the second part, the average speed is

v

km/h. Hence the average speed for the whole journey is

\frac{1}{2}(u + v)

km/h.
Which of the following examples of car journeys provide(s) a counterexample to the statement?
I In the first part of the journey, the car travels at a constant speed of 50 km/h for 100 km. In the second part of the journey, the car travels at a constant speed of 40 km/h for 100 km.
II In the first part of the journey, the car travels at a constant speed of 50 km/h for one hour. In the second part of the journey, the car travels at a constant speed of 40 km/h for one hour.
III In the first part of the journey, the car travels at a constant speed of 50 km/h for 80 km. In the second part of the journey, the car travels at a constant speed of 40 km/h for 100 km.

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: F

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Question 4

1 mark

The non-zero real number

c

is such that the equation

\cos x = c

has two solutions for

0 < x < \frac{3}{2}\pi

.
How many solutions of the equation

\cos^2 2x = c^2

are there in the range

0 < x < \frac{3}{2}\pi

?

A.2
B.3
C.4
D.6
E.7
F.8

Answer: D

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Question 5

1 mark

The two diagonals of the quadrilateral

Q

are perpendicular.
Consider the following statements:
I One of the diagonals of

Q

is a line of symmetry of

Q

.
II The midpoints of the sides of

Q

are the vertices of a square.
Which of these statements is/are necessarily true for the quadrilateral

Q

?

A.neither of them
B.I only
C.II only
D.I and II

Answer: A

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Question 6

1 mark

Which one of the following functions provides a counterexample to the statement:
if

f'(x) > 0

for all real

x

, then

f(x) > 0

for all real

x

.

A. $f(x) = x^2 + 1$
B. $f(x) = x^2 - 1$
C. $f(x) = x^3 + x + 1$
D. $f(x) = 1 - x$
E. $f(x) = 2^x$

Answer: C

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Question 7

1 mark

Sequence 1 is an arithmetic progression with first term 11 and common difference 3.
Sequence 2 is an arithmetic progression with first term 2 and common difference 5.
Some numbers that appear in Sequence 1 also appear in Sequence 2. Let

N

be the 20th such number.
What is the remainder when

N

is divided by 7?

A.0
B.1
C.2
D.3
E.4
F.5
G.6

Answer: B

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Question 8

1 mark

The diagram shows an example of a mountain profile.

This consists of upstrokes which go upwards from left to right, and downstrokes which go downwards from left to right. The example shown has six upstrokes and six downstrokes. The horizontal line at the bottom is known as sea level.
A mountain profile of order

n

consists of

n

upstrokes and

n

downstrokes, with the condition that the profile begins and ends at sea level and never goes below sea level (although it might reach sea level at any point). So the example shown is a mountain profile of order 6.
Mountain profiles can be coded by using U to indicate an upstroke and D to indicate a downstroke. The example shown has the code UDUUUDUDDUDD. A sequence of U's and D's obtained from a mountain profile in this way is known as a valid code.
Which of the following statements is/are true?
I If a valid code is written in reverse order, the result is always a valid code.
II If each U in a valid code is replaced by D and each D by U, the result is always a valid code.
III If U is added at the beginning of a valid code and D is added at the end of the code, the result is always a valid code.

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: D

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Question 9

1 mark

Consider the following attempt to solve the equation

4x\sqrt{2x - 1} = 10x - 5

:

4x\sqrt{2x - 1} = 5(2x - 1)

(I)

16x^2(2x - 1) = 25(2x - 1)^2

(II)

16x^2 = 25(2x - 1)

(III)

16x^2 - 50x + 25 = 0

(IV)

(8x - 5)(2x - 5) = 0

(V)
The solutions of the original equation are

x = \frac{5}{8}

and

x = \frac{5}{2}

.
Which one of the following is true?

