TMUA 2021 D513/02: Questions & Worked Solutions

Question 1

1 mark

Find the value of

\int_{1}^{4} \left(3\sqrt{x} + \frac{4}{x^2}\right) dx

A.-0.75
B.7.125
C.11
D.17
E.18
F.21.875
G.34.5

Answer: D

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Question 2

1 mark

A(0, 2)

and

C(4,0)

are opposite vertices of the square

ABCD

.
What is the equation of the straight line through

B

and

D

?

A. $y = -2x+5$
B. $y = -\frac{1}{2}x - 3$
C. $y = -\frac{1}{2}x + 2$
D. $y=x$
E. $y = 2x - 3$
F. $y = 2x +2$

Answer: E

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Question 3

1 mark

A student is chosen at random from a class. Each student is equally likely to be
chosen.
Which of the following conditions is/are necessary for the probability that the
student wears glasses to equal

\frac{4}{15}

?
I Exactly 11 students in the class do not wear glasses.
II The number of students in the class is divisible by 3.
III The class contains 30 students, and 8 of them wear glasses.

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

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Question 4

1 mark

Consider the following claim about positive integers

a

,

b

and

c

:
if

a

is a factor of

bc

, then

a

is a factor of

b

or

a

is a factor of

c

Which of the following provide(s) a counterexample to this claim?
I

a = 5

,

b = 10

,

c = 20

II

a = 8

,

b = 4

,

c = 4

III

a = 6

,

b = 7

,

c = 12

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

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Question 5

1 mark

On which line is the first error in the following argument?
A

\sin^2x + \cos^2 x = 1

for all values of

x

.
B Therefore

\cos x = \sqrt{1 - \sin^2x}

for all values of

x

.
C Hence

1 + \cos x = 1+\sqrt{1-\sin^2x}

for all values of

x

.
D Thus

(1 + \cos x)^2 = (1+\sqrt{1-\sin^2 x})^2

for all values of

x

.
E Substituting

x = \pi

gives

0 = 4

.

A.LIne A
B.Line B
C.Line C
D.Line D
E.Line E

Answer: B

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Question 6

1 mark

Consider the following two statements about the polynomial

f(x)

:
P:

f(x) = 0

for exactly three real values of

x

Q:

f'(x) = 0

for exactly two real values of

x

Which one of the following is correct?

A.P is necessary but not sufficient for Q.
B.P is sufficient but not necessary for Q.
C.P is necessary and sufficient for Q.
D.P is not necessary and not sufficient for Q.

Answer: D

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Question 7

1 mark

A circle has equation

(x – 9)^2 + (y + 2)^2 = 4

A square has vertices at

(1,0)

,

(1, 2)

,

(-1, 2)

and

(-1,0)

.
A straight line bisects both the area of the circle and the area of the square.
What is the

x

-coordinate of the point where this straight line meets the

x

-axis?

A.2
B.3
C.4
D.4.5
E.5
F.6
G.The straight line is not uniquely determined by the information given, so there is
more than one possible point of intersection.
H.There is no straight line that bisects both the area of the circle and the area of
the square.

Answer: B

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Question 8

1 mark

Consider the following statement about the polynomial

p(x)

, where

a

and

b

are real
numbers with

a < b

:

(*)

There exists a number

c

with

a < c < b

such that

p'(c) = 0

.
Which one of the following is true?

A.The condition $p(a) = p(b)$ is necessary and sufficient for $(*)$
B.The condition $p(a) = p(b)$ is necessary but not sufficient for $(*)$
C.The condition $p(a) = p(b)$ is sufficient but not necessary for $(*)$
D.The condition $p(a) = p(b)$ is not necessary and not sufficient for $(*)$

Answer: C

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Question 9

1 mark

Consider the following statements about a polynomial

f(x)

:
I

f(x) = px^3 + qx^2 + rx + s

, where

p ≠ 0

.
II There is a real number

t

for which

f'(t) = 0

.
III There are real numbers

u

and

v

for which

f(u) f(v) < 0

.
Which of these statements is/are sufficient for the equation

f(x) = 0

to have a real solution?

A.Statement I is
sufficient: Yes, Statement II is
sufficient: Yes, Statement III is
sufficient: Yes
B.Statement I is
sufficient: Yes, Statement II is
sufficient: Yes, Statement III is
sufficient: No
C.Statement I is
sufficient: Yes, Statement II is
sufficient: No, Statement III is
sufficient: Yes
D.Statement I is
sufficient: Yes, Statement II is
sufficient: No, Statement III is
sufficient: No
E.Statement I is
sufficient: No, Statement II is
sufficient: Yes, Statement III is
sufficient: Yes
F.Statement I is
sufficient: No, Statement II is
sufficient: Yes, Statement III is
sufficient: No
G.Statement I is
sufficient: No, Statement II is
sufficient: No, Statement III is
sufficient: Yes
H.Statement I is
sufficient: No, Statement II is
sufficient: No, Statement III is
sufficient: No

Answer: C

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Question 10

1 mark

The first seven terms of a sequence of positive integers are:

u_1 = 15

u_2 = 21

u_3 = 30

u_4 = 37

u_5 = 44

u_6 = 51

u_7 = 59

Consider the following statement about this sequence:

(*)

If

n

is a prime number, then

u_n

is a multiple of 3 or

u_n

is a multiple
of 5.
What is the smallest value of

n

that provides a counterexample to

(*)

?

