2006 Mathematical Methods (CAS) Exam 2

SUPERVISOR TO ATTACH PROCESSING LABEL HERE
Figures
Words
STUDENT NUMBER
Letter
Victorian Certificate of Education
2006
MATHEMATICAL METHODS (CAS)
Written examination 2
Monday 6 November 2006
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 book
Section Number of
questions
Number of questions
to be answered
Number of
marks
1
2
22
4
22
4
22
58
Total 80
• 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 scientiÞc
calculator. For approved computer based 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 answer book of 21 pages with a detachable sheet of miscellaneous formulas in the
centrefold.
• Answer sheet for multiple-choice questions.
Instructions
• Detach the formula sheet from the centre of this book during reading time.
• Write your student number in the space provided above on this page.
• Check that your name and student number as printed on your answer sheet for multiple-choice
questions are correct, and sign your name in the space provided to verify this.
• All written responses must be in English.
At the end of the examination
• Place the answer sheet for multiple-choice questions inside the front cover of this book.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic
devices into the examination room.
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2006
2006 MATHMETH(CAS) EXAM 2 2
SECTION 1 – continued
Question 1
The graph with equation y = x
2
is translated 3 units down and 2 units to the right.
The resulting graph has equation
A. y = (x − 3)
2
+ 2
B. y = (x − 2)
2
+ 3
C. y = (x − 2)
2
− 3
D. y = (x + 2)
2
− 3
E. y = (x + 2)
2
+ 3
Question 2
The smallest positive value of x for which tan (2x) = 1 is
A. 0
B.
π
8
C.
π
4
D.
π
2
E. π
Question 3
The range of the function f : [–2, 7) → R, f (x) = 5 – x is
A. (–2, 7]
B. [–2, 7)
C. (–2, ∞)
D. (–2, 7)
E. R
SECTION 1
Instructions for Section I
Answer all questions in pencil on the answer sheet provided for multiple-choice questions.
Choose the response that is correct for the question.
A correct answer scores 1, an incorrect answer scores 0.
Marks will not be deducted for incorrect answers.
No marks will be given if more than one answer is completed for any question.
3 2006 MATHMETH(CAS) EXAM 2
SECTION 1 – continued
TURN OVER
Question 4
For the system of simultaneous linear equations
z = 3
x + y = 5
x − y = 1
an equivalent matrix equation is
A.
x
y
z

¸



1
]
1
1
1 −

¸



1
]
1
1
1
·

¸



1
]
1
1
1
0 0 1
1 1 0
1 1 0
3
5
1
B.
1 1 0
1 1 0
0 0 1
5
1
3
−

¸



1
]
1
1
1

¸



1
]
1
1
1
·

¸



1
]
1
1
1
x
y
z
C.
x
y
z

¸



1
]
1
1
1
· −

¸



1
]
1
1
1

¸



1
]
1
1
1
2
0 1 1
0 1 1
2 0 0
3
5
1
D.
1 0 0
0 1 0
0 0 1
3
5
1

¸



1
]
1
1
1

¸



1
]
1
1
1
·

¸



1
]
1
1
1
x
y
z
E.
1 1 0
1 1 0
0 0 1
3
5
1
−

¸



1
]
1
1
1

¸



1
]
1
1
1
·

¸



1
]
1
1
1
x
y
z
Question 5
A bag contains three white balls and seven yellow balls. Three balls are drawn, one at a time, from the bag,
without replacement.
The probability that they are all yellow is
A.
3
500
B.
27
1000
C.
21
100
D.
7
24
E.
243
1000
Question 6
The function f : [a, ∞) → R, with rule f (x) = log
e
(x
4
), will have an inverse function if
A. a < 0
B. a ≤ – 1
C. a ≤ 1
D. a > 0
E. a ≥ – 1
2006 MATHMETH(CAS) EXAM 2 4
SECTION 1 – continued
Question 7
The function g has rule g(x) = log
e
|x – b|, where b is a real constant.
The maximal domain of g is
A. R
+
B. R\{b}
C. R
D. (–b, b)
E. (b, ∞)
Question 8
The average value of the function y = cos (x) over the interval [0,
π
2
] is
A.
1
π
B.
π
4
C. 0.5
D.
2
π
E.
π
2
Question 9
If y = 3a
2x
+ b, then x is equal to
A.
1
6
log
a
y b − ( )
B.
1
2 3
log
a
y b − ¸
¸

