TMUA 2023 D513/02: Questions & Worked Solutions

Question 1

1 mark

Given that

\frac{1}{\sqrt{x}-6} - \frac{1}{\sqrt{x}+6} = \frac{3}{11}

what is the value of x?

A. $2\sqrt{15}$
B. $4\sqrt{5}$
C. $5\sqrt{2}$
D. $\sqrt{58}$
E.50
F.58
G.60
H.80

Answer: H

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 2

1 mark

Evaluate

\int_{9}^{16} \left( \frac{1}{\sqrt{x}} + \sqrt{x} \right)^2 dx - \int_{9}^{16} \left( \frac{1}{\sqrt{x}} - \sqrt{x} \right)^2 dx

A.0
B.2
C.4
D.7
E.14
F.28
G.75
H.175

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 3

1 mark

Consider the claim:
For all positive real numbers x and y,

\sqrt{x^{y}}= x^{\sqrt{y}}

Which of the following is/are a counterexample to the claim?
I

x = 1, y = 16

II

x = 2, y = 8

III

x = 3, y = 4

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: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 4

1 mark

A student attempts to answer the following question.
What is the largest number of consecutive odd integers that are all prime?
The student's attempt is as follows:
I There are two consecutive odd integers that are prime (for example: 17, 19).
II Any three consecutive odd integers can be written in the form

n-2, n, n+2

for some n.
III If n is one more than a multiple of 3, then

n+2

is a multiple of 3.
IV If n is two more than a multiple of 3, then

n-2

is a multiple of 3.
V The only other possibility is that n is a multiple of 3.
VI In each case, one of the integers is a multiple of 3, so not prime.
VII Therefore the largest number of consecutive odd integers that are all prime is two.
Which of the following best describes this attempt?

A.It is completely correct.
B.It is incorrect, and the first error is on line I.
C.It is incorrect, and the first error is on line II.
D.It is incorrect, and the first error is on line III.
E.It is incorrect, and the first error is on line IV.
F.It is incorrect, and the first error is on line V.
G.It is incorrect, and the first error is on line VI.
H.It is incorrect, and the first error is on line VII.

Answer: G

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 5

1 mark

Consider the two statements
R: k is an integer multiple of

\pi

S:

\int_{0}^{k} \sin 2x dx = 0

Which of the following statements is true?

A.R is necessary and sufficient for S.
B.R is necessary but not sufficient for S.
C.R is sufficient but not necessary for S.
D.R is not necessary and not sufficient for S.

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 6

1 mark

Consider the following equation where a is a real number and

a > 1

:
(*)

a^x = x

Which of the following equations must have the same number of real solutions as (*)?
I

\log_a x = x

II

a^{2x} = x^2

III

a^{2x} = 2x

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

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 7

1 mark

The graph of the line

ax + by = c

is drawn, where a, b and c are real non-zero constants.
Which one of the following is a necessary but not sufficient condition for the line to have a positive gradient and a positive y-intercept?

A. $\frac{c}{b} > 0$ and $\frac{a}{b} < 0$
B. $\frac{c}{b} < 0$ and $\frac{a}{b} > 0$
C. $a>b>c$
D. $a<b<c$
E.a and c have opposite signs
F.a and c have the same sign

Answer: E

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 8

1 mark

A student draws a triangle that is acute-angled or obtuse-angled but not right-angled.
The student counts the number of straight lines that divide the triangle into two triangles, at least one of which is right-angled.
Which of the following statements is/are true?
I The student can draw a triangle for which there is exactly 1 such straight line.
II The student can draw a triangle for which there are exactly 2 such straight lines.
III The student can draw a triangle for which there are exactly 3 such straight lines.

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

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 9

1 mark

Consider the following statement about a pentagon P:
(*) If at least one of the interior angles in P is

108^{\circ}

, then all the interior angles in P form an arithmetic sequence.
Which of the following is/are true?
I The statement (*)
II The contrapositive of (*)
III The converse of (*)

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

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 10

1 mark

Here is an attempt to solve the inequality

x^4 - 2x^2 - 3 < 0

by completing the square:

x^4 - 2x^2 - 3 < 0

I if and only if

x^4 - 2x^2 + 1 < 4

II if and only if

(x^2 - 1)^2 < 4

III if and only if

-2 < x^2 - 1 < 2

IV if and only if

x^2 - 1 < 2

V if and only if

x^2 < 3

VI if and only if

-\sqrt{3} < x < \sqrt{3}

Which of the following statements is true?

A.The argument is completely correct.
B.The first error occurs in line I.
C.The first error occurs in line II.
D.The first error occurs in line III.
E.The first error occurs in line IV.
F.The first error occurs in line V.
G.The first error occurs in line VI.

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 11

1 mark

In this question, k is a positive integer.
Consider the following theorem:
If

2^k+1

is a prime, then k is a power of 2. (*)
Which of the following statements, taken individually, is/are equivalent to (*)?
I If k is a power of 2, then

2^k+1

is prime.
II

2^k+1

is not prime only if k is not a power of 2.
III A sufficient condition for k to be a power of 2 is that

2^k+1

is prime.

