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Unit 2 Maths Methods (CAS) Exam 2013
Monday November 18 - 1.50 pm
Reading time: 10 Minutes
Writing time: 80 Minutes
Instruction to candidates:
Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners,
rulers, a single bound exercise book containing notes and class-work, CAS calculator.
Materials Supplied:
Question and answer booklet, detachable multiple choice answer sheet at end of booklet.
Instructions:• Write your name and that of your teacher in the spaces provided.• Answer all short answer questions in this booklet where indicated.• Always show your full working where spaces are provided.• Answer the multiple choice questions on the detachable answer sheet.
Section A Section B Total exam
/20 /30 /50
1
Section A – Multiple choice questions (20 marks)
Question 1
Which of the intervals correctly describes what shown on the number-line below?
a) (!4,!2)" (0,#)b) [!4,!2]" [0,#]c) [!4,!2)" [0,#)d) (!4,!2]" [0,#]e) [!4,!2]" (0,#)
Question 2
What is the equation of the circle shown here?
a) (x +1)2 + (y !1)2 = 4
b) (x +1)2 ! (y +1)2 = 4
c) (x !1)2 + (y !1)2 = 4
d) (x +1)2 + (y +1)2 = 2
e) (x +1)2 + (y +1)2 = 4
2
Question 3
The angle 210° in radians is equal to:
a)7!6
b)6!7
c) 210!
d)210!
e)!210
Question 4
For an angle (θ) in the second quadrant (!2<" < ! ) , which of the following statements is incorrect?
a) sin(! ) > 0
b) cos (! ) < 0
c) tan(!) > 0
d) sin ! "#( ) = sin #( )
e) sin 2! "#( ) = "sin #( )
Question 5
The graph shown here could be described by the equation:a) y = 2 ! 4sin2x
b) y = 2 ! 2sin2x
c) y = !4sin x2
d) y = 2 ! 2sin x2
e) y = 2 ! 2sin" x2
3
Question 6
The solutions to the equation !1= 2 sin 2"( )where 0 <! < 2" are:
a)5!8, 7!8,13!8,15!8
b)1!8, 3!8, 9!8,11!8
c)5!8, 7!8
d)!4, 3!4
e) undefined
Question 7
The graph here shows the value of the US dollar (in Australian dollars) from the start of January to the start of
September.
During this 8 month period, the average value of the dollar:
a) Increased by 0.5 cents per month.b) Decreased by 2.5 cents per month.
c) Decreased by 0.625 cents per month.d) Decreased by 0.5 cents per month.e) Remained constant.
Question 8
The cubic function y = 2x2 (x ! 3) has stationary points at:
a) (0,0) and 3,0( )
b) (0,0) and 3,3( )
c) (0,0) and 2,0( )
d) (0,0) and 2,!8( )
e) (0,0) and 3,!8( )
4
Question 9
At the point x = 3 , the gradient of the curve y = x2 + 6x ! 2 is equal to:
a) 12b) -12
c) 14d) 10e) 25
Question 10
The function f (x) = ax3 + bx2 + cx + d has the derivative function:
a) f '(x) = !3ax2 ! 2bx2 ! cx ! d
b) f '(x) = ax2 + bx + c
c) f '(x) = 3ax2 + 2bx2 + cx + d
d) f '(x) = 3ax2 + 2bx + c
e) f '(x) = 0
Question 11
The derivative of the function y = x3 is:
a) 3x
b)3x2
c)x!23
3
d) x23
e)2x
23
3
Question 12
The tangent to the curve y = x2 ! 2x at the point 3,3( ) is given by the equation:
a) y = 4x +1
b) y = !4x +1c) y = 4x ! 9
d) y = 4x !1
e) y = 6x !15
5
Question 13
A possible anti-derivative of the function f (x) = 7x3 + 2x2 could be:
a) F(x) = 21x4 + 4x3
b) F(x) = x3
4+x2
2
c) F(x) = 7x3 + 2x2 + c
d) F(x) = 7x4 ! 2x3 + c
e) F(x) = 7x4
4+2x3
3! 4
Question 14
Which of the following correctly shows the
calculation of the area (A) between the curve
y = !x2 + 2x and the x axis?
a) A = x3
3! x2
"
#$
%
&'0
2
=43
b) A = !x3
3+ x2
"
#$
%
&'0
2
=43
c) A = !x2 + 2x"# $%02= 0
d) A = x3
3! x2
"
#$
%
&'2
0
= !43
e) A = 2x ! 2[ ]02= 2
Question 15
For the matrix multiplication shown below, what is the value of x?
2 x 5!" #$
x34
!
"
%%%
#
$
&&&
= 60[ ]
a) 5
b) 8c) 10
d) 16e) 34
6
Question 16
Which the following matrix operations cannot be performed?
