TMUA 2016 D513/02: Questions & Worked Solutions

Question 1

1 mark

Find the value of

\int_1^2 \left(x^2 - \frac{4}{x^2}\right)^2 dx

A. $\frac{43}{15}$
B.3
C. $\frac{97}{15}$
D. $\frac{103}{15}$
E. $\frac{163}{15}$
F.18

Answer: A

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Question 2

1 mark

Let

f(x) = \frac{(x^2 + 5)(2x)}{\sqrt[4]{x^3}},\ x > 0

. Which one of the following is equal to

f'(x)

?

A.
$8x^4 + \frac{40}{3}x^{\frac{1}{4}}$
B.
$\frac{9}{2}x^{\frac{5}{4}} + \frac{5}{2}x^{-\frac{3}{4}}$
C.
$8x^{\frac{9}{4}} + \frac{40}{3}x^{-\frac{1}{4}}$
D.
$\frac{8}{13}x^{\frac{13}{4}} + 8x^{\frac{5}{4}}$

Answer: B

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Question 3

1 mark

What is the value, in radians, of the largest angle

x

in the range

0 \le x \le 2\pi

that satisfies the equation

8 \sin^2 x + 4 \cos^2 x = 7

?

A.
$\frac{2\pi}{3}$
B.
$\frac{5\pi}{6}$
C.
$\frac{4\pi}{3}$
D.
$\frac{5\pi}{3}$
E.
$\frac{7\pi}{4}$
F.
$\frac{11\pi}{6}$

Answer: D

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Question 4

1 mark

Five sealed urns, labelled P, Q, R, S, and T, each contain the same (non-zero) number of balls. The following statements are attached to the urns.
Urn P This urn contains one or four balls.
Urn Q This urn contains two or four balls.
Urn R This urn contains more than two balls and fewer than five balls.
Urn S This urn contains one or two balls.
Urn T This urn contains fewer than three balls.
Exactly one of the urns has a true statement attached to it.
Which urn is it?

A.Urn P
B.Urn Q
C.Urn R
D.Urn S
E.Urn T

Answer: C

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Question 5

1 mark

Consider the statement:
(*) A whole number

n

is prime if it is 1 less or 5 less than a multiple of 6.
How many counterexamples to (*) are there in the range

0 < n < 50

?

A.2
B.3
C.4
D.5
E.6

Answer: C

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Question 6

1 mark

The sequence of functions

f_1(x)

,

f_2(x)

,

f_3(x)

, ... is defined as follows:

f_1(x) = x^{10}

f_{n+1}(x) = xf_n'(x)

for

n \ge 1

where

f_n'(x) = \frac{df_n(x)}{dx}

.
Find the value of

\sum_{n=1}^{20} f_n(x)

A.
$\frac{x^{10}(x^{20} – 1)}{x-1}$
B.
$\frac{x^{10} (x^{21} – 1)}{x-1}$
C.
$\left(\frac{10^{20}-1}{9}\right)x^{10}$
D.
$\left(\frac{10^{21}-1}{9}\right)x^{10}$
E.
$\left(\frac{(10!)^{20}-1}{9}\right)x^{10}$
F.
$\left(\frac{(10!)^{21}-1}{9}\right)x^{10}$
G.
$x^{10} + x^9 + x^8 + … + x + 1$
H.
$x^{10} + 10x^9 + (10 \times 9)x^8 + … + (10 \times 9 \times ... \times 2)x + (10 \times 9 \times ... \times 2 \times 1)$

Answer: C

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Question 7

1 mark

The four real numbers

a, b, c

, and

d

are all greater than 1.
Suppose that they satisfy the equation

\log_c d = (\log_a b)^2

.
Use some of the lines given to construct a proof that, in this case, it follows that
(*)

\log_b d = (\log_a b)(\log_a c)