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

Answer: E

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Question 11

1 mark

A student attempts to solve the following problem, where

a

and

b

are non-zero real
numbers:
Show that if

a^2 – 4b^3 > 0

then there exist real numbers

x

and

y

such that

a = xy(x + y)

and

b = xy

.
Consider the following attempt:

(x - y)^2 \geq 0

(I)
so

x^2 + y^2 - 2xy \geq 0

(II)
so

(x + y)^2 - 4xy \geq 0

(III)
so

x^2y^2(x + y)^2 – 4x^3y^3 > 0

(IV)
SO

a^2 - 4b^3 \geq 0

(V)
Which of the following best describes this attempt?

A.It is completely correct.
B.It is incorrect, but it would be correct if written in the reverse order.
C.It is incorrect, but the student has correctly proved the converse.
D.It is incorrect because there is an error in line (II).
E.It is incorrect because there is an error in line (III).
F.It is incorrect because there is an error in line (IV).

Answer: C

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Question 12

1 mark

Which of the following statements about polynomials

f

and

g

is/are true?
I If

f(x) \geq g(x)

for all

x \geq 0

, then

\int_{0}^{x} f(t) dt \geq \int_{0}^{x} g(t) dt

for all

x \geq 0

.
II If

f(x) \geq g(x)

for all

x \geq 0

, then

f'(x) \geq g'(x)

for all

x \geq 0

.
III If

f'(x) \geq g'(x)

for all

x \geq 0

, then

f(x) \geq g(x)

for all

x \geq 0

.

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

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Question 13

1 mark

A region

R

in the

(x, y)

-plane is defined by the simultaneous inequalities

y - x < 3

y - x^2 < 1

Which of the following statements is/are true for every point in

R

?
I

-1 < x < 2

II

(y - x)(y – x^2) < 3

III

y < 5

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 14

1 mark

Consider the following simultaneous equations, where

p

is a real number:

p^{2x} + \log_2 y = 2

2^{2x} + \log_2 y = 1

What is the complete range of

p

for which these simultaneous equations have a real
solution

(x, y)

?

A. $p<1$
B. $p ≠ 1$
C. $p>1$
D. $p<1$ or $p > 2$
E. $p ≠ 1$ and $p < 2$
F. $p> 1$ and $p < 2$
G. $p>2$
H.All real values of $p$

Answer: C

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Question 15

1 mark

A circle has equation

x^2 + ax + y^2 + by + c = 0

where

a

,

b

and

c

are non-zero real constants.
Which one of the following is a necessary and sufficient condition for the circle to
be tangent to the

y

-axis?

A. $a^2 = 4c$
B. $b^2 = 4c$
C. $\frac{a}{2} = \sqrt{\frac{a^2 + b^2}{4} - c}$
D. $\frac{b}{2} = \sqrt{\frac{a^2 + b^2}{4} - c}$
E. $-\frac{a}{2} = \sqrt{\frac{a^2 + b^2}{4} - c}$
F. $-\frac{b}{2} = \sqrt{\frac{a^2 + b^2}{4} - c}$

Answer: B

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Question 16

1 mark

p

and

q

are real numbers, and the equation

x |x| = px + q

has exactly

k

distinct real solutions for

x

.
Which one of the following is the complete list of possible values for

k

?

A.0, 1, 2
B.0, 1, 2, 3
C.0, 1, 2, 3, 4
D.0, 2, 4
E.1, 2, 3
F.1, 2, 3, 4

Answer: E

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Question 17

1 mark

Consider the following functions defined for

x > 1

:

f(x) = \log_2(\log_2 \sqrt{x})

g(x) = \log_2(\sqrt{\log_2x})

Which one of the following is true for all values of

x > 1

?

A. $0 \leq f(x) \leq g(x)$ or $g(x) \leq f(x) \leq 0$
B. $0 \leq g(x) \leq f(x)$ or $f(x) \leq g(x) \leq 0$
C. $\frac{1}{2} \leq f(x) \leq g(x)$ or $g(x) \leq f(x) \leq \frac{1}{2}$
D. $\frac{1}{2} \leq g(x) \leq f(x)$ or $f(x) \leq g(x) \leq \frac{1}{2}$
E. $1 \leq f(x) \leq g(x)$ or $g(x) \leq f(x) \leq 1$
F. $1 \leq g(x) \leq f(x)$ or $f(x) \leq g(x) \leq 1$

Answer: F

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Question 18

1 mark

A student chooses two distinct real numbers

x

and

y

with

0 < x < y < 1

.
The student then attempts to draw a triangle

ABC

with:

AB = 1

\sin A = x

\sin B = y

Which of the following statements is/are correct?
I For some choice of

x

and

y

, there is exactly one triangle the student
could draw.
II For some choice of

x

and

y

, there are exactly two different triangles
the student could draw.
III For some choice of

x

and

y

, there are exactly three different
triangles the student could draw.
(Note that congruent triangles are considered to be the same.)

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

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Question 19

1 mark

The angle

\theta

can take any of the values

1^{\circ}

,

2^{\circ}

,

3^{\circ}

, ...,

359^{\circ}

,

360^{\circ}

.
For how many of these values of

\theta

is it true that

\sin \theta \sqrt{1 + \sin \theta} \sqrt{1 - \sin \theta} + \cos \theta \sqrt{1 + \cos \theta} \sqrt{1 - \cos \theta} = 0

A.0
B.1
C.2
D.4
E.93
F.182
G.271
H.360

Answer: F

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Question 20

1 mark

A sequence of functions

f_1, f_2, f_3, ...

is defined by

f_1(x) = |x|

f_{n+1}(x) = |f_n(x) + x|

for

n \geq 1

Find the value of

\int_{-1}^{1} f_{99}(x) dx

A.0
B.0.5
C.1
D.49.5
E.50
F.99
G.99.5
H.100

Answer: E

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