A.The solution is correct.
B.Only one of $x = \frac{5}{8}$ and $x = \frac{5}{2}$ is correct and the error arises as a result of step (II).
C.Only one of $x = \frac{5}{8}$ and $x = \frac{5}{2}$ is correct and the error arises as a result of step (III).
D.Only one of $x = \frac{5}{8}$ and $x = \frac{5}{2}$ is correct and the error arises as a result of step (IV).
E.There is another value of $x$ that satisfies the original equation and the error arises as a result of step (II).
F.There is another value of $x$ that satisfies the original equation and the error arises as a result of step (III).
G.There is another value of $x$ that satisfies the original equation and the error arises as a result of step (IV).

Answer: F

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Question 10

1 mark

The function

f(x)

is defined for all real numbers.
Consider the following three conditions, where

a

is a real constant:
I

f(a - x) = f(a + x)

for all real

x

.
II

f(2a - x) = f(x)

for all real

x

.
III

f(a - x) = f(x)

for all real

x

.
Which of these conditions is/are necessary and sufficient for the graph of

y = f(x)

to have reflection symmetry in the line

x = a

?

A.Condition I: yes, Condition II: yes, Condition III: yes
B.Condition I: yes, Condition II: yes, Condition III: no
C.Condition I: yes, Condition II: no, Condition III: yes
D.Condition I: yes, Condition II: no, Condition III: no
E.Condition I: no, Condition II: yes, Condition III: yes
F.Condition I: no, Condition II: yes, Condition III: no
G.Condition I: no, Condition II: no, Condition III: yes
H.Condition I: no, Condition II: no, Condition III: no

Answer: B

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Question 11

1 mark

Consider the equation

2^x = mx + c

, where

m

and

c

are real constants.
Which of the following statements is/are true?
I The equation has a negative real solution only if

c > 1

.
II The equation has two distinct real solutions if

c > 1

.
III The equation has two distinct positive real solutions if and only if

c \le 1

.

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: A

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Question 12

1 mark

Consider the following statement:
For any positive integer

N

there is a positive integer

K

such that

N(Km + 1) - 1

is not prime for any positive integer

m

.
Which one of the following is the negation of this statement?

A.For any positive integer $N$ there is a positive integer $K$ such that there is a positive integer $m$ for which $N(Km + 1) - 1$ is prime.
B.For any positive integer $N$ there is a positive integer $K$ such that there is a positive integer $m$ for which $N(Km + 1) - 1$ is not prime.
C.For any positive integer $N$ there is a positive integer $K$ such that for any positive integer $m$ , $N(Km + 1) - 1$ is not prime.
D.For any positive integer $N$ , any positive integer $K$ and any positive integer $m$ , $N(Km + 1) - 1$ is not prime.
E.There is a positive integer $N$ such that for any positive integer $K$ there is a positive integer $m$ for which $N(Km + 1) - 1$ is not prime.
F.There is a positive integer $N$ such that for any positive integer $K$ there is a positive integer $m$ for which $N(Km + 1) - 1$ is prime.
G.There is a positive integer $N$ such that for any positive integer $K$ and any positive integer $m$ , $N(Km + 1) - 1$ is prime.
H.There is a positive integer $N$ and a positive integer $K$ for which there is no positive integer $m$ for which $N(Km + 1) - 1$ is prime.

Answer: F

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Question 13

1 mark

The following is an attempted proof of the conjecture:

\textbf{if } \tan\theta > 0, \textbf{ then } \sin\theta + \cos\theta > 1.

Suppose

\tan\theta > 0

, so in particular

\cos\theta ≠ 0

.

Since

\tan \theta = \frac{\sin\theta}{\cos \theta}

, then

\sin\theta\cos\theta = \tan\theta \cos^2\theta > 0

.

\quad

(I)

It follows that

1+2\sin\theta\cos\theta > 1

.

\quad

(II)

Therefore

\sin^2\theta + 2\sin\theta\cos\theta + \cos^2\theta > 1

,

\quad

(III)

which factorises to give

(\sin\theta + \cos\theta)^2 > 1

.

\quad

(IV)

Therefore

\sin\theta + \cos\theta > 1

.

\quad

(V)

A.The proof is correct.
B.The proof is incorrect, and the first error occurs in line (I).
C.The proof is incorrect, and the first error occurs in line (II).
D.The proof is incorrect, and the first error occurs in line (III).
E.The proof is incorrect, and the first error occurs in line (IV).
F.The proof is incorrect, and the first error occurs in line (V).

