TMUA 2016 D513/01: Questions & Worked Solutions

Question 1

1 mark

It is given that the expansion of

(ax + b)^3

is

8x^3 – px^2 + 18x – 3\sqrt{3}

, where

a

,

b

and

p

are real constants.

What is the value of

p

?

A. $-12\sqrt{3}$
B. $-6\sqrt{3}$
C. $-4\sqrt{3}$
D. $-\sqrt{3}$
E. $\sqrt{3}$
F. $4\sqrt{3}$
G. $6\sqrt{3}$
H. $12\sqrt{3}$

Answer: H

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Question 2

1 mark

The expression

3x^3 + 13x^2 + 8x + a

, where

a

is a constant, has

(x + 2)

as a factor.

Which one of the following is a complete factorisation of the expression?

A. $(x + 2)(x − 1)(3x – 2)$
B. $(x + 2)(x + 1)(3x - 2)$
C. $(x + 2)(x + 1)(3x + 2)$
D. $(x + 2)(x − 3)(3x + 2)$
E. $(x + 2)(x + 3)(3x - 2)$
F. $(x + 2)(x + 3)(3x + 2)$

Answer: E

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Question 3

1 mark

A line is drawn normal to the curve

y = \frac{2}{x^2}

at the point on the curve where

x = 1

.

This line cuts the x-axis at

P

and the y-axis at

Q

.

The length of

PQ

is

A. $\frac{3\sqrt{5}}{2}$
B. $\frac{3\sqrt{17}}{4}$
C. $\frac{7\sqrt{17}}{4}$
D. $\frac{35}{4}$
E. $\frac{35\sqrt{5}}{2}$
F. $\frac{3\sqrt{17}}{2}$

Answer: C

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Question 4

1 mark

The sequence

a_n

is defined by the rule:

a_n = (-1)^n – (-1)^{n-1} + (-1)^{n+2}

for

n \geq 1

.

Find the value of

\sum_{n=1}^{39} a_n

A. $-39$
B. $-3$
C. $-1$
D. $0$
E. $1$
F. $3$
G. $39$

Answer: B

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Question 5

1 mark

What is the total area enclosed between the curve

y = x^2 – 1

, the x-axis and
the lines

x = -2

and

x = 2

?

A. $\frac{4}{3}$
B. $\frac{8}{3}$
C. $4$
D. $\frac{16}{3}$
E. $12$
F. $16$

Answer: C

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Question 6

1 mark

P, Q, and R are each mixtures of red and white paint.

The percentage by volume of red paint in P is 30%.

The percentage by volume of red paint in Q is 20%.

The mixtures P, Q, and R are combined in the proportion 12 : 5 : 3 respectively.

If the resulting mixture contains 25% by volume of red paint, what percentage by volume
of mixture R is red paint?

A. $25\%$
B. $23\%$
C. $13\frac{1}{2}\%$
D. $19\frac{1}{2}\%$
E. $9\frac{3}{4}\%$
F.It is impossible to achieve this result.

Answer: C

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Question 7

1 mark

60% of a sports club's members are women and the remainder are men.

This sports club offers the opportunity to play tennis or cricket. Every member plays
exactly one of the two sports.

\frac{2}{5}

of the male members of the club play cricket;

\frac{2}{3}

of the cricketing members of the club are women.

What is the probability that a member of the club, chosen at random, is a woman who
plays tennis?

A. $\frac{1}{5}$
B. $\frac{7}{25}$
C. $\frac{1}{3}$
D. $\frac{11}{25}$
E. $\frac{3}{5}$

Answer: B

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Question 8

1 mark

Find the maximum angle

x

in the range

0^\circ \leq x \leq 360^\circ

which satisfies the equation

\cos^2(2x) + \sqrt{3}\sin(2x) - \frac{7}{4} = 0

A. $30^\circ$
B. $60^\circ$
C. $120^\circ$
D. $150^\circ$
E. $210^\circ$
F. $240^\circ$
G. $300^\circ$
H. $330^\circ$

Answer: F

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Question 9

1 mark

The line segment joining the points

(3, 3)

and

(7, 5)

is a diameter of a circle.

This circle is translated by 3 units in the negative x-direction, then reflected in the x-axis,
and then enlarged by a scale factor of 4 about the centre of the resulting circle.

The equation of the final circle is

A. $(x - 2)^2 + (y – 4)^2 = 320$
B. $(x – 2)^2 + (y + 4)^2 = 320$
C. $(x - 2)^2 + (y – 4)^2 = 80$
D. $(x – 2)^2 + (y + 4)^2 = 80$
E. $(x – 2)^2 + (y – 4)^2 = 20$
F. $(x - 2)^2 + (y + 4)^2 = 20$

Answer: D

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Question 10

1 mark

How many solutions does the equation

x \tan x = 1

have in the interval

-2\pi \leq x \leq 2\pi

?