.
(1) Let

x = \log_a b

and

y = \log_a c

(2)

d = (c^x)^2

(3)

d = c^{(x^2)}

(4)

d = b^{xy}

(5)

d = (a^y)^{(x^2)}

(6)

d = ((a^x)^y)^2

(7)

d = (a^x)^{xy}

(8)

d = a^{(y^2x)}

(9)

d = a^{(x^2y)}

A.(1). Then (2), so (6), so (8), so (7), and therefore (4), hence (*) as required.
B.(1). Then (2), so (7), so (8), so (6), and therefore (4), hence (*) as required.
C.(1). Then (3), so (5), so (9), so (7), and therefore (4), hence (*) as required.
D.(1). Then (3), so (7), so (9), so (5), and therefore (4), hence (*) as required.
E.(1). Then (4), so (5), so (9), so (7), and therefore (3), hence (*) as required.
F.(1). Then (4), so (6), so (8), so (7), and therefore (2), hence (*) as required.
G.(1). Then (4), so (7), so (8), so (6), and therefore (2), hence (*) as required.
H.(1). Then (4), so (7), so (9), so (5), and therefore (3), hence (*) as required.

Answer: C

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Question 8

1 mark

A region is defined by the inequalities

x + y > 6

and

x - y > -4

Consider the three statements:
1

x > 1

2

y > 5

3

(x + y)(x - y) > -24

Which of the above statements is/are true for every point in the region?

A.none
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 9

1 mark

Triangles

ABC

and

XYZ

have the same area.
Which of these extra conditions, taken independently, would imply that they are congruent?
(1)

AB = XY

and

BC = YZ

(2)

AB = XY

and

\angle ABC = \angle XYZ

(3)

\angle ABC = \angle XYZ

and

\angle BCA = \angle YZX

A.Condition (1): Does not imply congruent; Condition (2): Does not imply congruent; Condition (3): Does not imply congruent
B.Condition (1): Does not imply congruent; Condition (2): Does not imply congruent; Condition (3): Implies congruent
C.Condition (1): Does not imply congruent; Condition (2): Implies congruent; Condition (3): Does not imply congruent
D.Condition (1): Does not imply congruent; Condition (2): Implies congruent; Condition (3): Implies congruent
E.Condition (1): Implies congruent; Condition (2): Does not imply congruent; Condition (3): Does not imply congruent
F.Condition (1): Implies congruent; Condition (2): Does not imply congruent; Condition (3): Implies congruent
G.Condition (1): Implies congruent; Condition (2): Implies congruent; Condition (3): Does not imply congruent
H.Condition (1): Implies congruent; Condition (2): Implies congruent; Condition (3): Implies congruent

Answer: D

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Question 10

1 mark

In this question

x

and

y

are non-zero real numbers.
Which one of the following is sufficient to conclude that

x < y

?

A. $x^4 < y^4$
B. $y^4 < x^4$
C. $x^{-1} < y^{-1}$
D. $y^{-1} < x^{-1}$
E. $x^{\frac{3}{5}} < y^{\frac{3}{5}}$
F. $y^{\frac{3}{5}} < x^{\frac{3}{5}}$

Answer: E

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Question 11

1 mark

f(x)

is a polynomial with real coefficients.
The equation

f(x) = 0

has exactly two real roots,

x = -p

and

x = p

, where

p > 0

.
Consider the following three statements:
1

f'(x) = 0

for exactly one value of

x

between

-p

and

p

2 The area between the curve

y = f(x)

, the

x

-axis and the lines

x = -p

and

x = p

is given by

2 \int_0^p f(x) dx

3 The graph of

y = -f(-x)

intersects the

x

-axis at the points

x = -p

and

x = p

only
Which of the above statements must be true?

A.none
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

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Question 12

1 mark

The first term of an arithmetic sequence is

a

and the common difference is

d

.
The sum of the first

n

terms is denoted by

S_n

.
If

S_8 > 3S_6

, what can be deduced about the sign of

a

and the sign of

d

?