A.Statement I is equivalent to (*): Yes, Statement II is equivalent to (*): Yes, Statement III is equivalent to (*): Yes
B.Statement I is equivalent to (*): Yes, Statement II is equivalent to (*): Yes, Statement III is equivalent to (*): No
C.Statement I is equivalent to (*): Yes, Statement II is equivalent to (*): No, Statement III is equivalent to (*): Yes
D.Statement I is equivalent to (*): Yes, Statement II is equivalent to (*): No, Statement III is equivalent to (*): No
E.Statement I is equivalent to (*): No, Statement II is equivalent to (*): Yes, Statement III is equivalent to (*): Yes
F.Statement I is equivalent to (*): No, Statement II is equivalent to (*): Yes, Statement III is equivalent to (*): No
G.Statement I is equivalent to (*): No, Statement II is equivalent to (*): No, Statement III is equivalent to (*): Yes
H.Statement I is equivalent to (*): No, Statement II is equivalent to (*): No, Statement III is equivalent to (*): No

Answer: G

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 12

1 mark

In this question, p is a real constant.
The equation

\sin x \cos^2 x = p^2 \sin x

has n distinct solutions in the range

0 \le x \le 2\pi

Which of the following statements is/are true?
I

n = 3

is sufficient for

p > 1

II

n = 7

only if

-1 < p < 1

A.none of them
B.I only
C.II only
D.I and II

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 13

1 mark

Let x be a real number.
Which one of the following statements is a sufficient condition for exactly three of the other four statements?

A. $x \ge 0$
B. $x = 1$
C. $x = 0$ or $x = 1$
D. $x \ge 0$ or $x \le 1$
E. $x \ge 0$ and $x \le 1$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 14

1 mark

Three lines are given by the equations:

ax + by + c = 0

bx + cy + a = 0

cx + ay + b = 0

where a, b and c are non-zero real numbers.
Which one of the following is correct?

A.If two of the lines are parallel, then all three are parallel.
B.If two of the lines are parallel, then the third is perpendicular to the other two.
C.If two of the lines are parallel, then the third is parallel to $y = x$ .
D.If two of the lines are parallel, then the third is perpendicular to $y = x$ .
E.If two of the lines are perpendicular, then all three meet at a point.
F.If two of the lines are perpendicular, then the third is parallel to $y = x$ .
G.If two of the lines are perpendicular, then the third is perpendicular to $y = x$ .

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 15

1 mark

The base 10 number 0.03841 has the value

0 \times 10^{-1} + 3\times10^{-2} + 8 \times 10^{-3} + 4 \times 10^{-4} + 1 \times 10^{-5} = 0.03841

Similarly, the base 2 number 0.01101 has the value

0\times2^{-1} + 1\times 2^{-2} + 1 \times 2^{-3} + 0 \times 2^{-4} + 1 \times 2^{-5} = \frac{13}{32}

What is the value of the recurring base 2 number

0.00\dot{1}\dot{1} = 0.001100110011...?

A. $\frac{1}{3}$
B. $\frac{1}{5}$
C. $\frac{1}{15}$
D. $\frac{2}{15}$
E. $\frac{4}{15}$
F. $\frac{3}{16}$
G. $\frac{5}{16}$
H. $\frac{6}{31}$

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 16

1 mark

A sequence is defined by:

u_1 = a

u_2 = b

u_{n+2} = u_n + u_{n+1}

for

n \ge 1

where a and b are positive integers. The highest common factor of a and b is 7.
Which of the following statements must be true?
I

u_{2023}

is a multiple of 7
II If

u_1

is not a factor of

u_2

, then

u_1

is not a factor of

u_n

for any

n > 1

III The highest common factor of

u_1

and

u_5

is 7

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: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 17

1 mark

The ceiling of

x

, written

\lceil x \rceil

, is defined to be the value of

x

rounded up to the nearest integer.
For example:

\lceil \pi \rceil = 4

,

\lceil 2.1 \rceil = 3

,

\lceil 8 \rceil = 8

What is the value of the following integral?

\int_0^{99} 2^{\lceil x \rceil} \, dx

A. $2^{99}$
B. $2^{99}-1$
C. $2^{99}-2$
D. $2^{100}$
E. $2^{100}-1$
F. $2^{100}-2$

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 18

1 mark

The equation

x^4 + bx^2 + c = 0

has four distinct real roots if and only if which of the following conditions is satisfied?

A. $b^2 > 4c$
B. $b^2 < 4c$
C. $c > 0$ and $b > 2\sqrt{c}$
D. $c > 0$ and $b < -2\sqrt{c}$
E. $c < 0$ and $b < 0$
F. $c < 0$ and $b > 0$

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 19

1 mark

In this question,

f(x)

is a non-constant polynomial, and

g(x) = xf'(x)

f(x) = 0

for exactly M real values of x.

g(x) = 0

for exactly N real values of x.
Which of the following statements is/are true?
I It is possible that

M < N

II It is possible that

M = N

III It is possible that

M > N

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: H

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 20

1 mark

Let f be a polynomial with real coefficients.
The integral

I_{p,q}

where

p < q

is defined by

I_{p,q} = \int_{p}^{q} (f(x))^2 - (f(|x|))^2 dx

Which of the following statements must be true?
1

I_{p,q} = 0

only if

0 < p

2

f'(x) < 0

for all x only if

I_{p,q} < 0

for all

p < q < 0

3

I_{p,q} > 0

only if

p < 0

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: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →