2005 Mathematical Methods (CAS) Exam 2.pdf

8/14/2019 2005 Mathematical Methods (CAS) Exam 2.pdf

1/15

SUPERVISOR TO ATTACH PROCESSING LABEL HERE

Figures

Words

STUDENT NUMBER Letter

Victorian Certicate of Education

2005

MATHEMATICAL METHODS (CAS)

PILOT STUDY

Written examination 2

(Analysis task)

Monday 7 November 2005

Reading time: 9.00 am to 9.15 am (15 minutes)

Writing time: 9.15 am to 10.45 am (1 hour 30 minutes)

QUESTION AND ANSWER BOOK

Structure of book

Number of

questions

Number of questions

to be answered

Number of

marks

4 4 55

Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers,

sharpeners, rulers, a protractor, set-squares, aids for curve sketching, up to four pages (two A4 sheets)

of pre-written notes (typed or handwritten) and one approved CAS calculator (memory DOES NOT

need to be cleared) and, if desired, one scientic calculator. For the TI-92, Voyage 200 or approved

computer based CAS, their full functionality may be used, but other programs or les are notpermitted.

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 12 pages, with a detachable sheet of miscellaneous formulas in the

centrefold.

Working space is provided throughout the book.

Instructions

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

Write your student numberin 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.

VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2005


2/15

MATH METH (CAS) EXAM 2 2MATH METH (CAS) EXAM 2 2

Question 1

Letf : [0, !) "R,f (t) = 2et.

a. i. State the range off.

ii. Find the rule for the inverse offand state its domain.

1 + 2 = 3 marks

b. Letg: [0, !) "R,g(t) = (t 1)2et.

Part of the graph ofgis shown.

Question 1 continued

Instructions

Answer allquestions 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 notdrawn to scale.

g(t)

tO


3/15

3 MATH METH (CAS) EXAM 2

i. The rule for the derivative ofgmay be expressed in the formg!(t) = (t2+ bt+ c)et.

Find the exact values of band c.

ii. The graph ofy=g(t) has stationary points (1,p) and (m, n).

Find the exact values ofp, mand n.

iii. For the function q: [0, !)"R, with rule q(t) = 2g(t) 5, state the exact coordinates of the stationary

points of the graph ofy= q(t).

3 + 2 + 2 = 7 marks


TURN OVER


4/15


5/15


Question 2

Oz-Online is an Internet provider company, with more than one million customers, and has two Internet access

plans.

The Dial-up Plan

The Cable-modem Plan

80% of customers have the Dial-up Plan and 20% of customers the Cable-modem Plan.

a. If 10 customers of Oz-Online are selected at random i. what is the probability, correct to four decimal places, that exactly eight customers are on the

Dial-up Plan

ii. how many customers would be expected to be on the Dial-up Plan?

2 + 1 = 3 marks

b. The marketing manager for Oz-Online has decided that the monthly hours used by Dial-up Plan customers

is a random variable with probability density function given by

f t( )=

00.05e0.05t for t>0

for t0

i. Find the mean number of hours used in a month by a randomly selected Dial-up Plan customer.

ii. What percentage, correct to the nearest per cent, of Dial-up Plan customers uses more than 30 hours

in a month?

iii. What is the probability, correct to three decimal places, that a randomly chosen Dial-up Plan customer

uses more than 40 hours per month, given that the customer uses more than 30 hours per month?

2 + 2 + 2 = 6 marks


TURN OVER


6/15

MATH METH (CAS) EXAM 2 6

c. To try to increase the percentage of customers using a cable modem, Oz-Online offers all its present

customers the Cable-modem Plan with unlimited hours, and at a reduced cost.

The marketing manager estimates that each month 20% of the customers on the Dial-up Plan will change

to the Cable-modem Plan, and 5% of the customers on the Cable-modem Plan will switch to the Dial-up

Plan. No presentcustomers stop using Oz-Online.

i. After four months what proportion of the presentcustomers, correct to two decimal places, will now

be on the Dial-up Plan?

ii. How many months would this change pattern need to continue for the proportion of customers on

the Dial-up Plan to be less than 0.25?

3 + 2 = 5 marks

Total 14 marks


7/15


Question 3

A hydroelectric authority is proposing to build a horizontal pipeline which will pass through a new tunnel

and over a bridge. The diagram below shows a cross-section of the proposed route with a tunnel through the

mountain and a bridge over the valley to carry the pipeline.

The boundary of the cross-section can be modelled by a function of the form

y= 100cos" x( )

400

600+ 50 , 0 #x#1600

whereyis the height, in metres, above the proposed bridge and xis the distance, in metres, from a point O

where the tunnel will start.

a. What is the height (in metres) of the top of the mountain above the bridge?

1 mark

b. How many metres below the bridge is the bottom of the valley?