_
,

C.
1
6
log
a
y
b
¸
¸

_
,

D.
1
2
3
log
a
y
b
¸
¸

_
,

E.
1
3
2
log
a
y b − ( )
Question 10
The radius of a sphere is increasing at a rate of 3 cm/min.
When the radius is 6 cm, the rate of increase, in cm
3
/min, of the volume of the sphere is
A. 432π
B. 48π
C. 144π
D. 108π
E. 16π
5 2006 MATHMETH(CAS) EXAM 2
SECTION 1 – continued
TURN OVER
Question 11
The value(s) of k for which | 2k + 1 | = k + 1 are
A. 0 only
B.
−
2
3
only
C. 0 or
2
3
D. 0 or
−
2
3
E.
−
1
2
or 0
Question 12
A fair coin is tossed 10 times.
The probability, correct to four decimal places, of getting 8 or more heads is
A. 0.0039
B. 0.0107
C. 0.0547
D. 0.9453
E. 0.9893
Question 13
The transformation T: R
2
→ R
2
, which maps the curve with equation y = log
e
(x) to the curve with equation
y = log
e
(2x−4) + 3, could have rule
A. T
x
y
x
y

¸

1
]
1
¸
¸

_
,

·

¸

1
]
1

¸

1
]
1
+

¸

1
]
1
0 5 0
0 1
2
3
.
B. T
x
y
x
y

¸

1
]
1
¸
¸

_
,

·

¸

1
]
1

¸

1
]
1
+

¸

1
]
1
0 5 0
0 1
4
3
.
C. T
x
y
x
y

¸

1
]
1
¸
¸

_
,

·

¸

1
]
1

¸

1
]
1
+
−
¸

1
]
1
1 0
0 2
4
3
D.
T
x
y
x
y

¸

1
]
1
¸
¸

_
,

·

¸

1
]
1

¸

1
]
1
+
−
¸

1
]
1
1 0
0 2
2
3
E.
T
x
y
x
y

¸

1
]
1
¸
¸

_
,

·

¸

1
]
1

¸

1
]
1
+

¸

1
]
1
2 0
0 1
4
3
2006 MATHMETH(CAS) EXAM 2 6
SECTION 1 – continued
Question 14
For the graph of y = f (x) shown above, f ′(x) is negative when
A. −3 < x < 3
B. −3 ≤ x ≤ 3
C. x < –3 or x > 3
D. x ≤ −3 or x ≥ 3
E. –5 < x < 1 or x > 4
Question 15
The total area of the shaded regions in the diagram is given by
A. f x dx ( )
−
∫
1
6
B. − +
−
∫ ∫
f x dx f x dx ( ) ( )
1
0
0
6
C.
f x dx f x dx ( ) ( )
1
4
4
6
2
∫ ∫
+
D.
− + −
−
∫ ∫ ∫
f x dx f x dx f x dx ( ) ( ) ( )
1
1
1
4
4
6
E.
− + −
−
∫ ∫ ∫
f x dx f x dx f x dx ( ) ( ) ( )
1
1
1
4
6
4
(–3, 5)
(3, –4)
O
–5 1 4
y
x
O
–1 1 4
y
x
6
y = f (x)
7 2006 MATHMETH(CAS) EXAM 2
SECTION 1 – continued
TURN OVER
Question 16
Let f ′(x) = g ′(x) + 3, f (0) = 2 and g(0) = 1.
Then f (x) is given by
A. f (x) = g(x) + 3x + 1
B. f (x) = g ′(x) + 3
C. f (x) = g(x) + 3x
D. f (x) = 1
E. f (x) = g(x) + 3
Question 17
The function f satisÞes the functional equation
f
x y f x f y + ¸
¸

