2009 Mathematical Methods (CAS) Exam 1

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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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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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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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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2009 Mathematical Methods (CAS) Exam 1