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Objective Response Bank
YEAR 12
Mathematics Extension 2
Topics
1. Graphs
2. Complex numbers
3. Conics
4. Integration
5. Volumes
6. Mechanics
7. Polynomials
8. Harder Extension 1 topics
Year 12 Mathematics Extension 2
Graphs
1
Solutions
The diagram shows the graph of the function y  f ( x) .
Which of the following is the graph of y 
f ( x) ?
(B)
(A)
(D)
(C)
2
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Year 12 Mathematics Extension 2
2
The diagram shows the graph of the function y  f ( x) .
Which of the following is the graph of y  f ( x)2 ?
(B)
(A)
(D)
(C)
3
Year 12 Mathematics Extension 2
3
The diagram shows the graph of the function y  f ( x) .
Which of the following is the graph of y  f ( x) ?
(B)
(A)
(D)
(C)
4
Year 12 Mathematics Extension 2
4
The diagram shows the graph of the function y  f ( x) .
Which of the following is the graph of y 
1
?
f  x
(A)
(B)
(C)
(D)
5
Year 12 Mathematics Extension 2
5
The diagram below shows the graph of the function y  f ( x) .
Which diagram represents the graph of y 2  f ( x) ?
(A)
(B)
(C)
(D)
6
Year 12 Mathematics Extension 2
6
What is the graph of y  f  x  given that f ( x)  4  x 2 ?
(A)
(B)
(C)
(D)
7
Which of the following is the sketch of y  log 2 x 
(A)
(B)
(C)
(D)
7
1
?
x
Year 12 Mathematics Extension 2
8
The diagram shows the graph of the function y  f ( x) .
Which of the following is the graph of y  f ( x) ?
(B)
(A)
(D)
(C)
8
Year 12 Mathematics Extension 2
9
The diagram shows the graph of the function y  f ( x) .
Which of the following is the graph of y 
f  x ?
(B)
(A)
(D)
(C)
9
Year 12 Mathematics Extension 2
10 The diagram shows the graph of the function y  f ( x) .
Which of the following is the graph of y 2  f x  ?
(B)
(A)
(D)
(C)
10
Year 12 Mathematics Extension 2
Complex numbers
Solutions
11 Let z  1  i and w  1  2i . What is the value of zw ?
(A) 1  i
1  i
(C) 3  i
(D) 3  i
(B)
12 Let z  3  4i and w  3  i . What is the value of
(A)
3 3  4 (4 3  3)i

4
4
(B)
3 3  4 (4 3  3)i

4
4
(C)
3 3  4 (4 3  3)i

2
2
(D)
3 3  4 (4 3  3)i

2
2
13 Let z  1  2i and w  2  i . What is the value of
(A)
z
?
w
5
?
iw
1  2i
1  2i
(C) 1  2i
(D) 1  2i
(B)
14 Let z  3  i . What is the value of iz ?
(A) 1  3i
1  3i
(C) 1  3i
(D) 1  3i
(B)
15 Let z  2  i and w  1  i . What is the value of 3z  iw ?
(A) 5  4i
5  4i
(C) 7  4i
(D) 7  4i
(B)
11
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Year 12 Mathematics Extension 2
16 What is the value of arg z given the complex number z  1  i 3 ?

3
2
(B) 
3

(C) 
3

(D)
3
(A)