_
,

·
+
2 2
( ) ( )
where x and y are any non-zero
real numbers.
A possible rule for the function is
A. f (x) = log
e
|x|
B. f (x) =
1
x
C. f (x) = 2
x
D. f (x) = 2x
E. f (x) = sin (2x)
Question 18
The discrete random variable X has the following probability distribution.
X –1 0 1
Pr(X = x) a b 0.4
If the mean of X is 0.3 then
A. a = 0.3 and b = 0.3
B. a = 0.2 and b = 0.4
C. a = 0.4 and b = 0.2
D. a = 0.1 and b = 0.5
E. a = 0.7 and b = 0.3
Question 19
The simultaneous linear equations (m – 2) x + 3y = 6 and 2x + (m – 3) y = m – 1 have no solution for
A. m ∈ R \{0, 5}
B. m ∈ R \{0}
C. m ∈ R \{6}
D. m = 5
E. m = 0
2006 MATHMETH(CAS) EXAM 2 8
SECTION 1 – continued
Question 20
Let f : R → R be a differentiable function.
Then for all x ∈ R, the derivative of f (sin (4x)) with respect to x is equal to
A. 4 cos (4x) f ′(x)
B. sin (4x) f ′(x)
C. f ′(sin (4x))
D. 4 f ′(sin (4x))
E. 4 cos (4x) f ′(sin (4x))
Question 21
The times (in minutes) taken for students to complete a university test are normally distributed with a mean of
200 minutes and standard deviation 10 minutes.
The proportion of students who complete the test in less than 208 minutes is closest to
A. 0.200
B. 0.212
C. 0.758
D. 0.788
E. 0.800
9 2006 MATHMETH(CAS) EXAM 2
Question 22
Which one of the following could be the graph of y = a log
e
(x – b) where a < 0 and b > 0?
x
x
y
x
y
y
O
O
O
O
O
x
y
x
y
A. B.
C.
E.
D.
END OF SECTION 1
TURN OVER
2006 MATHMETH(CAS) EXAM 2 10
Question 1
Consider the function f :[0, 2π] → R, f (x) = 2 sin (x). The graph of f is shown below, with tangents drawn at
points A and B.
a. i. Find f ′(x).
ii. Find the maximum and minimum values of | f ′(x)|.
1 + 2 = 3 marks
SECTION 2
Instructions for Section 2
Answer all 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.
SECTION 2 – Question 1 – continued
y
x
2
O
–2
π 2π
A
B
11 2006 MATHMETH(CAS) EXAM 2
b. i. The gradient of the curve with equation y = f (x), when x ·
π
3
, is 1. Find the other value of x for
which the gradient of the curve, with equation y = f (x), is 1. (The exact value must be given.)
ii. Find the equation of the tangent to the curve at
x ·
π
3
. (Exact values must be given.)
iii. Find the axes intercepts of the tangent found in b. ii. (Exact values must be given.)
1 + 2 + 3 = 6 marks
c. The two tangents to the curve at points A and B have gradient 1. A translation of m units in the positive
direction of the x-axis takes the tangent at A to the tangent at B. Find the exact value of m.
2 marks
SECTION 2 – Question 1 – continued
TURN OVER
2006 MATHMETH(CAS) EXAM 2 12
SECTION 2 – continued
d. Let h:R → R, h(x) = 2 |sin (x) |. Find the general solution, for x, of the equation h(x) = 1.
2 marks
Total 13 marks
13 2006 MATHMETH(CAS) EXAM 2
Question 2
Each night Kim goes to the gym or the pool. If she goes to the gym one night, the probability she goes to the
pool the next night is 0.4, and if she goes to the pool one night, the probability she goes to the gym the next
night is 0.7.
a. Suppose she goes to the gym one Monday night.
i. What is the probability that she goes to the pool on each of the next three nights? (The exact value
must be given.)
ii. What is the probability that she goes to the pool on exactly two of the next three nights? (The exact
value must be given.)
1 + 2 = 3 marks
b. In the long term, what proportion of nights does she go to the pool? (Answer correct to three decimal
places.)
1 mark
SECTION 2 – Question 2 – continued
TURN OVER
2006 MATHMETH(CAS) EXAM 2 14
When Kim goes to the gym, the time, T hours, that she spends working out is a continuous random variable
with probability density function given by
f t
t t t t
( ) ·
− + − ≤ ≤
¹
'
¹
¹
¹
4 24 44 24 1 2
0
3 2
if
otherwise
c. Sketch the graph of y = f (t) on the axes below. Label any stationary points with their coordinates, correct
to two decimal places.
3 marks
d. What is the probability, correct to three decimal places, that she spends less than 75 minutes working out
when she goes to the gym?
2 marks
y
t
O
SECTION 2 – Question 2 – continued
15 2006 MATHMETH(CAS) EXAM 2
SECTION 2 – continued
TURN OVER
e. What is the probability, correct to two decimal places, that she spends more than 75 minutes working out
on 4 out of the 5 next times she goes to the gym?
2 marks
