TMUA 2022 D513/01: Questions & Worked Solutions

Question 1

1 mark

How many real solutions are there to the equation

2 \cos^4 \theta – 5 \cos^2 \theta + 3 = 0

in the interval

0 \leq \theta \leq 2\pi

?

A.1
B.2
C.3
D.4
E.5
F.6
G.7
H.8

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 2

1 mark

Find the complete set of values of

p

for which the equation

x^2 - 2px + y^2 – 6y – p^2 + 8p + 9 = 0

describes a circle in the xy-plane.

A. $p<-\frac{9}{4}$
B. $0<p<4$
C. $-1<p<9$
D. $p<0$ or $p > 4$
E. $p<-1$ or $p>9$
F.all real values of $p$

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 3

1 mark

Given the following statements about a function

f

•

f''(x) = a

for all

x

•

f(0) = 1, f(1) = 2

•

\int_{0}^{1} f(x)dx = 1

find the value of

a

.

A.-6
B.-3
C.-2
D.2
E.3
F.6

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 4

1 mark

These sectors of circles are similar.
The arc length of the smaller sector is 6.
The difference between the areas of the sectors is 21.
Find the positive difference between the perimeters of the sectors.

A.4.5
B.7
C.8
D.9
E.10.5
F.14
G.15

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 5

1 mark

The terms

x_n

of a sequence follow the rule

x_{n+1} = \frac{x_n + p}{x_n + q}

where

p

and

q

are real numbers.
Given that

x_1 = 3, x_2 = 5

, and

x_3 = 7

, find the value of

x_4

A.-5
B.5
C. $\frac{51}{7}$
D. $\frac{15}{2}$
E. $\frac{23}{3}$
F.9
G.11
H.13

Answer: H

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 6

1 mark

Given that

\int_{\log_2 5}^{\log_2 20} x \,dx = \log_2 M

what is the value of

M

?

A.4
B.15
C.16
D.20
E.25
F.100
G.10000

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 7

1 mark

Find the finite area enclosed between the line

y = 0

and the curve

y = x^2 – 4|x| – 12

A. $\frac{128}{3}$
B. $\frac{176}{3}$
C. $\frac{256}{3}$
D.108
E.144
F.288

Answer: E

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 8

1 mark

A geometric sequence has first term

a

and common ratio

r

, where

a

and

r

are positive integers and

r

is greater than 1.
The sum of the first

n

terms of this sequence is denoted by

S_n

It is given that the terms of the sequence satisfy

S_{30}-S_{20} = kS_{10}

for some positive integer

k

.
What is the smallest possible value of

k

?

A. $2^{10}$
B. $2^{20}$
C. $2^{30}$
D. $\frac{2^{10}}{2^{10}-1}$
E. $2^{10}(2^{10}-1)$

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 9

1 mark

This question is about pairs of functions

f

and

g

that satisfy

f(x) - g(x) = 2\sin x

and

f(x) g(x) = \cos^2 x

for all real numbers

x

.
Across all solutions for

f(x)

, what is the minimum value that

f(x)

attains for any

x

?

A. $1-\sqrt{2}$
B. $-1-\sqrt{2}$
C.0
D.-1
E.-2
F.-3
G.-4

Answer: E

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 10

1 mark

A sequence of translations is applied to the graph of

y = x^3

Which of the following graphs could be the result of this sequence of translations?
I

y = x^3 – 3x^2 + 9x – 27

II

y = x^3 – 9x^2 + 27x - 3

III

y = 27x^3 – 9x^2 + x − 3

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 11

1 mark

Evaluate

\sum_{n=1}^{100} \log_{10} (3^{1-n})

A.-4950 $\log_{10} 3$
B.4950 $\log_{10} 3$
C.-5050 $\log_{10} 3$
D.5050 $\log_{10} 3$
E. $1-4950 \log_{10} 3$
F. $1+4950 \log_{10} 3$
G. $1-5050 \log_{10} 3$
H. $1+5050 \log_{10} 3$

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 12

1 mark

A family of quadratic curves is given by

y_k = 2\left(x - \frac{k}{2}\right)^2 + \frac{k^2}{2} + 4k + 3

where

k

is any real number and

y_k

is a function of

x

.
All these curves are sketched, and the point with the lowest y-coordinate among all the curves

y_k

is

(a,b)

.
Find the value of

a + b

A.-1
B.-3
C.-5
D.-7
E.-9

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 13

1 mark

Given that

\left(a^3 + \frac{2}{b^3}\right)\left(\frac{2}{a^3} - b^3\right) = \sqrt{2}

where

a

and

b

are real numbers, what is the least value of

ab

?