17 What is the value of
z1
given the complex numbers z1  2  2i and z2  1  i 3 ?
z2
(A)
3 1
3 1

i
2
2
(B)
1 3
3 1

i
2
2
(C)
3 1
3 1

i
4
4
(D)
1 3
3 1

i
4
4
18 It is given that 3  i is a root of P( z )  z 3  az 2  bz  10 where a and b are real numbers.
Which expression factorises P(z) over the real numbers?
(A) ( z  1)( z 2  6 z  10)
(B)
( z  1)( z 2  6 z  10)
(C)
( z  1)( z 2  6 z  10)
(D) ( z  1)( z 2  6 z  10)
19 What is the solution to the equation z 2  i z ?
(A) (0, 0) and (0,1)
(B)
(0, 0) and (0, 1)
(C)
(0, 0) , (0, 1) , (
(D) (0, 0) , (0,1) , (
3 1
3 1
, ) and ( 
, )
2 2
2 2
3 1
3 1
, ) and ( 
, )
2 2
2 2
12
Year 12 Mathematics Extension 2
20 Consider the Argand diagram below.
Which inequality could define the shaded area?
(A) 0  z  2
(B) 1  z  2
(C)
0  z 1  2
(D) 1  z  1  2
21 Consider the Argand diagram below.
y
3
2
(1,1)
1
-2
-1
1
-1
-2
Which inequality could define the shaded area?
(A) | z | 1 and | z  (1  i ) | 1
(B)
| z | 1 and | z  (1  i) | 1
(C) | z | 1 and | z  (1  i ) | 1
(D) | z | 1 and | z  (1  i) | 1
13
2
3
x
Year 12 Mathematics Extension 2
22 Consider the Argand diagram below.
Which inequality could define the shaded area?
(A)
z  1  2 and 0  arg( z  i ) 
(B)
z  1  2 and 0  arg( z  i) 
(C)
z  1  1 and 0  arg( z  i ) 
(D)
z  1  1 and 0  arg( z  i) 

4

4

4

4
23 Consider the Argand diagram below.
y
4
3
2
1
-2
-1
1
2
x
-1
-2
Which inequality could define the shaded area?
(A) | z  i | 2 and 0  arg( z  1) 
(C) | z  i | 2 and 0  arg( z  1) 
3
4
(B)

| z  i | 2 and 0  arg( z  1) 
(D) | z  i | 2 and 0  arg( z  1) 
4
14
3
4

4
Year 12 Mathematics Extension 2
24 What is 1  i expressed in modulus-argument form?
(A) (cos
(B)
(C)
(D)


 i sin )
4
4


 i sin )
4
4
3
3
(cos
 i sin )
4
4
2(cos
2(cos
3
3
 i sin )
4
4
25 What is  3  i expressed in modulus-argument form?
(A)
(B)
(C)
(D)
2(cos
2(cos


 i sin )
6
6

 i sin )
6
6
2(cos
2(cos

5
5
 i sin )
6
6
5
5
 i sin )
6
6
26 What is 2  2 3i expressed in modulus-argument form?
(A)
(B)
2
2
 i sin )
3
3
2
2
4(cos
 i sin )
3
3
2(cos
(C)
2(cos
(D)
4(cos




 i sin )
3
3
 i sin )
3
3
27 What is (1  3i ) 1 expressed in modulus-argument form?
(A)
1


(cos  i sin )
4
3
3
(B)
1


(cos  i sin )
4
3
3
(C)
1


(cos  i sin )
2
3
3
(D)
1


(cos  i sin )
2
3
3
15
Year 12 Mathematics Extension 2
28 What are the three roots of z 3  1  0 in modulus argument form?
2π
3
2π
2π
cis0, cis , cis 
3
3
(A) cis0, cis
(B)
(C)
cis0, cis
π
3
π
π
(D) cis0, cis , cis 
3
3
29 Which of the following complex numbers equals ( 3  i)4 ?
(A)
2 
2
(B)
8 
8
(C)
2  2 3i
(D)
8  8 3i
3
3
i
i
30 Let the point R represent the complex number z on an Argand diagram. Which of the
following describes the locus of R specified by | z || z  4 | ?
(A) Perpendicular bisector of (0,0) and (4,0)
(B) Perpendicular bisector of (0,0) and (4,0)
(C) Circle with a centre (0,0) and radius of 2
(D) Circle with a centre (0,0) and radius of 4
16
Year 12 Mathematics Extension 2
Conics
Solutions
31 For the ellipse with the equation
(A)
1
4
(C)
3
4
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x2 y 2

 1 . What is the eccentricity?
4
3
1
(B)
2
(D)
9
16
x2 y2

 1 where a  b  0 . The tangent at P meets the
a 2 b2
tangents at the ends of the major axis at R and T.
32 The point P lies on the ellipse
y
R
P
T
x
What is the equation of the tangent at P?
(A)
ax
by

 a 2  b2
cos sin 
(B)
ax
by

 a 2  b2
sec tan 
(C)
x
y
sec  tan   1
a
b
(D)
x
y
cos   sin   1
a
b
17
Year 12 Mathematics Extension 2
x2 y2