f. Find the median time, to the nearest minute, that she spends working out in the gym.
3 marks
Total 14 marks
2006 MATHMETH(CAS) EXAM 2 16
Question 3
Tasmania Jones’ wheat Þeld lies between two roads as shown in the diagram below.
Main Road lies along the x-axis and Side Road lies along the curve with equation y = 3 – e
x
– e
–x
.
a. The y-axis intercept of the graph representing Side Road is b.
Show that b = 1.
1 mark
b. Find the exact value of a.
1 mark
wheat field
Side Road
Main Road
y (km)
x (km)
b
a –a
O
SECTION 2 – Question 3 – continued
17 2006 MATHMETH(CAS) EXAM 2
c. Since a is close to 1, Tasmania Þnds an approximation to the area of the wheat Þeld by using rectangles
of width 0.5 km, as shown on the following diagram.
i. Complete the table of values for y, where y = 3 − e
x
− e
−x
, giving values correct to two decimal
places.
x –0.5 0 0.5
y
ii. Use the table to Þnd Tasmania’s approximation to the area of the wheat Þeld, measured in square
kilometres, correct to one decimal place.
iii. Tasmania uses this approximation to the area to estimate the value of the wheat in his Þeld at harvest
time. He estimates that he will obtain w kg of wheat from each square kilometre of Þeld. The current
price paid to growers is $m per kg of wheat. Write a formula for his estimated value, $V, of the wheat
in his Þeld.
1 + 2 + 1 = 4 marks
2
1.5
1
–2 –1 –0.5 0.5 O 1
2
–0.5
0.5
y (km)
x (km)
SECTION 2 – Question 3 – continued
TURN OVER
2006 MATHMETH(CAS) EXAM 2 18
SECTION 2 – continued
d. Tasmania Jones decides to Þnd another approximation to the area of the wheat Þeld. He approximates the
curve representing Side Road with a parabola which passes through the points (0, 1), (1, 0) and (–1, 0). He
Þnds the area enclosed by the parabola and the x-axis as an approximation to the area of his wheat Þeld.
i. Find the equation of this parabola.
ii. Find the area enclosed by the parabola and the x-axis, giving your answer correct to two decimal
places.
1 + 2 = 3 marks
e. Find the values of k, where k is a positive real number, for which the equation 3 – ke
x
– e
–x
= 0 has one or
more solutions for x.
4 marks
Total 13 marks
SECTION 2 – continued
TURN OVER
19 2006 MATHMETH(CAS) EXAM 2
Question 4
A part of the track for Tim’s model train follows the curve passing through A, B, C, D, E and F shown above.
Tim has designed it by putting axes on the drawing as shown. The track is made up of two curves, one to the
left of the y-axis and the other to the right.
B is the point (0, 7).
The curve from B to F is part of the graph of f (x) = px
3
+ qx
2
+ rx + s where p, q, r and s are constants and
f ′(0) = 4.25.
a. i. Show that s = 7.
ii. Show that r = 4.25.
1 + 1 = 2 marks
supertrain
A
B
y
x
C
F
E
O –2
D
SECTION 2 – Question 4 – continued
TURN OVER
2006 MATHMETH(CAS) EXAM 2 20
The furthest point reached by the track in the positive y direction occurs when x = 1. Assume p > 0.
b. i. Use this information to Þnd q in terms of p.
ii. Find f (1) in terms of p.
iii. Find the value of a in terms of p for which f ′(a) = 0 where a > 1.
iv. If a ·
17
3
, show that p = 0.25 and q = –2.5.
2 + 1 + 1 + 2 = 6 marks
For the following assume f (x) = 0.25x
3
– 2.5x
2
+ 4.25x + 7.
c. Find the exact coordinates of D and F.
2 marks
SECTION 2 – Question 4 – continued
21 2006 MATHMETH(CAS) EXAM 2
d. Find the greatest distance that the track is from the x-axis, when it is below the x-axis, correct to two
decimal places.
1 mark
The curve from A to B is part of the graph with equation g x
a
bx
( ) ·
− 1
, where a and b are positive real constants.
The track passes smoothly from one section of the track to the other at B (that is, the gradients of the curves
are equal at B).
e. Find the exact values of a and b.
3 marks
f. Find the area of the shaded section bounded by the track between x = −2 and D, correct to two decimal
places.
4 marks
Total 18 marks
END OF QUESTION AND ANSWER BOOK
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 2006
MATH METH & MATH METH (CAS) 2
This page is blank
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
3
Ah
curved surface area of a cylinder: 2π rh volume of a sphere:
4
3
3
πr
volume of a cylinder: π r
2
h area of a triangle:
1
2
bc A sin
volume of a cone:
1
3
2
πr h
Calculus
d
dx
x nx
n n
( )
=
−1

