TMUA 2018 D513/01: Questions & Worked Solutions

Question 1

1 mark

Find the value of

\int_1^4 \frac{3 - 2x}{x\sqrt{x}}\,dx

A. $-\frac{13}{2}$
B. $-\frac{85}{16}$
C. $-\frac{13}{8}$
D.-1
E. $-\frac{1}{4}$
F. $-\frac{7}{4}$
G.7

Answer: D

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Question 2

1 mark

An arithmetic progression has first term

a

and common difference

d

.
The sum of the first 5 terms is equal to the sum of the first 8 terms.
Which one of the following expresses the relationship between

a

and

d

?

A. $a = -\frac{38}{3}d$
B. $a = -7d$
C. $a = -6d$
D. $a = 6d$
E. $a = 7d$
F. $a = \frac{38}{3}d$

Answer: C

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Question 3

1 mark

Find the shortest distance between the two circles with equations:

(x + 2)^2 + (y - 3)^2 = 18

(x - 7)^2 + (y + 6)^2 = 2

A.0
B.4
C.16
D. $2\sqrt{2}$
E. $5\sqrt{2}$

Answer: E

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Question 4

1 mark

Consider the simultaneous equations

3x^2 + 2xy = 4

x + y = a

where

a

is a real constant.
Find the complete set of values of

a

for which the equations have two distinct real
solutions for

x

.

A.There are no values of $a$ .
B. $-2 < a < 2$
C. $-1 < a < 1$
D. $a = 0$
E. $a < -1$ or $a > 1$
F. $a < -2$ or $a > 2$
G.All real values of $a$

Answer: G

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Question 5

1 mark

The function

f

is defined by

f(x) = x^3 + ax^2 + bx + c

.

a

,

b

and

c

take the values 1, 2 and 3 with no two of them being equal and not
necessarily in this order.
The remainder when

f(x)

is divided by

(x + 2)

is

R

.
The remainder when

f(x)

is divided by

(x + 3)

is

S

.
What is the largest possible value of

R - S

?

A.-26
B.5
C.7
D.17
E.29

Answer: D

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Question 6

1 mark

Find the number of solutions of the equation

x\sin 2x = \cos 2x

with

0 \le x \le 2\pi

.

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

Answer: E

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Question 7

1 mark

The non-zero constant

k

is chosen so that the coefficients of

x^6

in the expansions of

(1 + kx^2)^7

and

(k + x)^{10}

are equal.
What is the value of

k

?

A. $\frac{1}{6}$
B.6
C. $\frac{\sqrt{6}}{6}$
D. $\sqrt{6}$
E. $\frac{\sqrt{30}}{30}$
F. $\sqrt{30}$

Answer: A

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Question 8

1 mark

The sum to infinity of a geometric progression is 6.
The sum to infinity of the squares of each term in the progression is 12.
Find the sum to infinity of the cubes of each term in the progression.

A.8
B.18
C.24
D. $\frac{216}{7}$
E.72
F.216

Answer: D

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Question 9

1 mark

Find the complete set of values of the constant

c

for which the cubic equation

2x^3 - 3x^2 - 12x + c = 0

has three distinct real solutions.

A. $-20 < c < 7$
B. $-7 < c < 20$
C. $c > 7$
D. $c > -7$
E. $c < 20$
F. $c < -20$

Answer: B

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Question 10

1 mark

x

and

y

satisfy

|2 - x| \le 6

and

|y + 2| \le 4

.
What is the greatest possible value of

|xy|

?

A.16
B.24
C.32
D.40
E.48
F.There is no greatest possible value.

Answer: E

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Question 11

1 mark

The line

y = mx + 5

, where

m > 0

, is normal to the curve

y = 10 - x^2

at the
point

(p, q)

.
What is the value of

p

?