 1 and
a 2 b2
the chord PQ subtends a right angle at (0,0) . Which of the following is the correct
expression?
33 The points P(a cos  , b sin  ) and Q (a cos  , b sin  ) lie on the ellipse
(A)
b2
tan  tan    2
a
(B)
tan  tan   
(C)
tan  tan  
b2
a2
(D)
tan  tan  
a2
b2
a2
b2
x2 y 2

 1 where a  b  0 . The points
a 2 b2
P (a sec  , b tan  ) and Q (a sec  , b tan  ) lie on the hyperbola and the chord PQ subtends
a right angle at the origin.
34 The diagram below shows the hyperbola
y
Q
O
x
P
Use the parametric representation of the hyperbola to determine which of the following
expressions is correct?
a2
(A) sin  sin    2
b
a2
b2
(B)
sin  sin  
(C)
tan  tan   
(D)
tan  tan  
a2
b2
a2
b2
18
Year 12 Mathematics Extension 2
c
c
) and Q (cq, ) lie on the same branch of the hyperbola xy  c 2 (p 
p
q
q). The tangents at P and Q meet at the point T. What is the equation of the normal to the
hyperbola at P?
35 The points P (cp,
(A)
p 2 x  py  c  cp 4  0
(B)
p3 x  py  c  cp 4  0
(C)
x  p 2 y  2c  0
(D)
x  p 2 y  2cp  0
36 Consider the hyperbola with the equation
x2 y 2

 1.
16 9
What is the eccentricity of the hyperbola?
(A)
3
4
(B)
5
4
(C)
9
16
(D)
25
16
37 Consider the hyperbola with the equation
x2 y 2

1.
144 25
What are the equations of the directrices?
(A)
x
13
144
(B)
x
13
25
(C)
x
25
13
(D)
x
144
13
19
Year 12 Mathematics Extension 2
x2 y 2

 1.
16 9
What are the coordinates of the foci of the hyperbola?
(A) ( 4, 0)
(B) (0, 4)
38 Consider the hyperbola with the equation
(C)
(0, 5)
(D) (5, 0)
x2 y 2
39 Consider the hyperbola with the equation

 1.
4 3
What are the coordinates of the vertex of the hyperbola?
(A) ( 2, 0)
(B) (0, 2)
(C)
(0, 4)
(D) ( 4, 0)


c
c
40 The points P cp,  and Q cq,  , p  q, lie on the same branch of the hyperbola
p
q


2
xy  c . The tangents at P and Q meet at the point T.
Which of the following expressions is the equation of the tangent to the hyperbola at Q?
(A)
x  q 2 y  2cq
(B)
x  q 2 y  2c 2
(C)
x  p 2 y  2cp
(D)
x  p 2 y  2c 2
20
Year 12 Mathematics Extension 2
Integration
Solutions
41 What is the value of

1
0
xe x dx ?
2
(A)
1 e
2e
(B)
e 1
2e
(C)
2e  1
e
(D)
1  2e
e
42 What is the value of
ex
0 1  e x dx ?
1
(A) loge 1  e 
(C)
log e
(B) 1
1  e 
(D) log e
2
43 Which of the following is an expression for
x

16  x 2
e
2
2
dx ?
(A)
2 16  x2  c
(B)
 16  x2  c
(C)
1
16  x 2  c
2
(D)

44 Which of the following is an expression for

Use the substitution u  4  sin x .
(A) 4ln | 4  sin x | c
(C)
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sin x cos x
dx ?
4  sin x
(B)
 sin x  4ln | 4  sin x | c
1
16  x 2  c
2
4ln | 4  sin x | c
(D) sin x  4ln | 4  sin x | c
45 Which of the following is an expression for

Use the substitution u  cos x .
cos3 x 3cos5 x 3cos 7 x cos9 x



c
3
5
7
9
(A)

(B)
 cos3 x  3cos5 x  3cos 7 x  cos9 x  c
(C)
cos3 x 3cos5 x 3cos 7 x cos9 x



c
3
5
7
9
(D) cos3 x  3cos5 x  3cos 7 x  cos9 x  c
21
cos2 x sin 7 xdx ?
Year 12 Mathematics Extension 2
46 Which of the following is an expression for

1n  x  3 
1n  x  3 
1n  x  3 
1

x 2  6 x  10
dx ?