A. $-\sqrt{2}$
B. $\sqrt{2}$
C. $-2\sqrt{2}$
D. $2\sqrt{2}$
E. $-\frac{\sqrt{2}}{2}$
F. $\frac{\sqrt{2}}{2}$
G. $-2^{\frac{1}{6}}$
H. $2^{\frac{1}{6}}$

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 14

1 mark

A circle has centre

O

and radius 6.

P, Q

and

R

are points on the circumference with angle

POQ \geq \frac{\pi}{2}

The area of the triangle

POQ

is

9\sqrt{3}

What is the greatest possible area of triangle

PRQ

?

A. $18+9\sqrt{3}$
B. $18\sqrt{3}$
C. $27+9\sqrt{3}$
D. $27\sqrt{3}$
E. $36+9\sqrt{3}$
F. $36\sqrt{3}$

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 15

1 mark

A rectangle is drawn in the region enclosed by the curves

p

and

q

, where

p(x) = 8 - 2x^2

and

q(x) = x^2 - 2

such that the sides of the rectangle are parallel to the x- and y-axes.
What is the maximum possible area of the rectangle?

A. $\frac{26}{9}$
B. $\frac{52}{9}$
C. $\frac{4\sqrt{6}}{3}$
D. $\frac{8\sqrt{6}}{3}$
E. $4\sqrt{2}$
F. $8\sqrt{2}$
G. $\frac{20\sqrt{10}}{9}$
H. $\frac{40\sqrt{10}}{9}$

Answer: H

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 16

1 mark

The solutions to

7x^4 - 6x^2 + 1 = 0

are

\pm\cos\theta

and

\pm\cos\beta

.
Which one of the following equations has solutions

\pm\sin\theta

and

\pm\sin\beta

?

A. $7x^4-8x^2- 5 = 0$
B. $7x^4-8x^2 + 2 = 0$
C. $7x^4-6x^2-2 = 0$
D. $7x^4-6x^2 + 1 = 0$
E. $7x^4 + 6x^2 – 1 = 0$
F. $7x^4 + 6x^2 + 5 = 0$

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 17

1 mark

Find the complete set of values of

x

for which there are two non-congruent triangles with the side lengths and angle as shown in the diagram.

A. $1<x<3$
B. $1<x<4$
C. $1<x<5$
D. $3<x<4$
E. $3<x<5$
F. $4<x<5$

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 18

1 mark

It is given that

f(x) = x^2(x – 1)^2(x – 2)

and

g(x) = -p(x – q)^2(x – r)^2

where

p, q

and

r

are positive and

q<r

.
Find the set of values of

q

and

r

that guarantees the greatest number of distinct real solutions of the equation

f(x) = g(x)

for all

p

.

A. $q<1$ and $r<1$
B. $q<1$ and $1<r<2$
C. $q<1$ and $r>2$
D. $1<q<2$ and $1<r<2$
E. $1<q<2$ and $r>2$
F. $q>2$ and $r>2$

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 19

1 mark

Circle

C_1

is defined as

x^2 + y^2 = 25

A second circle

C_2

has radius 4 and centre

(a, b)

where

-2\leq a \leq 2

and

-3\leq b \leq 3

If the centre of

C_2

is equally likely to be located anywhere within the given range, what is the probability that

C_2

intersects

C_1

?

A. $\frac{1}{25}$
B. $\frac{9}{25}$
C. $\frac{16}{25}$
D. $\frac{6-\pi}{6}$
E. $\frac{16 – \pi}{24}$
F. $\frac{24 – \pi}{24}$

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 20

1 mark

n

is the number of points of intersection of the graphs

y = |x^2 - a^2|

and

y = a^2 |x − 1|

where

a

is a real number.
What is the smallest value of

n

that is not possible?

A. $n=1$
B. $n=2$
C. $n=3$
D. $n=4$
E. $n=5$

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →