Centre Number

Student Number

SCEGGS Darlinghurst

2010 Preliminary Course Semester 2 Examination

Mathematics

Outcomes Assessed: P2 – P8 Task Weighting: 40%

General Instructions

Reading time – 5 minutes

Working time – 2 hours

Write using blue or black pen

Attempt all questions and show all

necessary working

Answer all questions on the pad paper

provided

Write your Student Number at the top

of each page

Begin each question on a new page

Marks will be deducted for careless or

badly arranged work

Mathematical templates, geometrical

equipment and scientific calculators

may be used

Total marks – 78

Attempt Questions 1 – 6

Question Calc Comm Reasoning Marks

1 /13

2 /13

3 /13

4 /13

5 /13

6 /13

TOTAL /78

Preliminary Course, Semester 2 Examination 2010 page 1 Mathematics

BLANK PAGE

Preliminary Course, Semester 2 Examination 2010 page 2 Mathematics

Marks

Question 1 (13 marks)

(a) Express

38.0 in the form b

a, where a and b are positive integers. 2

(b) The crowd attendance at a football game on the weekend was 15 660 which 2

was a decline of 10% from the previous weekend. How many people attended

the previous weekend?

(c) Factorise fully: 673 2 xx . 2

(d) Find the exact value of 225cos . 2

(e) Find the value of v in asuv 222 when 7104.6 u , 5.21a and 2

13107.5 s . Give your answer to 3 significant figures.

(f) Find the values of a and b if ba 25

3. 3

Preliminary Course, Semester 2 Examination 2010 page 3 Mathematics

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Marks

Question 2 (13 marks)

(a) Find the size of each interior angle of a regular nonagon (9-sided polygon). 2

(b) Solve each of the following:

(i) 752 x 2

(ii) 0931032 xx 3

(c) Solve 3

160tan for 3600 . 2

(d) and are the roots of the quadratic equation 0322 xx .

Find the values of:

(i) 1

(ii) 22 1

(iii) 22 2

Preliminary Course, Semester 2 Examination 2010 page 4 Mathematics

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Marks

Question 3 (13 marks)

(a) Differentiate each of the following with respect to x:

(i) 32

5 3 x

x 2

(ii) 932 xx 2

(b) Find the equation of the tangent to the curve 2

x

xy at the point where 4

1x . Give your answer in general form.

(c) Consider the quadratic function kxkxy 62 .

(i) Show that the function is positive definite when 3k . 3

(ii) Sketch the graph of 464 2 xxy clearly showing any intercepts 2

with the axes and the co-ordinates of the vertex.

Preliminary Course, Semester 2 Examination 2010 page 5 Mathematics

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Marks

Question 4 (13 marks)

(a) In the diagram, AD and DC are equal to 30 cm.

(i) Find the size of DBA giving reasons. 2

(ii) Hence find the length of AB to the nearest centimetre. 2

Question 4 continues on the next page

60

20

30 cm 30 cm

B C A

D NOT

TO

SCALE

Preliminary Course, Semester 2 Examination 2010 page 6 Mathematics

Marks

Question 4 (continued)

(b)

In the diagram A is the point 0,10 and line l has the equation 02043 yx .

(i) Find the co-ordinates of the point C where l intersects the y axis. 1

(ii) B is on the line l such that AB is perpendicular to l. 2

Find the distance AB.

(iii) Prove that ABCAOC . 2

(iv) Find the area of the quadrilateral OABC. 1

(v) Find the gradient of AB and hence find the angle which AB makes 3

with the positive x axis. Give your answer to the nearest degree.

O

C

0,10A x

y

02043: yxl

Preliminary Course, Semester 2 Examination 2010 page 7 Mathematics

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Marks

Question 5 (13 marks)

(a) 2

Find the value of x giving reasons.

(b) Find a, b and c such that cxbxaxx 11322 . 3

Question 5 continues on the next page

E

74º

F

B D

H

C

G

A

107º

xº

Preliminary Course, Semester 2 Examination 2010 page 8 Mathematics

Marks

Question 5 (continued)

(c) A boat leaves the Island of Ischia and travels 40 km on a bearing of T060

to Naples. It then turns to travel a further 45 km to the Island of Capri on

a bearing of T200 .

(i) Find the size of INC . 1

(ii) Find the distance between the islands of Ischia and Capri to the nearest 2

kilometre.

(iii) Find the area of .INC Give your answer in square kilometres to 2

2 significant figures.

Question 5 continues on the next page

Capri, C

Naples, N

Ischia, I

North

North

Preliminary Course, Semester 2 Examination 2010 page 9 Mathematics

Marks

Question 5 (continued)

(d)

(i) Copy and complete the sketch so that it is an odd function. 1

(ii) Write down the equation of the piecewise function drawn in part (i). 2

y

x O

3,2

Preliminary Course, Semester 2 Examination 2010 page 10 Mathematics

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Marks

Question 6 (13 marks)

(a) (i) Write down the factorised form of 33 ba . 1

(ii) Hence show that

cossin1

cossin

cossin 33

. 2

(b)

Write down a pair of inequations to describe the region shaded above. 4

(c) x

xf1

(i) State the domain and range of xf . 2

(ii) For what value of x does 2xf ? 1

(iii) Find xf ' and hence calculate 3'f leaving your answer in 2

simplified surd form with a rational denominator.

(iv) Show that xfxf '23

. 1

End of Paper

y

2

2 x