x  6 x  10   c
x  6 x  10   c
x  6 x  10   c
(A) 1n x  3  x 2  6 x  10  c
(B)
(C)
(D)
2
2
2
47 Which of the following is an expression for
 x 3
(A) sin 1 
c
 2 
(C)
1

7  6 x  x2
(B)
 x 3
sin 1 
c
 4 
dx ?
 x 3
sin 1 
c
 2 
 x 3
(D) sin 1 
c
 4 
48 Which of the following is an expression for
x
2
2
dx ?
 4 x  13
(A)
1 1 ( x  2)
tan
c
3
3
(B)
2
( x  2)
tan 1
c
3
3
(C)
1 1 ( x  2)
tan
c
9
9
(D)
2
( x  2)
tan 1
c
9
9
49 Which of the following is an expression for
(A) ln( x 2  1)  ln | x  2 | c
(B)
ln( x 2  1)  2ln | x  2 | c
(C)
ln( x 2  1)  3tan 1 x  ln | x  2 | c
(D) ln( x 2  1)  3tan 1 x  2ln | x  2 | c
22
 x
2
7x  4
dx ?
 1 ( x  2)
Year 12 Mathematics Extension 2
50 Which of the following is an expression for
(A)
1
4
ln x  1  ln x  4  c
5
5
(B)
1
4
 ln x  1  ln x  4  c
5
5
(C)
1
5
ln x  1  ln x  4  c
4
4
(D)
1
5
 ln x  1  ln x  4  c
4
4
51 Which of the following is an expression for
x
  x  1 ( x  4) dx ?
4 x2  5x  1
  x  3  x2  1 dx ?
(A) 1n  x  3  x 2  1  c
(B) 1n  x  3  x2  1  c
2
(C) 1n  x  3  x 2  1  tan -1 x  c
(D) 1n  x  3  x2  1  tan -1 x  c
2
52 What is the value of


2
0
(A) 0.322
(C) 1.107
1
d ? Use the substitution t  tan 2 .
cos   2sin   3
(B) 0.785
(D) 1.570
53 What is the value of   sec xdx ? Use the substitution t  tan 2x .
1 t
| c
t 1
t 1
| c
(D) ln |
t 1
(A) ln | (t  1)(t  1) |  c
(C)
(B)
ln | (1  t )(t  1) |  c
54 What is the value of
ln |
 x  x  2 dx ? Use the substitution u  x  2 .
3
5
1
(A)
1
7
(B)
2
7
(C)
1
3
(D)
2
3
23
Year 12 Mathematics Extension 2
55 Which of the following is an expression for
(A)
x2
x2
log e x   c
2
4
(B)
x2
x
log e x   c
2
2
(C)
(D)
x2
c
4
x
x log e x   c
2
(A)
x 4 log e x x5

c
4
20
(B)
x 4 log e x x5
 c
4
5
(C)
x5 log e x x5

c
5
20
(D)
x5 log e x x5
 c
5
5
57 What is the value of
(C)
e
x log e x 
56 Which of the following is an expression for
(A)
 x log xdx ? Use integration by parts.

12

4
log e xdx ? Use integration by parts.

6
0
x cos xdx ? Use integration by parts.
3 1

12
2

x
(B)
3
1
2
x
58 Let I n   cosntdt , where 0  x 
(D)

3 1

12
2

12

.
2
Which of the following is the correct expression for I n ?
0
(A)
 n 1 
In  
 I n  2 with n  2 .
 n 
(B)
 n 1
In  
 I n  2 with n  2 .
 n 
(C)
I n  n  n  1 I n  2 with n  2 .
(D)
I n  n  n  1 I n  2 with n  2 .
24
3
1
2
Year 12 Mathematics Extension 2