A = 4 3!4 7
"
#$
%
&' B = !4
3
"
#$
%
&' C = 1 0 1!" #$ D =
4 !49 !86 0
"
#
$$$
%
&
'''
a) ABb) BA
c) A-1
d) CD
e) A2
Question 17
Using a standard deck of cards (52 cards), the chance of dealing two cards that are both aces is:a) 1 in 26
b) 1 in 169c) 1 in 221d) 3 in 676
e) 1 in 1352
Question 18
A football player is practising kicking goals from the 50 metre line at a training session. He has found that if he kicks through the goal posts, then there is an 80% chance of the next kick also being successful.
However, there is only a 60% chance of kicking through the goalposts if the last kick missed. If the first kick of the session is successful, what is the chance that the fourth kick is also successful?
a) 20%b) 25%c) 75%
d) 76%e) 80%
Question 19
Two independent events have the probabilities Pr(A) = 0.5, Pr(B) = 0.7, Pr(A⋂B) = 0.2. Which of the following
statements is incorrect?
a) Pr(A∪B) = 1
b) Pr(A’∩B’) = 0
c) Pr(A’∩B) = 0.5
d) Pr(A∩B’) = 0.5
e) The the most likely outcome is that only event B occurs.
7
Question 20
The Venn diagram shown here gives the results of a survey of 80 girls at a school, about whether they play
basketball or netball.
Which of the following statements about the results is incorrect?
a) Netball is more popular than basketball.b) 8 of the the girls play both sports.
c) Most of the girls at the school don’t play either of these sports.d) There are 17 basketballers.e) There are 36 girls that play one or both games.
8
Section B – Short answer questions (30 marks)
Question 1
For the function f (x) = 1x+ 3 :
a) State the domain of the function. (1 mark)
b) State the range of the function. (1 mark)
c) Sketch the graph of the function f(x) below. You do not need to find intercepts. (1 mark)
f(x) f-1(x)
d) On the other axes, sketch the graph of the inverse function f !1(x) . (2 marks)
9
Question 2
The temperature inside a house varies periodically throughout the day. It can be approximated by the equation:
T (t) = 21! 3cos " t12#$%
&'(
(Time is measured in hours from midnight.)
a) State the minimum and maximum that the temperature reaches. (1 mark)
Minimum: ° Maximum: °
b) Find the time that it takes the temperature to return to the minimum. (1 mark)
hours
c) Calculate the times (to the nearest half hour) in the first 24 hours at which the temperature is 22°. (2 marks)
10
Question 3
The amount of water V(t) remaining in the dam for the first 100 days of the year has been modelled by the
equation V (t) = 10t3
3! 350t 2 + 6000t + 200000, 0 " t "100
a) Find the initial volume of the dam. (1 mark)
litres
b) Find the derivative V’(t) that gives the rate of change of volume with respect to time. (1 mark)
c) Use this derivative to find the days on which the volume is a minimum. (2 marks)
d) Calculate the minimum amount of water in the dam. (2 marks)
litres
e) Use a CAS calculator or other method to find the days on which the rate of water loss is 4000 litres/day. (2 marks)
11
Question 4
Two matrix transformations (A & B) are to be applied to the point (1,5). The point is to be reflected around the x axis by matrix A, then dilated by a factor of 3 away from the y axis by matrix B.
a) Give each of the matrices used for the transformation. (2 marks)
A =_ __ _
!
"##
$
%&&
B =_ __ _
!
"##
$
%&&
b) Find the new co-ordinates of the point. (1 mark)
Question 5
For the matrix A = 3 61 !1
"
#$
%
&' :
a) Find the determinant of A. (1 mark)
b) Find the inverse A. (1 mark)
c) Use this information to find the solution to the simultaneous equations 3x + 6y = 24 and x ! y = !1 .
(2 marks)
12
Question 6
A survey of 250 cyclists in the Warrnambool area was conducted about what type of riding they did. It was found that 80 of the riders surveyed took part in road racing (R) and 35 of those surveyed took part in mountain bike racing (M). 150 of the cyclists did not complete in either type of race.
a) Complete the table below, showing the number in each group in each group. (4 marks)
M M’
R
R’
250
b) What is the probability that a randomly selected cyclist races in either road or mountain bike events? (1 mark)
c) What is the probability that a randomly selected mountain bike racer also races on the road? (1 mark)
13
Answer sheet for section A
1. a b c d e
2. a b c d e
3. a b c d e
4. a b c d e
5. a b c d e
6. a b c d e
7. a b c d e
8. a b c d e
9. a b c d e
10. a b c d e
11. a b c d e
12. a b c d e
13. a b c d e
14. a b c d e
15. a b c d e
16. a b c d e
17. a b c d e
18. a b c d e
19. a b c d e
20. a b c d e
15
Answer sheet for section A
1. a b c d e
2. a b c d e
3. a b c d e
4. a b c d e
5. a b c d e
6. a b c d e
7. a b c d e
8. a b c d e
9. a b c d e
10. a b c d e
11. a b c d e
12. a b c d e
13. a b c d e
14. a b c d e
15. a b c d e
16. a b c d e
17. a b c d e
18. a b c d e
19. a b c d e
20. a b c d e
Unit 2 Maths Methods (CAS) Exam 2013 Solutions
1
Section B – Short answer questions (30 marks)
Question 1
For the function f (x) = 1x+ 3 :
a) State the domain of the function. (1 mark)
R\ 0{ } or !",0( )# 0,"( )
b) State the range of the function. (1 mark)
R\ 3{ } or !",3( )# 3,"( )
c) Sketch the graph of the function f(x) below. (1 mark)
f(x) f-1(x)
d) On the other axes, sketch the graph of the inverse function f !1(x) . (2 marks)
Unit 2 Maths Methods (CAS) Exam 2013 Solutions
2
Question 2
The temperature inside a house varies periodically throughout the day. It can be approximated by the
equation:
T (t) = 21! 3cos " t12#$%
&'(
(Time is measured in hours from midnight.)
