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7/27/2019 2009 Mathematical Methods (CAS) Exam 1
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2009 MATHMETH & MATHMETH(CAS) EXAM 1 2
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Question 1a. Differentiate x log e( x) with respect to x.
2 marks
b. For f ( x) =cos( ) x x2 2
nd f ( ).
3 marks
Instructions
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.
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2009 MATHMETH & MATHMETH(CAS) EXAM 1 4
Question 2
a. Find an anti-derivative of 1
1 2 xwith respect to x.
2 marks
b. Evaluate x dx 11
4
.
3 marks
Question 3
Let f : R \{0} m R where f ( x) =3
4 x
. Find f 1 , the inverse function of f .
3 marks
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Question 4
Solve the equation tan (2 x) = 3 for x , ,
4 4 4
3
4.
3 marks
Question 5Four identical balls are numbered 1, 2, 3 and 4 and put into a box. A ball is randomly drawn from the box, andnot returned to the box. A second ball is then randomly drawn from the box.a. What is the probability that the rst ball drawn is numbered 4 and the second ball drawn is numbered 1?
1 mark
b. What is the probability that the sum of the numbers on the two balls is 5?
1 mark c. Given that the sum of the numbers on the two balls is 5, what is the probability that the second ball drawn
is numbered 1?
2 marks
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2009 MATHMETH & MATHMETH(CAS) EXAM 1 6
Question 6Oil is leaking at a constant rate to form a circular puddle on the oor. The oil is being added to the puddle at therate of 10 mm 3 per minute causing the puddle to spread out evenly, with constant depth of 2 mm.When the radius of the puddle is r mm, the volume, V mm 3, of oil in the puddle is given by V = 2 r 2.Find the rate of change of the radius of the puddle when the radius is 30 mm. Give an exact answer, with unitsof mm per minute.
3 marks
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2009 MATHMETH & MATHMETH(CAS) EXAM 1 8
Question 8Let f : R m R, f ( x) = e x + k , where k is a real number. The tangent to the graph of f at the point where x = a
passes through the point (0, 0). Find the value of k in terms of a .
3 marks
Question 9
Solve the equation 2 log ( )e x log ( )e x 3 = log e12
for x.
4 marks
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Question 10a. Use the relationship f ( x + h) z f ( x) + hf ( x) for a small positive value of h, to nd an approximate value
for 8 063 . .
4 marksb. Explain why this approximate value is greater than the exact value for 8 063 . .
1 mark
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 2009
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MATH METH & MATH METH (CAS) 2
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END OF FORMULA SHEET
Mathematical Methods and Mathematical Methods (CAS)Formulas
Mensuration
area of a trapezium:12 a b h volume of a pyramid:
13 Ah
curved surface area of a cylinder: 2 rh volume of a sphere:43
3 r
volume of a cylinder: r 2h area of a triangle:12
bc Asin
volume of a cone: 13
2 r h
Calculusd
dx x nxn n 1
x dx
n x c nn n x
1
111 ,
d dx
e aeax ax e dx a e cax ax 1
d dx
x xe
log ( ) 1 1 x
dx x ce logd dx
ax a axsin( ) cos( ) sin( ) cos( )ax dx a ax c 1d dx
ax a axcos( ) = sin( ) cos( ) sin( )ax dx a ax c 1
d dx
axa
axa axtan( )
( ) =
cossec ( )2
2
product rule: d dx
uv udvdx
vdudx
quotient rule: d dx
uv
vdudx
udvdx
v
2
chain rule: dydx
dydu
dudx
approximation: f x h f x h f x z a
Probability
Pr( A) = 1 Pr( Aa) Pr( A B) = Pr( A) + Pr( B) Pr( A B)
Pr( A| B) =Pr
Pr A B
B
mean: = E( X ) variance: var( X ) = S 2 = E(( X )2) = E( X 2) 2
probability distribution mean variance
discrete Pr( X = x) = p( x) = x p( x) S 2 = ( x )2 p( x)
continuous Pr(a < X < b ) = f x dxa
b( ) d
d x f x dx( ) 2 2 dd ( ) ( ) x f x dx
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