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Copyright © 2010 Neap ABN 49 910 906 643 96–106 Pelham St Carlton VIC 3053 Tel: (03) 8341 8341 Fax: (03) 8341 8300 TEVMMU34EX1_QA_2010.FM

Trial Examination 2010

VCE Mathematical Methods (CAS) Units 3 & 4

Written Examination 1

Question and Answer Booklet

Reading time: 15 minutes

Writing time: 1 hour

Student’s Name: ______________________________

Teacher’s Name: ______________________________

Structure of Booklet

Number of questionsNumber of questions

to be answeredNumber of marks

10 10 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 of paper,

white out liquid/tape or a calculator of any type.

Materials supplied

Question and answer booklet of 9 pages, with a detachable sheet of miscellaneous formulas in the

centrefold.

Working space is provided throughout the booklet.

Instructions

Detach the formula sheet from the centre of this booklet during reading time.

Write your name and teacher’s 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.

Students are advised that this is a trial examination only and cannot in any way guarantee the content or the format of the 2010 VCE Mathematical Methods

(CAS) Units 3 & 4 Written Examination 1.

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VCE Mathematical Methods (CAS) Units 3 & 4 Trial Examination 1 Question and Answer Booklet

2 TEVMMU34EX1_QA_2010.FM Copyright © 2010 Neap

Question 1

Let .

a. Find the y-intercept of the graph of .

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1 mark

b. Find the x-intercepts of the graph of .

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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 booklet are not drawn to scale.

f 0 2 π,[ ] R, where f x( )→: sin2

2 x π

2---–

1–=

y f x( )=

y f x( )=

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Copyright © 2010 Neap TEVMMU34EX1_QA_2010.FM 3

Question 2

a. Given for , show that

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1 mark

b. Hence find the coordinates of the stationary point on the curve with equation and

determine the nature of this stationary point.

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4 marks

y x2loge

1

x---

= x 0> dy

dx------ 2 xloge x( )– x.–=

y x2loge

1

x---

=

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Question 3

Let .

a. State the largest value of a such that the graph of is one-to-one.

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1 mark

b. Find the rule of the inverse function .

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2 marks

c. Sketch the graph of on the axes below.

1 mark

∞ a,–( ): R, where f x( )→ x 2–( )24+=

y f x( )=

f 1–

y f 1–

f x( )( )=

y

x0

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Question 4

Consider the two functions f and g defined by the rules .

a. On the axes below, the graphs of f and g are shown over the domain .

Label each graph as f or g and write down the coordinates of P.

2 marks

b. Differentiate loge(cos( x)) and hence find an antiderivative of tan( x).

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2 marks

c. Find the area enclosed by the y-axis and the graphs of f and g.

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3 marks

f x( ) x( ) and g x( )tan 2 x( )cos= =

0 π

2---

,

y

x0

P

π

2---

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Question 5

Consider the independent events A and B where .

Find Pr( B).

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3 marks

Question 6

Solve the equation for x.

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2 marks

Pr A B( ) 3

4--- and Pr B( ) Pr A ′ B ′∩( )= =

e x

4– 5e x–

=

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Question 7

The time in hours, T , that a randomly chosen flight from a particular airport is delayed has a probability

density function given by:

The probability that a randomly chosen flight is delayed by 3 hours, at most, equals .

A person has already had their flight delayed so they have been waiting in the departure lounge for

3 hours.

What is the probability that this person will be delayed by up to a further 3 hours before departure?

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3 marks

Question 8

Let , where k is a real number.

Determine the equation of the tangent to the graph of f at the point where x = a, and hence show that k cannot

be negative if this tangent is to pass through the origin.

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3 marks

t ( )

t –

50

------ 1

5

--- for 0 t 10≤ ≤+

0 for t 0 or t 10><

=

51

100---------

f R R, f x( )→: x2

k +=

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Question 9

A transformation is described by the equation:

a. Find the image of point (2, 3).

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1 mark

b. Find the image of the line with equation under this transformation.

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c. Write in words, in the correct order, the transformations described by the equation .

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TX B+ X ′ where T 2 0

0 1–, B

4–

2, X

x

y and X ′ x ′

y ′= = = = =

y 2 x 4–=

TX B+ X ′=

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Question 10

a. Use the relationship for small positive values of h, to find the approximate

change in the volume of a cuboid with volume , given by , if x changes from 3 cm

to 3.02 cm.

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2 marks

b. Explain whether this approximation is either greater or less than the exact change.

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1 mark

END OF QUESTION AND ANSWER BOOKLET

f x h+( ) f x( ) hf ′ x( )+≈V cm

3V 6 x

3=

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