59 Let I n   x n sin xdx , where 0  x 

.
2
Which of the following is the correct expression for I n ?
0
(A)  n  n(n  1) I n  2
(B)
 n  n(n  1) I n  2
(C)  n  n(n  2) I n  2
(D)  n  n(n  2) I n  2
60 Let I n   x n e ax dx . Which of the following is the correct expression for I n ?
(A)
x neax
In 
 nI n 1
a
(B)
In 
x n eax n
 I n 1
a
a
(C)
In 
x neax
 nI n 1
a
(D)
In 
x neax n
 I n 1
a
a
25
Year 12 Mathematics Extension 2
Volumes
Solutions
Main Menu
61 The parabola y  x3 is rotated about the y axis {x : 0  x  2} to form a solid.
What is the volume of this solid using the method of slicing?
(A)
2
cubic units
5
(B)
3
cubic units
5
(C)
(D)
93
cubic units
5
96
cubic units
5
62 The area between the curve y  3x  x 2 , the x-axis, x  0 and x  3 , is rotated about the
y-axis to form a solid.
What is the volume of this solid using the method of slicing?
(A)
9
cubic units
4
(B)
9
cubic units
2
(C)
27
cubic units
4
(D)
27
cubic units
2
26
Year 12 Mathematics Extension 2
63 The region enclosed by the ellipse ( x  1) 2 
y2
 1 is rotated about the y axis to
4
form a solid.
What is the correct expression for volume of this solid using the method of slicing?
2
(A) V    1  y 2 dy
2
2
(B) V   2 1  y 2 dy
2
2
(C) V    4  y 2 dy
2
2
(D) V   2 4  y 2 dy
2
64 The region is bounded by the lines x  1 , y  1 , y  1 and by the curve x   y 2 . The
region is rotated through 360º about the line x  2 to form a solid. What is the correct
expression for volume of this solid?
(A) V     y 4  4 y 2  3dy
1
1
(B) V     y 4  4 y 2  3dy
1
1
(C) V     y 4  4 y 2  4dy
1
1
(D) V     y 4  4 y 2  4dy
1
1
27
Year 12 Mathematics Extension 2
65 What is the volume of the solid formed when the region bounded by the curves y  2 x3
and y  2 x is rotated about the x-axis? Use the method of slicing.
(A)
5
cubic units
14
(B)
10
cubic units
14
(C)
5
cubic units
7
(D)
10
cubic units
7
66 What is the volume of the solid formed when the region bounded by the curves y  x 2 ,
y  30  x 2 and the y-axis is rotated about the y-axis? Use the method of slicing.
What is the correct expression for volume of this solid using the method of cylindrical
shells?
5


(A) V   2 x 2  30  x 2 dx
0


5
(B) V   2 x x 2  30  x 2 dx
0
5
(C) V   2
0
5
(D) V   2 x
0


30  x 2  x 2 dx


30  x 2  x 2 dx
28
Year 12 Mathematics Extension 2
67 The region bounded by y  4 x 2  x 4 and 0  x  2 is rotated about the y axis to form a
solid.
What is the volume of this solid using the method of cylindrical shells?
(A)
16
units3
3
(B)
8
units3
3
(C)
20
units3
3
(D)
32
units3
3
68 A solid is formed by rotating the region enclosed by the parabola y 2  4ax , its vertex
(0,0) and the line x  a , about the x-axis.
y
x
What is the volume of this solid using the method of cylindrical shells?
(A)
7 a 3
units3
4
(B)
7 a 3
units3
8
(C)
7 a 3
units3
16
(D)
2 a 3 units3
29
Year 12 Mathematics Extension 2
69 The region enclosed by y  sin x , y  0 and x 

is rotated around the y-axis to
2
produce a solid. What is the volume of this solid using the method of cylindrical shells?
(A)  units3
(C)

units3
2
3
units3
(D)
2 units3
(B)
2
30
Year 12 Mathematics Extension 2
Mechanics
70
71
72
Solutions
Main Menu
A particle of mass m falls from rest under gravity and the resistance to its motion is
mkv 2 , where v is its speed and k is a positive constant. Which of the following is the
correct expression for square of the velocity where x is the distance fallen?
(A) v 2 
g
1  e2kx 
k
(B)
v2 
g
1  e 2 kx 

