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Determine the complex number in Cartesian form that is the result of an anticlockwise

19 / 01 / 2019 Maths

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Determine the complex number in Cartesian form that is the result of an anticlockwise

rotation of 5πœ‹

6

on 𝑧 = 2 + 𝑖. Express your answer in exact values and verify on an Argand

diagram.

2. Consider the complex number 𝑧 = cos πœƒ + 𝑖 sin πœƒ.

a. Determine 𝑧

3 by expanding and then simplifying (cos πœƒ + 𝑖 sin πœƒ)

3

.

(That is, use (π‘Ž + 𝑏)

3 = π‘Ž

3 + 3π‘Ž

2𝑏 + 3π‘Žπ‘

2 + 𝑏

3

.)Β 

b. Now, use DeMoivre’s theorem to determine 𝑧

3

. (1)

c. Using a. and b., prove that cos 3πœƒ = βˆ’3 cos πœƒ + 4 cos3 πœƒ.Β 

3. If 𝑧 βˆ’ (3 βˆ’ 𝑖) is a factor of 𝑃(𝑧) = 𝑧

3 βˆ’ 8𝑧

2 + 22𝑧 + π‘Ž, where

a Î R

, determine π‘Ž and all

the roots of 𝑃(𝑧) = 𝑧

3 βˆ’ 8𝑧

2 + 22𝑧 + π‘Ž. (7)

4. Determine the values of π‘₯ for which π‘₯

2 βˆ’ 3π‘₯ ≀ 4. (4)

5. Consider the function 𝑓(π‘₯) =

1

π‘₯βˆ’3

.

a. State the domain and range of 𝑓(π‘₯). (1)

a. Determine 𝑑𝑦

𝑑π‘₯

.Β 

b. Determine the equation of the tangent to the curve at the point (1, βˆ’2).

b. Determine 𝑓

βˆ’1

(π‘₯).Β 

e. Determine 𝑓 ∘ 𝑓

βˆ’1

and state the domain and range of 𝑓 ∘ 𝑓

βˆ’1

.Β 

6. Determine the following limits.

a. limπ‘₯β†’2

(π‘₯

2 βˆ’ 4)Β 

b. limπ‘₯β†’7

π‘₯βˆ’7

π‘₯

2βˆ’49

(2)

c. limπ‘₯β†’3

2

π‘₯βˆ’3

Β 

7. A psychologist proposes that the ability of a child to memorize during their first four years

can be modelled by 𝑓(π‘₯) = π‘₯lnπ‘₯ + 1, where π‘₯ is the age in years, 0 < π‘₯ ≀ 4.

a. During which month is the ability at a minimum?Β 

b. When is it at a maximum?Β 

DIRECTIONS:

All questions on this assignment must be submitted by: Tuesday 4 October, 2016 at 2:00 pm

Assignments can be submitted to the assignment submission machine on the third floor of the

Priestley Building (#67). Your assignment must include the cover sheet which has been

emailed to you. All assignment solutions are to be legibly handwritten. Remember, your

solutions MUST be YOUR OWN WORK. Assignment solutions will be available on Blackboard

approximately one week following the due date.Β 

2

8. The curve with equation 𝑦

2 = π‘₯

3 + 3π‘₯

2

is called the Tschirnhausen Cubic (see below).


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