A. $0$
B. $1$
C. $2$
D. $3$
E. $4$
F. $5$
G. $6$

Answer: E

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Question 11

1 mark

The real roots of the equation

4^{2x} + 12 = 2^{2x+3}

are

p

and

q

, where

p > q

.

The value of

p – q

can be expressed as

A. $\frac{3}{4}$
B. $1$
C. $4$
D. $-\frac{1}{2} + \log_{10} \frac{3}{2}$
E. $\frac{\log_{10} 3}{\log_{10} 4}$
F. $\frac{\log_{10} 3}{\log_{10} 2}$

Answer: E

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Question 12

1 mark

A right circular cylinder is contained within a sphere of radius 5 cm in such a way that the
whole of the circumferences of both ends of the cylinder are in contact with the sphere.

The diagram shows a planar cross section through the centre of the sphere and cylinder.

[diagram not to scale]

Find, in cubic centimetres, the maximum possible volume of the cylinder.

A. $250\pi$
B. $500\pi$
C. $1000\pi$
D. $\frac{250\sqrt{3}}{3}\pi$
E. $\frac{500\sqrt{3}}{9}\pi$
F. $\frac{1000\sqrt{3}}{9}\pi$

Answer: E

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Question 13

1 mark

How many real roots does the equation

3x^5 – 10x^3 – 120x + 30 = 0

have?

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

Answer: C

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Question 14

1 mark

The terms of an infinite series

S

are formed by adding together the corresponding terms in
two infinite geometric series,

T

and

U

.

The first term of

T

and the first term of

U

are each 4.

In order, the first three terms of the combined series

S

are

8

,

3

, and

\frac{5}{4}

.

What is the sum to infinity of

S

?

A. $\frac{32}{5}$
B. $\frac{20}{3}$
C. $\frac{64}{5}$
D. $\frac{40}{3}$
E. $16$
F. $32$

Answer: D

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Question 15

1 mark

The least possible value of the gradient of the curve

y = (2x + a)(x – 2a)^2

at the point
where

x = 1

, as

a

varies, is

A. $-\frac{49}{4}$
B. $-8$
C. $-\frac{25}{4}$
D. $-\frac{7}{4}$
E. $-\frac{47}{16}$

Answer: C

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Question 16

1 mark

Given the simultaneous equations

\log_{10} 2 + \log_{10}(y - 1) = 2 \log_{10} x

\log_{10}(y + 3 - 3x) = 0

the values of

y

are

A. $\frac{5}{2} \pm \frac{3\sqrt{5}}{2}$
B. $3\pm\sqrt{3}$
C. $7\pm 3\sqrt{3}$
D. $3,9$
E. $1, 13$

Answer: C

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Question 17

1 mark

It is given that

y = (1 + 2 \cos x) \cos 2x

for

0 < x < \pi

The complete set of values of

x

for which

y

is negative is

A. $0<x<\frac{2\pi}{3}$ or $\frac{3\pi}{4}<x<\pi$
B. $0<x<\frac{3\pi}{4}$ or $\frac{3\pi}{4}<x<\pi$
C. $0<x<\frac{2\pi}{3}$ or $\frac{3\pi}{4}<x<\pi$
D. $\frac{\pi}{4}<x<\frac{2\pi}{3}$ or $\frac{3\pi}{4}<x<\pi$
E. $\frac{\pi}{4}<x<\frac{2\pi}{3}$
F. $\frac{\pi}{4}<x<\frac{3\pi}{4}$

Answer: D

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Question 18

1 mark

The function

\frac{1-x}{\sqrt{x^2}}

is defined for all

x ≠ 0

.

The complete set of values of

x

for which the function is decreasing is

A. $x \leq −2, x > 0$
B. $-2 < x < 0$
C. $x \leq 1, x≠ 0$
D. $x \geq 1$
E. $-2 \leq x \leq 1, x ≠ 0$
F. $x \leq -2, x \geq 1$

Answer: A

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Question 19

1 mark

The coefficient of

x^3

in the expansion of

(1 + 2x + 3x^2)^6

is equal to twice the coefficient
of

x^4

in the expansion of

(1 – ax^2)^5

.

Find all possible values of the constant

a

.

A. $\pm 2\sqrt{2}$
B. $\pm \sqrt{17}$
C. $\pm \sqrt{34}$
D. $\pm 2\sqrt{17}$
E.There are no possible values of $a$ .

Answer: B

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Question 20

1 mark

The diagram shows a square-based pyramid with base

PQRS

and vertex

O

. All the edges
of the pyramid are of length 20 metres.

[diagram not to scale]

Find the shortest distance, in metres, along the outer surface of the pyramid from

P

to the
midpoint of

OR

.

A. $10\sqrt{5}-2\sqrt{3}$
B. $10\sqrt{3}$
C. $10\sqrt{5}$
D. $10\sqrt{7}$
E. $10\sqrt{5}+ 2\sqrt{3}$

Answer: D

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