k
(C)
v2 
g
1  e 2 kx 

k
(D) v 2 
g
1  e2kx 
k
A rock is projected vertically upwards from ground level. Assume air resistance is kv,
where v is the velocity of the rock and k is a positive constant. The rock falls back to
ground level under the influence of g, the acceleration due to gravity. Consider the
rock’s motion starting from maximum height. Let y be the displacement and t be the
time elapsed after the rock has reached maximum height. Assume the rock has a unit
mass. Which of the following is the correct expression for velocity?
(A) v 
g kt
(e  1)
k
(B)
v
g kt
(e  1)
k
(C)
v
k kt
(e  1)
g
(D) v 
k kt
(e  1)
g
A particle of mass m is moving in a straight line under the action of a force.
m
(6  10 x)
x3
What of the following is an expression for its velocity in any position, if the particle
starts from rest at x  1 ?
F
(A) v  
1
(3  10 x  7 x 2 )
x
(B)
v   x (3  10 x  7 x 2 )
(C)
v
1
2(3  10 x  7 x 2 )
x
(D) v   x 2(3  10 x  7 x 2 )
31
Year 12 Mathematics Extension 2
73
A particle of mass m is projected vertically upwards with an initial velocity of
u ms-1 in a medium in which the resistance to the motion is proportional to the square
of the velocity v ms-1 of the particle or mkv2. Let x be the displacement in metres of the
particle above the point of projection, O, so that the equation of motion is
x    g  kv 2  where g ms-2 is the acceleration due to gravity. Assume k = 10 and the
acceleration due to gravity is 10 ms-2.
Which of the following gives the correct expression for the time taken?
74
(A) t 
1
 tan -1 u  tan -1 v 
10
(B)
t
1
tan -1 v  tan -1 u 

10
(C)
t
1
tan -1 u  tan -1 v 

10
(D) t 
1
tan -1 v  tan -1 u 

10
A conical pendulum consists of a body P of mass m kg and a string of length l metres.
Point A is fixed and the body P rotates in a horizontal circle of radius r and centre O at
a constant angular velocity of  radians per second. OA is vertical and has a length of
h metres. The angle OAP is  radians. The body, P, is subject to a gravitational force
of mg newtons. The tension in the string is T newtons. Which of the following gives
the correct resolution of forces on P in the horizontal and vertical directions?
(A) T sin   mg  0 and T cos  mr 2
(B)
T sin   mg  0 and T cos  mr 2
(C) T cos   mg  0 and T sin   mr 2
(D) T cos  mg  0 and T sin   mr 2
32
Year 12 Mathematics Extension 2
75
Two light inextensible strings are attached to a particle of mass m. The particle
describes a horizontal circle with constant angular velocity  . Which of the following
gives the correct resolution of forces in the horizontal and vertical directions?
(A) T sin   T sin   m 2 r and T cos   T cos   mg
(B)
T sin   T cos   m 2 r and T sin   T cos   mg
(C) T sin   T sin   m 2 r and T cos   T cos   mg
(D) T sin   T cos   m 2 r and T sin   T cos   mg
76
A body of mass m kg is attached by two light rods AB and BC. Both rods are l metres
in length. Rod AB is hinged at point A and makes an angle  with the vertical shaft.
Rod BC is attached to a ring that can slide freely along the vertical shaft.
T1
T2
What are the tensions in the rods?
1
1
mg sec  ml 2  and T2   ml 2  mg sec 

2
2
1
1
T1   mg sin   ml 2  and T2   ml 2  mg sin  
2
2
(A) T1 
(B)
(C) T1 
1
1
mg sec  ml 2  and T2   ml 2  mg sec 

2
2
(D) T1 
1
1
mg sin   ml 2  and T2   ml 2  mg sin  

2
2
33
Year 12 Mathematics Extension 2
77
Two light inextensible strings PQ and QR each of length l are attached to a particle of
mass m at Q. The other ends P and R are fixed to two points in a vertical line such that
P is a distance l above R. The particle describes a horizontal circle with constant
angular velocity  .
P
l

