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NAME:
MATHEMATICAL METHODS (CAS) UNIT 4
School Assessed Coursework (SAC4)
Analysis Task (Integral Calculus)
Technology active
Reading time: 1 0 minutes
Writing time: 8 0 M i n u t e s
QUESTION AND ANSWER BOOK
Structure of book
Section
1
Number of
questions
4
Number of questions
to be answered
Number of
marks
4
60
• Students are permitted to bring into the examination room: pens, pencils,
highlighters, erasers, sharpeners, rulers, a protractor, set-squares, aids for curve
sketching, one bound reference, one approved CAS calculator (memory DOES NOT
need to be cleared) and, if desired, one scientific calculator. For approved computerbased CAS, their full functionality may be used.
• Students are NOT permitted to bring into the examination room: blank sheets of paper
and/or white out liquid/tape.
Materials supplied
• Question and a detachable sheet of miscellaneous formulas.
Instructions
• Write your name 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.
W.J.Dilan Fernando ST ID 18120466
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Question 3
The diagram below shows part of the graph of the function f : R+ → R,
f (x) =
7
.
x
y
C
A
O
x
b
1
a
The line segment CA is drawn from the point C(1, f (1)) to the point A(a, f (a)) where a > 1.
a.
i. Calculate the gradient of CA in terms of a.
ii.
At what value of x between 1 and a does the tangent to the graph of f have the same gradient as
CA?
2+ 3 = 5 marks
W.J.Dilan Fernando ST ID 18120466
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b. i. Calculate
e
∫ f x dx
1
Let b be a positive real number less than one. Find the exact value of b such that
ii.
1 𝑓(𝑥)𝑑𝑥
∫𝑏
is equal to 7.
1 + 2 = 3 marks
c.
i. Express the area of the region bounded by the line segment CA, the x-axis, the line x = 1 and
the line x = a in terms of a.
ii.
For what exact value of a does this area equal 7?
2 + 2 =4 marks
W.J.Dilan Fernando ST ID 18120466
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Question 4 (16 marks)
16 −x 2
Part of the graph of a function g: R →R, g(x) =
is shown below.
4
y
C
y = g(x)
A
B
O
a.
x
Points B and C are the positive x-intercept and y-intercept of the graph of g, respectively, as
shown in the diagram above. The tangent to the graph of g at the point A is parallel to the line
segment BC.
i. Find the equation of the tangent to the graph of g at the point A.
ii.
The shaded region shown in the diagram above is bounded by the graph of g, the
tangent at the point A, and the x-axis and y-axis.
Evaluate the area of this shaded region.
W.J.Dilan Fernando ST ID 18120466
3 marks
3 marks
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b. i Find the maximum area A(k)max of the shaded region and the value of k for which this occurs
where A(k) = 𝑘 2/2×√𝑘 − 4
3 marks
The tangent to the graph of g at a point P has a negative gradient and intersects the y-axis at
point D(0, k), where 5 ≤ k ≤ 8.
y
D(0, k)
C
P
y = g(x)
O
Find the gradient of the tangent in terms of k.
b. ii.
W.J.Dilan Fernando ST ID 18120466
B
x
3marks
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Mathematical Methods (CAS) Formulas
Mensuration
area of a trapezium:
1 ab h


2
volume of a pyramid:
curved surface area of a cylinder:
2rh
volume of a sphere:
volume of a cylinder:
 r 2h
area of a triangle:
volume of a cone:
1 2
r h
3
1
Ah
3
4 3
r
3
1
bcsin A
2
Calculus
1 n1
x  c, n −1
n 1
1
e ax dx  e ax  c
x
d n
n−1
dx
d ax
e ae ax
dx
1
d
 loge (x)   x
dx
 
 sin(ax)  a
dx
product rule:
chain rule:
dx 
1
x dx loge x  c
cos(ax)
dx
d
 cos(ax)  = − a sin(ax)
dx
 tan(ax)  =
n
2
1
sin(ax)dx − cos(ax)  c
1
cos(ax)dx  a sin(ax)  c
a sec 2 (ax)
cos (ax)
d
dv
du
 uv   u  v
dx
dx
dx
dy
dx
dy du
du dx
W.J.Dilan Fernando ST ID 18120466
quotient rule:
approximation:
d  u 
dx v 
du
dx
dv
dx
v2
f  x h   f  x  h f  x 
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References
1.VCAA study design
Vcaa.vic.edu.au,. (2014). Pages - Mathematical Methods CAS Index. Retrieved 8 August 2014, from
http://www.vcaa.vic.edu.au/Pages/vce/studies/mathematics/cas/casindex.aspx
2. VCAA Maths Methods past exams
Vcaa.vic.edu.au,. (2014). Pages - Mathematical Methods (CAS) – Exams and Examination Reports.
Retrieved 8 August 2014, from
http://www.vcaa.vic.edu.au/Pages/vce/studies/mathematics/cas/casexams.aspx
5. assessment hand book and rubric
Vcaa.vic.edu.au,. (2014). Pages - Mathematical Methods CAS Index. Retrieved 9 August 2014, from
http://www.vcaa.vic.edu.au/Pages/vce/studies/mathematics/cas/casindex.aspx
W.J.Dilan Fernando ST ID 18120466
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W.J.Dilan Fernando ST ID 18120466