1 mark

c. What is the exact length of

i. the tunnel

ii. the bridge?

1 + 1 = 2 marks


TURN OVER

x

y

tunnel

600 m

Obridge

valley

mountain


8/15


d. What would be the length (correct to the nearest metre) of the tunnelif it were built 20 m higher up the

mountain?

2 marks

A second proposal is to build a solid concrete dam instead of a bridge. The shaded area in the diagram below

also shows a cross-section of the dam wall.

e. i. Write a denite integral, the value of which is the area of the cross-section of the dam wall.

ii. Find the area of the cross-section of the dam wall, correct to the nearest square metre.

2 + 1 = 3 marks


x

y

tunnel

600 m

Odam wall


9/15


A third proposal is to build the tunnel and bridge above the original proposed position.

f. Suppose the tunnel is built at a height such that it starts at a point on the mountain when

x= k, 0 < k< 400.

i. Find the length of the tunnel in terms of k.

ii. Find the length of the bridge in terms of k.

iii. The estimated total cost, Cthousand dollars, of building the tunnel and bridge for this third proposal

is equal to the sum of the square of the length (in metres) of the tunnel and the square of the length(in metres) of the bridge.

Write down an expression for the estimated total cost of building the tunnel and the bridge if the

tunnel starts whenx= k, in terms of k.


TURN OVER

x

y

tunnel

600 m

O

bridge

k


10/15


11/15


Question 4

The pollution level,yunits, along a straight road between two factories, A and B, which are 10 km apart, is

given by

yp

x

q

x=

+ +

1 11, where 0 #x#10

wherexkm is the distance from Factory A, andpand qare positive constants.a. On a particular day the values ofpand qare measured to bep= 9 and q= 4.

i. Finddy

dx

ii. Find the exact value of xat which the pollution level yis a minimum and the exact value of this

minimum.

iii. Sketch the graph ofy=9

1

4

11x x+ +

, where 0 #x#10, on the axes below. Clearly label the endpoints

and turning point with their coordinates. Exact values are required.


TURN OVER

y

O x

10


12/15


END OF QUESTION AND ANSWER BOOK

iv. Jack travels from Factory A to Factory B along the road. For what length of his journey (in kilometres

correct to three decimal places) is the pollution level less than 5?

1 + 3 + 2 + 1 = 7 marks

b. On another day only the value of qis known, q= 4. The pollution level,yunits, is given by

y=p

x x+ +

14

11, where 0 #x#10 andpis a positive constant.

For what values ofpdoes

i. the maximum value ofyoccur whenx= 10

ii. the minimum value ofyoccur whenx= 10?

3 + 2 = 5 marks

Total 12 marks


13/15

MATHEMATICAL METHODS (CAS)

PILOT STUDY

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 2005


14/15

MATH METH (CAS) PILOT STUDY 2

Mathematical Methods CAS Formulas

Mensuration

area of a trapezium:1

2a b h+( ) volume of a pyramid:

1

3Ah

curved surface area of a cylinder: 2!rh volume of a sphere:4

3

3!r

volume of a cylinder: !r2h area of a triangle: 12

bc Asin

volume of a cone:1

3

2!r h

Calculus

d

dxx nxn n( )= 1

x dx

nx c nn n=

+ + +

1

111 ,

d

dxe aeax ax( )= e dx ae c

ax ax= +1

d

dx x xelog ( )( )=

1

1

xdx x c

e= +log

d

dxax a axsin( ) cos( )( )= sin( ) cos( )ax dx a ax c= +

1

d

dxax a axcos( )( ) = sin( )

cos( ) sin( )ax dx

a ax c= +

1

d

dxax

a

axa axtan( )

( )( ) ==

cossec ( )

2

2 product rule:d

dxuv u

dv

dxvdu

dx( )= +

approximation: f x h f x h f x+( ) ( )+ ( ) chain rule:dy

dx

dy

du

du

dx=

average value: 1b a

f x dxa

b

( ) quotient rule:d

dx

u

v

vdu

dx

udv

dxv =

2

Probability

Pr(A) = 1 Pr(A) Pr(AB) = Pr(A) + Pr(B) Pr(AB)

Pr(A|B) =Pr

Pr

A B

B

( )( )

transition matrices: Sn= Tn S

0

mean: = E(X) variance: var(X) = 2= E((X)2) = E(X2) 2

Discrete distributions

Pr(X=x) mean variance

general p(x) = "xp(x) 2 = "(x)2p(x)

= "x2p(x) 2

binomial nCxpx(1 p)nx np np(1 p)

hypergeometric

Dx

N Dn x

Nn

C C

C

n

D

Nn

D

N

D

N

N n

N1

1

Continuous distributions

Pr(a


15/15

2005 Mathematical Methods (CAS) Exam 2.pdf

Documents