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MATHEMATICAL METHODS (CAS)
Written examination 2Wednesday 9 November 2011
Reading time: 11.45 am to 12.00 noon (15 minutes) Writing time: 12.00 noon to 2.00 pm (2 hours)
QUESTION AND ANSWER BOOK
Structure of bookSection Number of
questionsNumber of questions
to be answeredNumber of
marks
12
224
224
2258
Total 80
Materials supplied
Instructions
student numbername student number
and
At the end of the examination
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
SUPERVISOR TO ATTACH PROCESSING LABEL HERE
Figures
Words
STUDENT NUMBER
Letter
2011
SECTION 1
Question 1d
A. d2
52
, −⎛⎝⎜
⎞⎠⎟
B.
C. d −⎛
⎝⎜
⎞⎠⎟
52
0,
D. 0 52
, −⎛⎝⎜
⎞⎠⎟
d
E. 5
20+⎛
⎝⎜
⎞⎠⎟
d ,
Question 2perpendicular
A. 12
B. –2
C. 12
D. 4
E. 2
Question 3x + a x3 – 13x2 – ax a a
A. – 4B. –3C. – 1D. 1E. 2
SECTION 1
Instructions for Section 1all
correct
not
SECTION 1TURN OVER
Question 4
e f x x
A. f xf x
( )( )
B. 2 f xf x
( )( )
C. f xf x( )( )2
D. e f x
E. e f x
Question 5g R g x 2 4x
A. g–1 R g–1 xx2 4
2
B. g–1 R g–1 x x 2
C. g–1 R g–1 xx2
4
D. g–1 R g–1 xx2 4
2
E. g–1 R R g–1 xx2 4
2
Question 6X
f xx x e
elsewheree( )
log ( )=
≤ ≤⎧⎨⎩
10
X E XA.B.C. 1D.E.
SECTION 1
Question 7
f x 3 x f h f hf 3 8 5.A.B.C.D.E.
Question 8f R R f x x x
g 32
5,⎡⎣⎢
⎞⎠⎟ R g x x
h f + g hA.
B. −⎡⎣⎢
⎤⎦⎥
34
10,
C. −⎛⎝⎜
⎤⎦⎥
34
154
,
D. 34
13,⎡⎣⎢
⎞⎠⎟
E. 32
13,⎡⎣⎢
⎞⎠⎟
SECTION 1TURN OVER
Question 9y f x
4
–5 –3 O
y
x
y f x
–3–5 O
4
A.
–3 O
B.
–3 O
D.
–3–5 O
4
E.
–3–5 O
C.
x x x
x x
y y
y y
y
SECTION 1
Question 10
A. 45
B. 25
C. 23
D. 13
E. 34
Question 11f x e x
A. e
B. 13
6log ( )e
C. loge3125
43⎛
⎝⎜
⎞⎠⎟ −
D. 13
31254
3loge⎛⎝⎜
⎞⎠⎟ −
E. 5 5 2 2 3
3log ( ) log ( )e e
Question 12X
a X > aa
A.B.C.D.E.
SECTION 1TURN OVER
Question 13
A.B.C.D.E.