Answer: F

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Question 14

1 mark

In the triangle

PQR

,

PR = 2

,

QR = p

and

\angle RPQ = 30^{\circ}

.
What is the set of all the values of

p

for which this information uniquely determines the length of

PQ

?

A. $p=1$
B. $p=\sqrt{3}$
C. $1 \le p < 2$
D. $\sqrt{3} \le p < 2$
E. $p=1$ or $p \ge 2$
F. $p=\sqrt{3}$ or $p \ge 2$
G. $p<2$
H. $p\ge 2$

Answer: E

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Question 15

1 mark

It is given that

f(x) = x^3 + 3qx^2 + 2

, where

q

is a real constant.
The equation

f(x) = 0

has 3 distinct real roots.
Which of the following statements is/are necessarily true?
I The equation

f(x) + 1 = 0

has 3 distinct real roots.
II The equation

f(x + 1) = 0

has 3 distinct real roots.
III The equation

f(-x) - 1 = 0

has 3 distinct real roots.

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: G

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Question 16

1 mark

In this question,

x_1, x_2, x_3, \ldots

is an arithmetic progression, all of whose terms are integers.
Let

n

be a positive integer. If the median of the first

n

terms of the sequence is an integer, which of the following three statements must be true?
I The median of the first

n + 2

terms is an integer.
II The median of the first

2n

terms is an integer.
III The median of

x_2, x_4, x_6, \ldots, x_{2n}

is an integer.

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: F

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Question 17

1 mark

A positive integer is called a squaresum if and only if it can be written as the sum of the squares of two integers. For example, 61 and 9 are both squaresums since

61 = 5^2 + 6^2

and

9 = 3^2 + 0^2

.
A prime number is called awkward if and only if it has a remainder of 3 when divided by 4. For example, 23 is awkward since

23 = 5 \times 4 + 3

.
A (true) theorem due to Fermat states that:
A positive integer is a squaresum if and only if each of its awkward prime factors occurs to an even power in its prime factorisation.
It follows that

5 \times 23^2

is a squaresum, since 23 occurs to the power 2, but

5 \times 23^3

is not, since 23 occurs to the power 3.
Which one of the following statements is not true?

A.Every square number is a squaresum.
B.If $N$ and $M$ are squaresums, then so is $NM$ .
C.If $NM$ is a squaresum, then $N$ and $M$ are squaresums.
D.If $N$ is not a squaresum, then $kN$ is a squaresum for some number $k$ which is a product of awkward primes.

Answer: C

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Question 18

1 mark

f(x)

is a polynomial function defined for all real

x

.
Which of the following is a necessary condition for the inequality

\frac{f(a)+f(b)}{2} \ge f\left(\frac{a+b}{2}\right)

to be true for all real numbers

a

and

b

with

a < b

?

A. $f(x) \ge 0$ for all real $x$
B. $f'(x) \ge 0$ for all real $x$
C. $f''(x) \ge 0$ for all real $x$
D. $f(x) \le 0$ for all real $x$
E. $f'(x) \le 0$ for all real $x$
F. $f''(x) \le 0$ for all real $x$

Answer: C

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Question 19

1 mark

Three real numbers

x, y

and

z

satisfy

x > y > z > 1

.
Which one of the following statements must be true?

A. $\frac{2^{z+1}}{2^x} > \frac{2^x + 2^z}{2^y}$
B. $2 > \frac{3^x + 3^z}{3^y}$
C. $\frac{2 \times 5^x}{5^z} > \frac{5^x + 5^z}{5^y}$
D. $2 < \frac{7^x + 7^z}{7^y}$

Answer: C

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Question 20

1 mark

It is given that the equation

\sqrt{x + p} + \sqrt{x} = p

has at least one real solution for

x

, where

p

is a real constant.
What is the complete set of possible values for

p

?

A. $p=0$ or $p=1$
B. $p=0$ or $p \ge 1$
C. $p \ge -x$
D. $p \ge \sqrt{x}$
E. $p \ge 0$
F. $p \ge 1$

Answer: B

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