mbiology 2011 source analysis task sheet 1

8/6/2019 mBiology 2011 Source Analysis Task Sheet 1

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Tuesday 20 May 2008 : 11:30 a.m.Time: 2 hours

Examination material: one 21-page question bookletApproved dictionaries, notes, calculators and computer software may be used

Instructions to Candidates1. You will have 5 minutes to read the paper. You must not write in your question booklet or use a

calculator during this reading time but you may make notes on the scribbling paper provided.

2. Answer each question in the space provided in this question booklet. You may write on pages 19and 20 if you need more space, making sure to label each answer clearly

3. The total mark is approximately 94. The allocation of marks is shown below:Question 1 2 3 4 5 6 7 8 9

Marks 13 5 5 6 8 9 12 13 23

4. Appropriate steps of logic and correct answers are required for full marks.5. Show all working in this booklet. (You are strongly advised not to use scribbling paper. Work that

you consider incorrect should be crossed out with a single line.)

6. Use only black or blue pens for all work other than graphs and diagrams, for which you may use asharp dark pencil.

7. State all answers correct to three significant figures, unless otherwise stated or as appropriate.8. Diagrams, where given, are not necessarily drawn to scale.9. The list of mathematical formulae is on page 21. You may remove the page from this booklet

before the examination begins.

10.Write your name, student number, lecturers name and group in the space provided at the top of thispage.

Name : _____________________________________

Student No. : ____________________________________

Group : ____________________________________

Lecturers Name : ____________________________

MATHEMATICAL STUDIES

SOUTH AUSTRALIAN MATRICULATION

Pages : 21

Questions: 9

Mid Year Examination 2008Semester 1 Examination (January 2008 Intake)

Semester 2 Examination (July 2007 Intake)


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QUESTION 1

There is no need to simplify your answers to parts (a), (b), (c) and (d).

(a) Finddx

dyif 3223 24 xxxxy ! .

(3 marks)

(b) Finddx

dyif

xe

xy

2

2

! .

(3 marks)


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(c) Finddx

dyfor xeexy 2ln! by first using logarithm laws to simplify the function completely.

(4 marks)

(d) Find .132

12 3 dx

xx

.

(3 marks)


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QUESTION 2

Find, from first principles, the derivative of9

53)(

!

x

xxf .

(5 marks)


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QUESTION 3

Find the value ofkalgebraically given that the area of the shaded region below is equal to3

8units

2.

(5 marks)

x

y

4

0

24 xxy !

k


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QUESTION 4

A tank in the shape of an inverted cone is filled with

water so that the radius of the water surface r (inmetres) is always equal to a quarterof its depthx (in

metres). Water is running out of the bottom of the

tank at a constant rate of 2 cubic metres per minute.

For parts (a), (b) and (c) below, leave all your

answers in T form.

(a) Show that the volume of the waterV is given by the function 48

3xxV

T! .

(2 marks)

(b) Find an expression fordx

dVin terms ofx.

(1 mark)

r

x


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(c) How fast is the water level falling at the point when the water is 8 metres deep?

(Hint:dt

dx

dx

dV

dt

dV! )

(3 marks)


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QUESTION 5

Given 2ln352 22 ! xyxyx .

(a) Show that xyx

xyx

dx

dy

56

5412

! .

(4 marks)

(b) Is the tangent at the point 0,1 a horizontal tangent? Prove your answer.

(2 marks)


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(c) Hence find the equation of the tangent.

(2 marks)


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QUESTION 6

A vase made of glass of uniform thickness has a square base of side

x cm and a fixed volume of 192 cm3.

This vase is to be made of two different types of coloured glass

joined together, the upper portion being glass with a floral motif,

and the lower to be glass of a solid colour.

The height of the floral upper portion of the vase is to be twice theheight of the solid coloured lower portion. Let the height of the

solid coloured portion be h cm.

Additionally, the cost of the glass with the floral motif is 1.5 times

the cost of the solid coloured glass (you may ignore the thickness ofthe glass and the cost of joining the two types of glass).

(a) Show that the height of the solid coloured portion is given by2

64

xh ! cm.

(1 mark)

(b) Given that the cost of producing the solid coloured portion isp dollars per cm2, show that the cost

of making the whole vase, xC , is given by

!

xxpxC

10242dollars.

(3 marks)

x

x

open


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(c) What value ofx would minimize the cost of producing the vase? Show all working to obtainxbelow and prove that your answer does give the minimum cost.

(3 marks)

(d) In the space provided below, sketch the diagram of the vase and label it with the dimensions that

would minimize its production cost.

(2 marks)


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QUESTION 7

An object is launched upwards at a speed of 19.6 metres per second )ms( 1 from a 58.8 metre tall

platform. The function for the object's height,s at time tseconds after launch is

8.586.199.4 2 ! ttts , wheres is in metres.

(a) Find the expressions for its velocity and acceleration at any instant and draw sign diagrams for bothfunctions.

(4 marks)

(b) At what time interval(s) is the objects:

(i) speed decreasing?

(1 mark)

(ii) velocity increasing?

(1 mark)


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(c) When does the object strike the ground?

(2 marks)

(d) Draw the motion diagram for the object, and calculate the total distance travelled by the object

when it strikes the ground.

(4 marks)


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QUESTION 8

Consider the function 21232 23 ! xxxxf .

(a) Show algebraically that xf has two stationary points, and find and classify these points.

(6 marks)

(b) Find the inflection point(s) of xf .

(2 marks)


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(c) Sketch the graph of xfy ! in the space provided below, clearly labelling all the points foundabove.

(3 marks)

(d) On the graph you have drawn in (c),

(i) label one point Mwhere 0"dxf . (1 mark)

(ii) label one point Nwhere 0

xf . (1 mark)


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QUESTION 9

(a) Show that

!

0

1 2 3

4ln4

4

8dx

x

x.

(4 marks)

(b) Consider the function 4

482

2

!

x

xxxf .

(i) Show that xf has no stationary point. (Your working should include the sign diagram of xfd ).

(5 marks)


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(ii) Express xf in the form of 42

!x

BxAxf and hence identify the vertical asymptotes of

xf .

(2 marks)

(c) Sketch the graph of 4

482

2

!

x

xxxf in the space provided below. Clearly show the axes

intercepts and the vertical asymptotes on your graph.

y

(3 marks)


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(d) For the interval 01 ee x ,

(i) draw two upper rectangles with an equal width of 0.5 units each onto the graph in (c).

(1 mark)

(ii) calculate the total area of the upper rectangles you have drawn in (c), bringing your answer to 2decimal places.

(2 marks)

(e) Using your results from (a) to help you, find the exact area below the graph of xf for 01 ee x .

(4 marks)

(f) Comment on the answers you obtained in (d)(ii) and (e) by making a comparison and justifyingyour conclusion.

(2 marks)


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You may write on this page if you need more space to finish your answers. Make sure to label eachanswer carefully (e.g Question 4(c) continued).


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You may write on this page if you need more space to finish your answers. Make sure to label eachanswer carefully (e.g Question 4(c) continued).


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mbiology 2011 source analysis task sheet 1

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