Question 14
y
xO 1
1
e 2
y = e 2x
i. 1
2
0
xe dx
ii. e e dxx2 2
0
1
iii. 2
2
1
eye dy
iv. 2
1
log ( )2
ee x
dx
A. ii. B. ii. iii.C. i. ii. iii. iv.D. ii. iv.E. i. iv.
SECTION 1
Question 15
3
–16
23
y
xO
A. y x= +⎛⎝⎜
⎞⎠⎟ +2
61cos π
B. y x= −⎛⎝⎜
⎞⎠⎟ +2 4
61cos π
C. y x= −⎛⎝⎜
⎞⎠⎟ −4 2
121sin π
D. y x= +⎛⎝⎜
⎞⎠⎟ −3 2
61cos π
E. y x= +⎛⎝⎜
⎞⎠⎟ −2 4 2
31sin π
Question 16not
f R R f x |x2
A. fB. fC. f x x.D. f x 2x x E. f x 2x x
Question 17
y x x= +32
A. 4x y
B. 4y + x
C. y x= +
41
D. x – 4y –5
E. 4y + 4x 20
SECTION 1TURN OVER
Question 18x3 x2 + 15x + w x
A. w < 25B. wC. wD. w w > 25E. w > 1
Question 19f R R f x x2
y
y
x
x1 2 3 4 5 6
1
O
O 2 3 4 5 6
p p
A. 10B. 15C. 20D. 25E. 30
2011 MATHMETH(CAS) EXAM 2 10
SECTION 1 – continued
Question 20A part of the graph of g: R R, g(x) = x2 – 4 is shown below.
A
B
y
xaO
The area of the region marked A is the same as the area of the region marked B.The exact value of a isA. 0B. 6
C. 6 D. 12E. 2 3
Question 21For two events, P and Q, Pr(P Q) = Pr(P Q).P and Q will be independent events exactly whenA. Pr(P ) = Pr(Q)
B. Pr(P Q ) = Pr(P Q)
C. Pr(P Q) = Pr(P) + Pr(Q)
D. Pr(P Q ) = Pr(P Q)
E. Pr(P) = 12
11
END OF SECTION 1TURN OVER
Question 22
c a a b b c
A. 1 1 1
log ( ) log ( ) log ( )c a ba b c
B. 1 1 1
log ( ) log ( ) log ( )a b cc a b
C. 1 1 1
log ( ) log ( ) log ( )a b cb c a
D. 1 1 1
log ( ) log ( ) log ( )a b ca b c
E. 1 1 1
log ( ) log ( ) log ( )c b aab ac cb
2011 MATHMETH(CAS) EXAM 2 12
SECTION 2 – Question 1 – continued
Question 1Two ships, the Elsa and the Violet, have collided. Fuel immediately starts leaking from the Elsa into the sea.The captain of the Elsa estimates that at the time of the collision his ship has 6075 litres of fuel on board and
he also forecasts that it will leak into the sea at a rate of t2
5 litres per minute, where t is the number of minutes
that have elapsed since the collision.a. At this rate how long, in minutes, will it take for all the fuel from the Elsa to leak into the sea?
3 marks
SECTION 2
Instructions for Section 2Answer all questions in the spaces provided.In all questions where a numerical answer is required an exact value must be given unless otherwise
In questions where more than one mark is available, appropriate working must be shown.Unless otherwise indicated, the diagrams in this book are not drawn to scale.
13
SECTION 2 – Question 1TURN OVER
is not
t r
b.103
SECTION 2 – Question 1
g R g xπx80
⎛⎝⎜
⎞⎠⎟
y (m)
CO
x (m)
h
h R h x − ⎛⎝⎜
⎞⎠⎟60
40sin π x
c. y h x
15 2011 MATHMETH(CAS) EXAM 2
SECTION 2 – continuedTURN OVER
d.
i. Find sin π x dx40
0
40⎛⎝⎜
⎞⎠⎟∫
ii.
1 + 2 = 3 marks
Total 11 marks
SECTION 2 – Question 2
Question 2A
BX A,
Y B
f y
yy y
e yy
( )
. . ( )
=
<
≤ ≤
>
⎧
⎨⎪⎪
⎩⎪⎪ − −
0 0
160 4
0 25 40 5 4
a. i. X
ii. Y
b. Y
SECTION 2 – Question 2TURN OVER
c. i. Y
ii. a Y a
d. Pr( )Y ≤ =3 932
produced by machine B
2011 MATHMETH(CAS) EXAM 2 18
SECTION 2 – continued
All of the chocolates produced by machine A and machine B are stored in a large bin. There is an equal number of chocolates from each machine in the bin.It is found that if a chocolate, produced by either machine, takes longer than 3 seconds to produce then it can
e. A chocolate is selected at random from the bin. It is found to have taken longer than 3 seconds to produce. Find, correct to four decimal places, the probability that it was produced by machine A.