x dx
n
x c n
n n
=
+
+ ≠ −
+
∫
1
1
1
1
,
d
dx
e ae
ax ax
( )
=

e dx
a
e c
ax ax
= +
∫
1
d
dx
x
x
e
log ( ) ( ) =
1

1
x
dx x c
e
= +
∫
log
d
dx
ax a ax sin( ) cos( ) ( ) =

sin( ) cos( ) ax dx
a
ax c = − +
∫
1
d
dx
ax a ax cos( ) ( ) − = sin( )

cos( ) sin( ) ax dx
a
ax c = +
∫
1
d
dx
ax
a
ax
a ax tan( )
( )
( ) = =
cos
sec ( )
2
2

product rule:
d
dx
uv u
dv
dx
v
du
dx
( ) = + quotient rule:
d
dx
u
v
v
du
dx
u
dv
dx
v






=
−
2
chain rule:
dy
dx
dy
du
du
dx
= approximation: f x h f x hf x + ( ) ≈ ( ) + ′
( )
Probability
Pr(A) =1 – Pr(A′) Pr(A ∪ B) =Pr(A) +Pr(B) – Pr(A ∩ B)
Pr(A|B) =
Pr
Pr
A B
B
∩ ( )
( )
mean: µ =E(X) variance: var(X) =σ
2
=E((X – µ)
2
) =E(X
2
) – µ
2
probability distribution mean variance
discrete Pr(X =x) =p(x) µ =∑ x p(x) σ
2
=∑ (x – µ)
2
p(x)
continuous Pr(a < X < b) =
f x dx
a
b
( )
∫
µ =
−∞
∞
∫
x f x dx ( ) σ µ
2 2
= −
−∞
∞
∫
( ) ( ) x f x dx