A.both $a$ and $d$ are negative
B. $a$ is positive, $d$ is negative
C. $a$ is negative, $d$ is positive
D.a is negative, but the sign of $d$ cannot be deduced
E. $d$ is negative, but the sign of $a$ cannot be deduced
F.neither the sign of $a$ nor the sign of $d$ can be deduced

Answer: F

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Question 13

1 mark

In this question,

a, b

, and

c

are positive integers.
The following is an attempted proof of the false statement:
If

a

divides

bc

, then

a

divides

b

or

a

divides

c

.
['

a

divides

bc

' means '

a

is a factor of

bc

']
Which line contains the error in this proof?
1. The statement is equivalent to ‘if

a

does not divide

b

and

a

does not divide

c

then

a

does not divide

bc

'.
2. Suppose

a

does not divide

b

and

a

does not divide

c

. Then the remainder when dividing

b

by

a

is

r

, where

0 < r < a

, and the remainder when dividing

c

by

a

is

s

, where

0 < s < a

.
3. So

b = ax + r

and

c = ay + s

for some integers

x

and

y

.
4. Thus

bc = a(axy + xs + yr) + rs

.
5. So the remainder when dividing

bc

by

a

is

rs

.
6. Since

r > 0

and

s > 0

, it follows that

rs > 0

.
7. Hence

a

does not divide

bc

.

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

Answer: E

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Question 14

1 mark

f(x) = ax^4 + bx^3 + cx^2 + dx + e

, where

a, b, c, d

, and

e

are real numbers.
Suppose

f(x) = 1

has

p

distinct real solutions,

f(x) = 2

has

q

distinct real solutions,

f(x) = 3

has

r

distinct real solutions, and

f(x) = 4

has

s

distinct real solutions.
Which one of the following is not possible?

A. $p = 1, q = 2, r = 4$ and $s = 3$
B. $p = 1, q = 3, r = 2$ and $s = 4$
C. $p = 1, q = 4, r = 3$ and $s = 2$
D. $p = 2, q = 4, r = 3$ and $s = 1$
E. $p = 4, q = 3, r = 2$ and $s = 1$

Answer: B

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Question 15

1 mark

Consider the quadratic

f(x) = x^2 – 2px + q

and the statement:
(*)

f(x) = 0

has two real roots whose difference is greater than 2 and less than 4.
Which one of the following statements is true if and only if (*) is true?

A. $q < p^2 < q + 4$
B. $\sqrt{q+1} < p < \sqrt{q + 4}$
C. $q - 3 \le p^2 - 4 \le q$
D. $q < p^2 - 1 < q + 3$
E. $q - 2 < p^2 - 3 < q + 2$

Answer: D

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Question 16

1 mark

In the figure,

PQRS

is a trapezium with

PQ

parallel to

SR

.
The diagonals of the trapezium meet at

X

.

U

lies on

SP

and

T

lies on

RQ

such that

UT

is a line segment through

X

parallel to

PQ

.
The length of

PQ

is 12 cm and the length of

SR

is 3 cm.
What, in centimetres, is the length of

UT

?

A.4.2
B.4.5
C.4.8
D.5.25
E.6

Answer: C

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Question 17

1 mark

Consider these simultaneous equations, where

c

is a constant:

y = 3 \sin x + 2

y = x + c

Which of the following statements is/are true?
1 For some value of

c

: there is exactly one solution with

0 \le x \le \pi

and there is at least one solution with

- \pi < x < 0

.
2 For some value of

c

: there is exactly one solution with

0 \le x \le \pi

and there are no solutions with

- \pi < x < 0

.
3 For some value of

c

: there is exactly one solution with

0 \le x \le \pi

and there are no solutions with

x > \pi

.

A.none
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: H

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Question 18

1 mark

Consider this statement about a function

f(x)

:
(*) If

(f(x))^2 \le 1

for all

-1 \le x \le 1

then

\int_{-1}^1 (f(x))^2 dx \le \int_{-1}^1 f(x) dx

Which one of the following functions provides a counterexample to (*)?

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

Answer: D

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Question 19

1 mark

Some identical unit cubes are used to construct a three-dimensional object by gluing them together face to face.
Sketches of this object are made by looking at it from the right-hand side, from the front and from above. These sketches are called the side elevation, the front elevation, and the plan view respectively.