A. $\frac{\sqrt{2}}{6}$
B. $-\frac{\sqrt{2}}{6}$
C. $\frac{3\sqrt{2}}{2}$
D. $-\frac{3\sqrt{2}}{2}$
E. $\sqrt{5}$
F. $-\sqrt{5}$

Answer: C

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Question 12

1 mark

A curve has equation

y = f(x)

, where

f(x) = x(x - p)(x - q)(r - x)

with

0 < p < q < r

.
You are given that:

\int_0^r f(x) \text{d}x = 0

\int_0^q f(x) \text{d}x = -2

\int_p^r f(x) \text{d}x = -3

What is the total area enclosed by the curve and the

x

-axis for

0 \le x \le r

?

A.0
B.1
C.4
D.5
E.6
F.10

Answer: F

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Question 13

1 mark

The function

f(x)

has derivative

f'(x)

.
The diagram below shows the graph of

y = f'(x)

.
Which point corresponds to a local minimum of

f(x)

?

A.A
B.B
C.C
D.D
E.E
F.F

Answer: C

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Question 14

1 mark

The line

y = mx + 4

passes through the points

(3, \log_2p)

and

(\log_2 p, 4)

.
What are the possible values of

p

?

A. $p=1$ and $p=4$
B. $p=1$ and $p=16$
C. $p=\frac{1}{4}$ and $p=4$
D. $p=\frac{1}{4}$ and $p=64$
E. $p=\frac{1}{64}$ and $p=4$
F. $p=\frac{1}{64}$ and $p=16$

Answer: B

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Question 15

1 mark

Find the sum of the real solutions of the equation:

3^x - (\sqrt{3})^{x+4} + 20 = 0

A.1
B.4
C.9
D. $\log_3 20$
E. $2 \log_3 20$
F. $4 \log_3 20$

Answer: E

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Question 16

1 mark

The curve

C

has equation

y = x^2 + bx + 2

, where

b \ge 0

.
Find the value of

b

that minimises the distance between the origin and the
stationary point of the curve

C

.

A. $b=0$
B. $b=1$
C. $b=2$
D. $b = \frac{\sqrt{6}}{2}$
E. $b = \sqrt{2}$
F. $b = \sqrt{6}$

Answer: F

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Question 17

1 mark

There are two sets of data: the mean of the first set is 15, and the mean of the
second set is 20.
One of the pieces of data from the first set is exchanged with one of the pieces of
data from the second set.
As a result, the mean of the first set of data increases from 15 to 16, and the mean of
the second set of data decreases from 20 to 17.
What is the mean of the set made by combining all the data?

A. $16\frac{1}{4}$
B. $16\frac{1}{3}$
C. $16\frac{1}{2}$
D. $16\frac{2}{3}$
E. $16\frac{3}{4}$

Answer: A

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Question 18

1 mark

What is the smallest positive value of

a

for which the line

x = a

is a line of
symmetry of the graph of

y = \sin \left(2x - \frac{4\pi}{3}\right)

?

A. $\frac{\pi}{12}$
B. $\frac{5\pi}{12}$
C. $\frac{7\pi}{12}$
D. $\frac{11\pi}{12}$
E. $\frac{19\pi}{12}$

Answer: B

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Question 19

1 mark

A triangle

ABC

is to be drawn with

AB = 10\text{cm}

,

BC = 7\text{cm}

and the angle at

A

equal to

\theta

, where

\theta

is a certain specified angle.
Of the two possible triangles that could be drawn, the larger triangle has three times
the area of the smaller one.
What is the value of

\cos \theta

?

A. $\frac{5}{7}$
B. $\frac{151}{200}$
C. $\frac{2\sqrt{2}}{5}$
D. $\frac{\sqrt{17}}{5}$
E. $\frac{\sqrt{51}}{8}$
F. $\frac{\sqrt{34}}{8}$

Answer: D

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Question 20

1 mark

Find the value of

\sin^2 0^\circ + \sin^2 1^\circ + \sin^2 2^\circ + \sin^2 3^\circ + \cdots + \sin^2 87^\circ + \sin^2 88^\circ + \sin^2 89^\circ + \sin^2 90^\circ

A.0.5
B.1
C.1.5
D.45
E.45.5
F.46

Answer: E

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