l
r
Q

l
mg
R
What is the tension in the strings?
(A) T1 
m 2
m
(lw  2 g ) and T2  (lw2  g )
2
2
T1 
m 2
m
(lw  2 g ) and T2  (lw2  g )
2
2
(B)
(C) T1  m(lw2  2 g ) and T2  m(lw2  g )
(D) T1  m(lw2  2 g ) and T2  m(lw2  g )
78
79
What is the angle at which a road must be banked so that a car may round a curve with
a radius of 200 metres at 100 km/h without sliding? Assume that the road is smooth.
(A)
21.49
(B)
22.49
(C)
23.49
(D)
24.49
A conical pendulum consists of a body P of mass m kg and a string of length l metres.
A is fixed and the body P rotates in a horizontal circle of radius r and centre O at a
constant angular velocity of  radians per second. OA is vertical and OA = h metres.
The angle OAP is . The body, P, is subject to a gravitational force of mg newtons.
The tension in the string is T newtons. What is the angular velocity?
(A)
g
h
(B)
h
g
(C)
2
g
h
(D)
2
h
g
34
Year 12 Mathematics Extension 2
Polynomials
80
81
82
83
Solutions
Main Menu
Let  ,  and  be roots of the equation x3  x 2  2 x  5  0 . Which of the following
polynomial equations have roots   2 ,   2 and   2 ?
(A)
x3  7 x 2  14 x  3  0
(B)
x3  7 x 2  21x  3  0
(C)
x3  x 2  6 x  9  0
(D)
x3  2 x 2  6 x  9  0
The polynomial equation x3  5 x 2  6  0 has roots  ,  and  . Which of the
following polynomial equations have roots   1 ,   1 and   1 ?
(A)
x3  8 x 2  13x  0
(B)
x3  8 x 2  7 x  0
(C)
x3  3x 2  7 x  2  0
(D)
x3  2 x 2  7 x  2  0
The polynomial equation x3  3x 2  x  2  0 has roots  ,  and  . Which of the
following polynomial equations have roots 2     ,   2   and     2 ?
(A)
x3  6 x 2  44 x  49  0
(B)
x3  12 x 2  44 x  49  0
(C)
x3  3x 2  36 x  5  0
(D)
x3  6 x 2  36 x  5  0
Let  ,  and  be roots of the equation x3  3x 2  4  0 . Which of the following
polynomial equations have roots  2 ,  2 and  2 ?
(A)
x3  9 x 2  24 x  4  0
(B)
x3  9 x 2  12 x  4  0
(C)
x3  9 x 2  24 x  16  0
(D)
x3  9 x 2  12 x  16  0
35
Year 12 Mathematics Extension 2
84
The polynomial equation x3  3x 2  x  2  0 has roots  ,  and  . Which of the
1 1
1
following polynomial equations have roots ,
and ?

85
(A)
x3  x 2  3x  1  0
(B)
x3  2 x 2  3x  1  0
(C)
2 x3  x 2  3x  1  0
(D)
2 x3  2 x 2  3x  1  0


The polynomial equation P( x)  2 x 4  3x3  2 x 2  7 x  3 has roots  ,  ,  and  .
1 1 1
1
Which of the following polynomial equations have roots ,
,
and ?

(A)
2 x 4  3 x3  x 2  5 x  4
(B)
2 x 4  3 x3  x 2  5 x  4
(C)
3 x 4  7 x3  2 x 2  3 x  2
 

(D) 3x4  7 x3  2 x2  3x  2
86
87
The polynomial equation x3  x 2  2 x  5  0 has roots  ,  and  . Which of the
following polynomial equations have roots  2 ,  2 and  2 ?
(A)
x3  5 x 2  6 x  25  0
(B)
x3  5 x 2  14 x  25  0
(C)
x3  4 x 2  5 x  1  0
(D)
x3  4 x 2  5 x  1  0
The polynomial equation x3  5 x 2  6  0 has roots  ,  and  . Which of the
following polynomial equations have roots  2 ,  2 and  2 ?
(A)
x3  25 x 2  60 x  36  0
(B)
x3  25 x 2  60 x  12  0
(C)
x3  x 2  12 x  36  0
(D)
x3  x 2  12 x  12  0
36
Year 12 Mathematics Extension 2
88
The equation 24 x3  12 x 2  6 x  1 has roots  ,  and  .
What is the value of  if      ?
89
90
(A)

