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SUPERVISOR TO ATTACH PROCESSING LABEL HERE
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Words
STUDENT NUMBER Letter
Victorian Certicate of Education
2007
MATHEMATICAL METHODS (CAS)
Written examination 1
Friday 9 November 2007
Reading time: 9.00 am to 9.15 am (15 minutes)
Writing time: 9.15 am to 10.15 am (1 hour)
QUESTION AND ANSWER BOOK
Structure of book
Number of
questions
Number of questions
to be answered
Number of
marks
12 12 40
Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers,
sharpeners, rulers.
Students are NOT permitted to bring into the examination room: notes of any kind, blank sheets ofpaper, white out liquid/tape or a calculator of any type.
Materials supplied
Question and answer book of 9 pages, with a detachable sheet of miscellaneous formulas in the
centrefold.
Working space is provided throughout the book.
Instructions
Detach the formula sheet from the centre of this book during reading time.
Write yourstudent number in the space provided above on this page.
All written responses must be in English.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic
devices into the examination room.
VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2007
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2007 MATHMETH & MATHMETH(CAS) EXAM 1 2
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Question 1
Let f(x) =x
x
3
sin ( ). Find f(x).
2 marks
Question 2
a. Solve the equation loge
(3x + 5) + loge(2) = 2, forx.
2 marks
b. Let g(x) = loge(tan (x)). Evaluateg
4
.
2 marks
Instructions
Answerall questions in the spaces provided.
A decimal approximation will not be accepted if an exact answer is required to a question.
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.
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2007 MATHMETH & MATHMETH(CAS) EXAM 1 4
Question 3
The diagram shows the graph of a function with domainR.
a. For the graph shown above, sketch on the same set of axes the graph of the derivative function.
3 marks
b. Write down the domain of the derivative function.
1 mark
Question 4
A wine glass is being lled with wine at a rate of 8 cm3/s. The volume, Vcm3, of wine in the glass when the
depth of wine in the glass isx cm is given by V x= 432 . Find the rate at which the depth of wine in the glass is
increasing when the depth is 4 cm.
3 marks
3 3O
y
x
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Question 5
It is known that 50% of the customers who enter a restaurant order a cup of coffee. If four customers enter the
restaurant, what is the probability that more than two of these customers order coffee? (Assume that what any
customer orders is independent of what any other customer orders.)
2 marks
Question 6
Two events,A andB, from a given event space, are such that Pr (A) =1
5and Pr (B) =
1
3.
a. Calculate Pr ( ) A B when Pr ( ) =A B1
8.
1 mark
b. Calculate Pr ( ) A B whenA andB are mutually exclusive events.
1 mark
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2007 MATHMETH & MATHMETH(CAS) EXAM 1 6
Question 7
Iff(x) =x cos (3x), then f(x) = cos (3x) 3x sin (3x).
Use this fact to nd an antiderivative ofx sin (3x).
3 marks
Question 8
Let f:RR, f(x) = sin2
3
x
.
a. Solve the equation sin2
3
x
=
3
2forx[ ]0, 3 .
2 marks
b. Letg:RR,g(x) = 3f(x 1) + 2.
Find the smallest positive value ofx for whichg(x) is a maximum.
2 marks
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Question 9
The graph off:RR, is shown. The normal to the graph off where it crosses they-axis
is also shown.
a. Find the equation of the normal to the graph off where it crosses they-axis.
2 marks
b. Find the exact area of the shaded region.
3 marks
y
xO
y = f(x)
f x e
x
( ) = +2 1
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2007 MATHMETH & MATHMETH(CAS) EXAM 1 8
Question 10
The area of the region bounded by the curve with equation y kx=12 , where kis a positive constant, thex-axis
and the line with equationx = 9 is 27. Find k.
3 marks
Question 11
There is a daily ight from Paradise Island to Melbourne. The probability of the ight departing on time, given
that there is ne weather on the island, is 0.8, and the probability of the ight departing on time, given that the
weather on the island is not ne, is 0.6.
In March the probability of a day being ne is 0.4.
Find the probability that on a particular day in March
a. the ight from Paradise Island departs on time
2 marks
b. the weather is ne on Paradise Island, given that the ight departs on time.
2 marks
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Question 12
Pis the point on the line 2x +y 10 = 0 such that the length ofOP, the line segment from the origin O toP, is
a minimum. Find the coordinates ofPand this minimum length.
4 marks
END OF QUESTION AND ANSWER BOOK
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MATHEMATICAL METHODS AND
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 2007
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MATH METH & MATH METH (CAS) 2
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3 MATH METH & MATH METH (CAS)
END OF FORMULA SHEET
Mathematical Methods and Mathematical Methods CAS
Formulas
Mensuration
area of a trapezium:1
2 a b h+( ) volume of a pyramid:1
3Ah
curved surface area of a cylinder: 2rh volume of a sphere:4
3
3r
volume of a cylinder: r2h area of a triangle:1
2bc Asin
volume of a cone:1
3
2r h
Calculus
d
dx
x nxn n( ) = 1
x dx
n
x c nn n=
+
+ +
1
1
11 ,
d
dxe aeax ax( ) = e dx a e c
ax ax= +1
d
dxx
xelog ( )( ) =
1
1
xdx x ce= + log
d
dxax a axsin( ) cos( )( ) = sin( ) cos( )ax dx a ax c= +
1
d
dxax a axcos( )( ) = sin( )
cos( ) sin( )ax dx
aax c= +
1
d
dxax
a
axa axtan( )
( )( ) ==
cossec ( )
2
2
product rule:d
dxuv u
dv
dxv
du
dx( ) = + quotient rule:
d
dx
u
v
vdu
dxu
dv
dx
v
=
2
chain rule:dy
dx
dy
du
du
dx= approximation: f x h f x h f x+( ) ( ) + ( )
Probability
Pr(A) = 1 Pr(A) Pr(AB) = Pr(A) + Pr(B) Pr(AB)
Pr(A|B) =Pr
Pr
A B
B
( )( )
mean: = E(X) variance: var(X) = 2 = E((X)2) = E(X2) 2
probability distribution mean variance
discrete Pr(X=x) =p(x) = xp(x) 2 = (x )2p(x)
continuous Pr(a< X < b) = f x dxa
b( ) =
x f x dx( ) 2 2=
( ) ( )x f x dx