(a) Find the modulus, argument and express following complex number in Euler form:

(i) 𝒛 = 𝟏 − √𝟑𝒊

(ii) z = ((1+i)/(1−i ))⁷

(b) Find poles and order of the function 𝑓(𝑧) = 1/(z³-5z²+8z-4)

(i) |𝒛 − 𝟏 − 𝒊| ≤ 𝟐

(ii) 𝟐 < |𝒛 − 𝟑| < 4

Rubrics for Question-1:

Marks	Level	Descriptions
5	Excellent	Gives a complete response with a clear, coherent and suitable explanation including strong arguments; identifies all the important elements of the problem with proper examples
4	Very Good	Gives a complete response including strong arguments; identifies all the important elements of the problem without examples.
3	Good	Completes the problem but the explanation may be muddled; argumentation may be incomplete; may not include examples.
2	Average	Completes the problem with some minor computational errors, may include wrong examples.
1	Poor	Description is not understandable; may make major computational errors, include wrong examples.

(a) Find the center and radius of the circle |𝑧 + 1 − 2𝑖| = √π

(b) By using De Moivre’s theorem, compute all roots of the complex function 𝒛^𝟏𝟐 = (−√𝟑 − 𝒊).

Rubrics for Question-2:

Marks	Level	Descriptions
5	Excellent	Gives clear explanations with appropriate diagrams (if necessary); identifies all the important elements of the problem
4	Very Good	Gives clear explanations without appropriate diagrams; understands the underlying mathematical ideas shortly
3	Good	Completes the problem but the explanation may be muddled; diagram may be inappropriate or unclear, understands the underlying mathematical ideas shortly
2	Average	Completes the problem with some minor computational errors and mathematical ideas is not clearly stated.
1	Poor	Unable to indicate which information is appropriate to the problem.

(a) Verify that the Cauchy-Riemann equations are satisfied for the function 𝒇(𝒛) = 𝒆^𝒙+𝒊y

(b) An unbiased coin is tossed 20 times. Find the probability of

(i) Just 5 heads

(ii) At least one head or at best two head

Rubrics for Question-3:

Marks	Level	Descriptions
5	Excellent	A high level of mathematical thinking which includes exceptional skills and appropriate mathematical tools and techniques in the resolution of problems in task(s).
4	Very Good	Apply excepted methods and mathematical tools and techniques in the resolution of problems in task(s) with minor computational errors.
3	Good	Illustrates the essential elements but some ideas are missing, a limited variety of tools and techniques used to resolve the situation presented in the task(s).
2	Average	Demonstrates the essential elements with limited variety of tools and techniques. Lack of mathematical concepts leads to wrong outputs.
1	Poor	Contains irrelevant responses that have no valid relationship to the task(s). Lack of mathematical concepts and unsuccessful attempt to justify results.

(a) By using Cauchy’s integral formula, evaluate ∮ 𝒛^𝟐𝒅𝒛/(𝒛 − 𝒛^2)(z + i) , where C is the circle |𝒛| = 𝟐 described in the positive sense.

(b) Evaluate the contour integral ∫𝟐𝝅0 𝟏d𝜽(𝟓 − 𝟒𝐜𝐨𝐬 𝜽).

Rubrics for Question-4:

Marks	Level	Descriptions
5	Excellent	A high level of mathematical thinking which includes exceptional skills and appropriate mathematical tools and techniques in the resolution of problems in task(s).
4	Very Good	Apply excepted methods and mathematical tools and techniques in the resolution of problems in task(s) with minor computational errors.
3	Good	Illustrates the essential elements but some ideas are missing, a limited variety of tools and techniques used to resolve the situation presented in the task(s).
2	Average	Demonstrates the essential elements with limited variety of tools and techniques. Lack of mathematical concepts leads to wrong outputs.
1	Poor	Contains irrelevant responses that have no valid relationship to the task(s). Lack of mathematical concepts and unsuccessful attempt to justify results.