Determine the complex number in Cartesian form that is the result of an anticlockwise
rotation of 5𝜋
on 𝑧 = 2 + 𝑖. Express your answer in exact values and verify on an Argand
2. Consider the complex number 𝑧 = cos 𝜃 + 𝑖 sin 𝜃.
a. Determine 𝑧
3 by expanding and then simplifying (cos 𝜃 + 𝑖 sin 𝜃)
(That is, use (𝑎 + 𝑏)
3 = 𝑎
3 + 3𝑎
2𝑏 + 3𝑎𝑏
2 + 𝑏
b. Now, use DeMoivre’s theorem to determine 𝑧
c. Using a. and b., prove that cos 3𝜃 = −3 cos 𝜃 + 4 cos3 𝜃.
3. If 𝑧 − (3 − 𝑖) is a factor of 𝑃(𝑧) = 𝑧
3 − 8𝑧
2 + 22𝑧 + 𝑎, where
a Î R
, determine 𝑎 and all
the roots of 𝑃(𝑧) = 𝑧
3 − 8𝑧
2 + 22𝑧 + 𝑎. (7)
4. Determine the values of 𝑥 for which 𝑥
2 − 3𝑥 ≤ 4. (4)
5. Consider the function 𝑓(𝑥) =
a. State the domain and range of 𝑓(𝑥). (1)
a. Determine 𝑑𝑦
b. Determine the equation of the tangent to the curve at the point (1, −2).
b. Determine 𝑓
e. Determine 𝑓 ∘ 𝑓
and state the domain and range of 𝑓 ∘ 𝑓
6. Determine the following limits.
2 − 4)
7. A psychologist proposes that the ability of a child to memorize during their first four years
can be modelled by 𝑓(𝑥) = 𝑥ln𝑥 + 1, where 𝑥 is the age in years, 0 < 𝑥 ≤ 4.
a. During which month is the ability at a minimum?
b. When is it at a maximum?
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.
8. The curve with equation 𝑦
2 = 𝑥
3 + 3𝑥
is called the Tschirnhausen Cubic (see below).