a) State the minimum and maximum that the temperature reaches. (1 mark)
Minimum: 18 ° Maximum: 24 °
b) Find the time that it takes the temperature to return to the minimum. (1 mark)
Time period: T =
2!! /12
= 24 h
24 hours
c) Calculate the times (to the nearest half hour) in the first 24 hours at which the temperature is 22°.
(2 marks)
22 = 21 !3cos "t
12
#
$%
&
'(
!
13= cos "t
12
#
$%
&
'(
1.91 = !t
12
t = 7.3 and 16.7
7.30 am 4.30 pm
Unit 2 Maths Methods (CAS) Exam 2013 Solutions
3
Question 3
The amount of water V(t) remaining in the dam for the first 100 days of the year has been modelled by the
equation V (t) = 10t3
3! 350t 2 + 6000t + 200000, 0 " t "100
a) Find the initial volume of the dam. (1 mark)
V (0)=200,000 200,000 litres
b) Find the derivative V’(t) that gives the rate of change of volume with respect to time. (1 mark)
V '(t)=10t2!700t+6000
c) Use this derivative to find the day on which the volume is a minimum. (2 marks)
0=10t2!700t+6000
0=10(t2!70t+600)
0=10(t!10)(t!60)
t =10 and 60
The minimum is at t = 60
60
d) Calculate the minimum amount of water in the dam. (2 marks)
V (60)=10!603
3"350!602+6000!60+200,000
V (60)=20,000
20,000 litres
e) Use a CAS calculator or other method to find the days on which the rate of water loss is 4000 litres/day. (2 marks)
!4000=10t2!700t+6000
0=10t2!700t+10,000=10(t2!70t+1000)
0=10(t!20)(t!50)
t =20 and50
20 and 50
Unit 2 Maths Methods (CAS) Exam 2013 Solutions
4
Question 4
Two matrix transformations (A & B) are to be applied to the point (1,5).
The point is to be reflected around the x axis by matrix A, then dilated by a factor of 3 away from the y axis
by matrix B.
a) Give each of the matrices used for the transformation. (2 marks)
A= 1 0
0 !1"
#$
%
&'
B = 3 0
0 1!
"#
$
%&
b) Find the new co-ordinates of the point. (1 mark)
3 00 1
!
"#
$
%&
1 00 '1
!
"#
$
%&
15
!
"#
$
%&=
3'5
!
"#
$
%& The point is now at (3,-5)
Question 5
For the matrix A = 3 61 !1
"
#$
%
&' :
a) Find the determinant of A. (1 mark)
det A= (3!"1)"(6!1)="9
b) Find the inverse A. (1 mark)
3 61 !1
"
#$
%
&'
!1
=!19
!1 !6!1 3
"
#$
%
&'=
19
23
19
!13
"
#
$$$$
%
&
''''
c) Use this information to find the solution to the simultaneous equations 3x + 6y = 24 and x ! y = !1 .
(2 marks)
3 61 !1
"
#$
%
&'
xy
"
#$
%
&'= 24
1"
#$
%
&'
xy
!
"#
$
%&= 3 6
1 '1!
"#
$
%&
'124'1
!
"#
$
%&=
19
1 61 '3
!
"#
$
%&
24'1
!
"#
$
%&=
19
1827
!
"#
$
%&=
23
!
"#
$
%&
x =2 y =3
Unit 2 Maths Methods (CAS) Exam 2013 Solutions
5
Question 6
A survey of 250 cyclists in the Warrnambool area was conducted about what type of riding they did. It was
found that 80 of the riders surveyed took part in road racing (R) and 35 of those surveyed took part in
mountain bike racing (M). 150 of the cyclists did not complete in either type of race.
a) Complete the table below, showing the number in each group in each group. (4 marks)
M M’
R 15 65 80
R’ 20 150 170
35 215 250
b) What is the probability that a randomly selected cyclist races in either road or mountain bike events? (1 mark)
Pr(R!M )=Pr(R )+Pr(M )"Pr(R#M )=
80+35"15250
100250
=40%
c) What is the probability that a randomly selected mountain bike racer also races on the road? (1 mark)
Pr(R |M )=
n(R!M )n(M )
=1535
1535
=43%
Unit 2 Maths Methods (CAS) Exam 2013 Solutions
6