(C)
1
2
1
2
(B)
1
4
(D) 1
What are the zeros of the equation x 4  x 2  6 x  4  0 over the complex field given
that it has a rational zero of multiplicity 2?
(A)
1 , 1  5i and 1  5i
(B)
1 , 1  3i and 1  3i
(C)
1 , 1  5i and 1  5i
(D)
1 , 1  3i and 1  3i
The polynomial P( x)  x 4  ax 2  bx  28 has a double root at x  2 .
What are the values of a and b?
(A) a  11 and b  12
a  5 and b  12
(C) a  11 and b  12
(D) a  5 and b  12
(B)
91
92
The polynomial P( x)  x5  3x4  4 x3  4 x2  3x 1 has x  1 as a root of multiplicity
3 and x  i as a root. Which of the following expressions is a factorised form of P ( x )
over the complex numbers?
(A)
P( x)  ( x  1)3 ( x 1)( x  1)
(B)
P( x)  ( x 1)3 ( x 1)( x  1)
(C)
P( x)  ( x  1)3 ( x  i)( x  i)
(D)
P( x)  ( x 1)3 ( x  i)( x  i)
What are the values of real numbers p and q such that 1  i is a root of the equation
z 3  pz  q  0 ?
(A)
p  2 and q  4
(B)
p  2 and q  4
(C)
p  2 and q  4
(D)
p  2 and q  4
37
Year 12 Mathematics Extension 2
Harder Extension 1 topics
93
Solutions
Main Menu
Two equal circles touch externally at B. XB is a diameter of one circle. XZ is the
tangent from X to the other circle and cuts the first circle at Y.
Z
Y
X
B
A
Which is the correct expression that relates XZ to XY?
(A) 3XZ  4 XY
(B) XZ  2 XY
94
(C)
2 XZ  3XY
(D)
2 XZ  5 XY
What is the derivative of sin 1 x  1  x2 ?
(A)
(B)
(C)
(D)
1 x
1 x
1 x
1 x
1 x
1 x
1 x
1 x
38
C
Year 12 Mathematics Extension 2
n
95
Using the binomial theorem (1  x) n  nC0  nC1 x1  nC2 x 2  ...  nCn x n   nCk x k
k 0
which of the following expressions is correct?
n
1
n(n  1)(n  2)...(n  k ) 1

(A) (1  ) n  
n
nk
k!
k 0
(B)
n
1
n(n  1)(n  2)...(n  k )
1
(1  )n  

k
n
n
(k  1)!
k 0
(C)
n
1
n(n  1)(n  2)...(n  k  1) 1
(1  ) n  

n
nk
k!
k 0
n
1
n(n  1)(n  2)...(n  k  1)
1

(D) (1  ) n  
k
n
(k  1)!
n
k 0
96
The labor party conducted a survey for the 2010 election. The ratio of the votes in
three seats X, Y and Z was 4:3:2 respectively. The percentage of votes for Ms Gillard in
these seats was 60%, 30% and 90% respectively. Ten voters were chosen at random,
what is the probability that Ms Gillard gained at least eight votes?
(A) 0.1672897536
(B) 0.2509346304
(C) 0.3345795072
(D) 0.418224384
97
A coin is tossed 20 times. What is the probability of obtaining at most 3 heads?
(A) 0.0000029
(B) 0.0002
(C) 0.0013
(D) 0.0059
98
1
What is the solution to the equation tan -1 4 x   tan -1 3x   tan -1   ?
7
(A)
x
1
2
or x 
7
7
(B)
x
1
2
or x 
3
3
(C)
x
1
1
or x 
3
4
(D)
x  3 or x  4
39
Year 12 Mathematics Extension 2
99 What is the solution to the inequation 2sin3x  1 if 0  x  2 ?
Hint: Use a sketch.
(A)
(B)
(C)
(D)

6

6

18

18
x
5 13
17 25
29
x
x
,
,
6
6
6
6
6
x
7 13
20 25
31
x
x
,
,
6
6
6
6
6
x
5 13
17 25
29
x
x
,
,
18 18
18
18
18
x
7 13
20 25
31
x
x
,
,
18 18
18
18
18
100 What is the solution to the inequation
x(5  x)
 3 ?
x4
2  x  4 or x  6
(B) 1  x  4 or x  5
(A)
4  x  6 or x  2
(D) 4  x  5 or x  1
(C)
101 A rock is projected to just clear two poles of height h metres at distances of b and c
metres from the point of projection. If v is the velocity of the projection at an angle 
to the horizontal. Which of the following is the correct expression for square of the
velocity?
(A) v 2 
2(b  c) tan 
g sec2 
(B)
v2 
2(b  c) tan 
g sec2 
(C)
(b  c) g sec2 
v 
2 tan 
2
(D) v 2 
(b  c) g sec2 
2 tan 
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