3 marks
Total 15 marks
SECTION 2TURN OVER
CONTINUES OVER PAGE
SECTION 2 – Question 3
Question 3a. f R R f x x3 + 5x i. f x
ii. f x x
b. p p R R p x ax3 + bx2 + cx + k a b c k i. p m m
ii. p m
c. q q R R q x x3
i. q–1 x
21
SECTION 2TURN OVER
ii. y q x y q–1 x
d. g g R R g x x3 + 2x2 + cx + k c k i. g c
ii. y g x y g–1 x k
2011 MATHMETH(CAS) EXAM 2 22
SECTION 2 – Question 4 – continued
Question 4Deep in the South American jungle, Tasmania Jones has been working to help the Quetzacotl tribe to get drinking water from the very salty water of the Parabolic River. The river follows the curve with equation y = x2 – 1, xTasmania has his camp site at (0, 0) and the Quetzacotl tribe’s village is at (0, 1). Tasmania builds a desalination plant, which is connected to the village by a straight pipeline.
y
x1
Parabolic River
village (0, 1)
desalination plant (m, n)
camp(0, 0)
(–1, 0)
pipeline
a. If the desalination plant is at the point (m, n) show that the length, L kilometres, of the straight pipeline that carries the water from the desalination plant to the village is given by
L m m= − +4 23 4.
3 marks
23
SECTION 2 – Question 4TURN OVER
b.
i.dLdm
ii.
SECTION 2 – Question 4
72
34
,⎛
⎝⎜⎜
⎞
⎠⎟⎟
x y x 7
2 .
y
c.
T x x k x= − + + −( )12
1 14
7 44 2 2 k
k
d. k1
2 13
i. dTdx
25
ii.
e. kk
f. k
END OF QUESTION AND ANSWER BOOK
MATHEMATICAL METHODS (CAS)
Written examinations 1 and 2
FORMULA SHEET
Directions to students
Detach this formula sheet during reading time.
This formula sheet is provided for your reference.
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2011
MATH METH (CAS) 2
This page is blank
3 MATH METH (CAS)
END OF FORMULA SHEET
Mathematical Methods (CAS)Formulas
Mensuration
area of a trapezium: 12a b h+( ) volume of a pyramid:
13Ah
curved surface area of a cylinder: 2 rh volume of a sphere: 43
3r
volume of a cylinder: r 2h area of a triangle: 12bc Asin
volume of a cone: 13
2r h
Calculusddx
x nxn n( )= -1
x dx
nx c nn n=
++ ≠ −+∫
11
11 ,
ddxe aeax ax( )= e dx a e cax ax= +∫
1
ddx
x xelog ( )( )= 1
1x dx x ce= +∫ log
ddx
ax a axsin( ) cos( )( )= sin( ) cos( )ax dx a ax c= − +∫
1
ddx
ax a axcos( )( ) -= sin( ) cos( ) sin( )ax dx a ax c= +∫
1
ddx
ax aax
a axtan( )( )
( ) ==cos
sec ( )22
product rule: ddxuv u dv
dxv dudx
( )= + quotient rule: ddx
uv
v dudx
u dvdx
v⎛⎝⎜
⎞⎠⎟ =
−
2
chain rule: dydx
dydududx
= approximation: f x h f x h f x+( ) ≈ ( ) + ′( )
ProbabilityPr(A) = 1 – Pr(A ) Pr(A B) = Pr(A) + Pr(B) – Pr(A B)
Pr(A|B) = Pr
PrA BB∩( )( ) transition matrices: Sn = Tn S0
mean: μ = E(X) variance: var(X) = 2 = E((X – μ)2) = E(X2) – μ2
probability distribution mean variance
discrete Pr(X = x) = p(x) μ = x p(x) 2 = (x – μ)2 p(x)
continuous Pr(a < X < b) = f x dxa
b( ) μ =
−∞
∞∫ x f x dx( ) σ μ2 2= −
−∞
∞∫ ( ) ( )x f x dx