Mathematics Extension 2 - aceh.b-cdn.net

Student Name:_____________________

Teacher:_________________________

HURLSTONE

AGRICULTURAL

HIGH

SCHOOL

Year 12

TASK ONE

2021

Mathematics Extension 2

Examiners

Mr G Rawson, Mrs P Biczo, Mr G Huxley

Total

marks: 33

Section 1 ¨C 3 marks

? Attempt Questions 1-3

? Allow about 4 minutes for this section

Section 2 ¨C 30 marks

? Attempt Questions 4-6

? Allow about 36 minutes for this section

General

Instructions

?

?

?

?

?

?

?

?

?

Reading time ¨C 3 minutes

Working time ¨C 40 minutes

Write using black pen.

Calculators approved by NESA may be used.

A reference sheet is provided.

A multiple choice answer sheet is provided at the back of the

paper.

In Questions 4-6 show relevant mathematical reasoning and/or

calculations.

Write your name at the top of each booklet. Answer the

questions in the space provided.

Extra booklets are available if you require more space.

Written by 2021 HAHS Maths Faculty, rewritten by Ariq Abdullah

Section 1

3 marks

Attempt Questions 1-3

1. Given ?? = 1 ? ??, which expression is equal to ?? 3 ?

a. ¡Ì2?? ?3????

4

b. 2¡Ì2?? ?3????

4

3????

c. ¡Ì2?? 4

d. 2¡Ì2??

3????

4

2. The complex number ?? satisfies arg ? ???2 ? = ? ??.

??+2??

2

What is the maximum value of |??|?

a. ¡Ì2

b. 2¡Ì2

c. 2 ? ¡Ì2

d. 2 + ¡Ì2

??

3. Which of the following is equivalent to ?

1 ¡Ì3

+

??

2

2

1 ¡Ì3

b.

+

??

2

2

¡Ì3 1

c. ?

+ ??

2

2

¡Ì3 1

d.

? ??

2

2

a.

?

7????

6

?? ????

??

HAHS Mathematics Extension 2 ¨C Task 1 ¨C 2021 HSC

2

Section 2

30 marks

Attempt Questions 4-6

Allow about 36 minutes for this section

Question 4 (10 marks)

(a) Let ?? = 4 + ?? and ?? = ???. Find, in the form ?? + ????,

(i) ??

(ii) ?? ? ??

(b) Let ?? = 1 + ??¡Ì3 and ?? = 1 + ??.

??

(i)

Find in the form ?? + ????

(ii) Express ?? in modulus-argument form.

(iii) Given that ?? has the modulus-argument form

??

??

?? = ¡Ì2 ?cos + ?? sin ?

4

4

??

Find the modulus-argument form of .

??

??

(iv) Hence, find the exact value of sin .

12

1

1???

is

1

1

1

??

(c) Let ?? = 2(cos ?? + ?? sin ??).

(i) Find ???????

1 ? ??.

(ii) Show that the real part of

Marks

1?2 cos ??

5?4 cos ??

.

Question 5 (10 marks)

(a)

(i) Express ?¡Ì3 ? ?? in modulus-argument form.

(ii) De Moivre¡¯s theorem states that (??????????)?? = ?? ?? ?????????? for any integer ??.

2

1

1

1

2

Marks

2

2

6

Show that ??¡Ì3 ? ??? is a rea number.

(b) Sketch the region in the complex plane where the inequalities 1 ¡Ü |??| ¡Ü 2 and

0 ¡Ü ?? + ??? ¡Ü 3 hold simultaneously.

(c)

Let ?? = 2(cos ?? + ?? sin ??).

Question 6 (10 marks)

(a) Given that ?? = 3 + ?? is a root of ?? 2 + ???? + ?? = 0, where ?? and ?? are real, find

the values of ?? and ??.

(b) Solve the equation ??? 2???? + ?? ?2???? ? = 1 where ??? ¡Ü ?? ¡Ü ??.

3

3

Marks

3

3

Question 6 continued on next page

HAHS Mathematics Extension 2 ¨C Task 1 ¨C 2021 HSC

3

(c)

Let the complex numbers ??(??), ??(??) and the origin form a triangle of area

1 ??2 on the Argand Diagram.

M is the midpoint of UV.

Let ¡Ï?????? = ??

(i) Show that |?? + ??||?? ? ??| sin ?? = 4.

??

(ii) If ?? = 2 , prove that (|??| ? |??|)(|??| + |??|) = 0.

2

2

End of Exam

HAHS Mathematics Extension 2 ¨C Task 1 ¨C 2021 HSC

4

Year 12

MULTIPLE CHOICE

Outcome

1. 1 ? i=

MEX12-4

Mathematics Extension 2

Solutions and Marking Guidelines

Solutions

12 + ( ?1) =

2

arg (1 ? i ) =

?

Marking Guidelines

2

1 mark

¦Ð

4

?

? ¦Ð?

? ¦Ð ??

2 ? cos ? ? ? + i sin ? ? ? ?

? 4?

? 4 ??

?

=

¡à1 ? i

=

(1 ? i )

3

?

? ¦Ð?

? ¦Ð ??

2 ? cos ? ? ? + i sin ? ? ? ?

? 4?

? 4 ??

?

( )

3

?

? 3¦Ð

= 2 2 ? cos ? ?

? 4

?

= 2 2e

MEX12-4

Ass Task 1 2021 HSC

?

3

?

? 3¦Ð ? ?

? + i sin ? ? ? ?

?

? 4 ??

3¦Ð i

4

¡àB

¦Ð

? z?2 ?

?= ?

2

? z + 2i ?

2. arg ?

¡à arg ( z ? 2 ) ? arg ( z ? ( ?2i ) ) =?

¦Ð

2

, and so z lies on a

semicircle (excluding the endpoints) whose diameter is the

line joining the points 2 and ?2i on the argand diagram.

Need to test whether top or bottom half of the semicircle.

Testing if z is at (0, 0):

1 mark

¦Ð

arg ( z ? 2 ) ? arg ( z + 2=

i ) 180¡ã ? 90¡ã ¡Ù ? .

2

If z is ( 2, ?2 ) , arg ( z ? 2 ) ? arg ( z + 2i ) = ?90¡ã ? 0¡ã = ?

¡à z is on the semicircle containing the point ( 2, ?2 ) .

¦Ð

2

.

From the diagram, the position of z that gives the maximum

value of z is when z is at ( 2, ?2 ) .

¡à z is d ( 0, 0 ) , ( 2, ?2 ) = 8 u

¡àB

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