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Question 1 (a) Express the following in real-imaginary form: 2i(1 – i) + (V3 + 1)° + (1 + i)(I+,i). (b) Let z = 4- 2i and w = 3+5i. Express z2/w in real-imaginary form. (c) Let z = 2- 2i and w-1-3i. Plot z, w and z on the complex plane.

Question 1 (a) Express the following in real-imaginary form: 2i(1 – i) + (V3 + 1)° + (1 + i)(I+,i). (b) Let z = 4- 2i and w = 3+5i. Express z2/w in real-imaginary form. (c) Let z = 2- 2i and w-1-3i. Plot z, w and z on the complex plane.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 1 (a) Express the following in real-imaginary form: 2i(1 – i) + (V3 + 1)? + (1+ i)(I +i). (b) Let z = 4 - 2i and w = 3+5i. Express z2/w in real-imaginary form. (c) Let z = 2– 2i and w = -1-3i. Plot z, w and z on the complex plane. Question 2 Let p(z) = z° – 2* + 10z° + 82? – 16z + 80. (a) Given that z - 1 – 3i is a factor of p(z), factorize p(z) into quadratic and cubic factors. (b) Find the cube roots of -8 in mod-arg form. (c) Hence factorize p(2) into linear factors. (You may leave complex numbers in mod-arg form if you wish.) Question 3 Find the sum to infinity of sin 0 + } sin 40 + į sin 70 + ... Hint: the formula for the sum of a geometric series is in the info shect. Question 4 (a) Evaluate In(-3- /3i), and sketch the values on the Complex Plane. Which is the principal value? (b) Use the definition of 2" to show that i' is always pure real. (c) Use the definition sin z = (e" -e-") to express sin z in real-imaginary form. (d) Express Im(e(z+tv)) in terms of z and y.
Transcribed Image Text:Question 1 (a) Express the following in real-imaginary form: 2i(1 – i) + (V3 + 1)? + (1+ i)(I +i). (b) Let z = 4 - 2i and w = 3+5i. Express z2/w in real-imaginary form. (c) Let z = 2– 2i and w = -1-3i. Plot z, w and z on the complex plane. Question 2 Let p(z) = z° – 2* + 10z° + 82? – 16z + 80. (a) Given that z - 1 – 3i is a factor of p(z), factorize p(z) into quadratic and cubic factors. (b) Find the cube roots of -8 in mod-arg form. (c) Hence factorize p(2) into linear factors. (You may leave complex numbers in mod-arg form if you wish.) Question 3 Find the sum to infinity of sin 0 + } sin 40 + į sin 70 + ... Hint: the formula for the sum of a geometric series is in the info shect. Question 4 (a) Evaluate In(-3- /3i), and sketch the values on the Complex Plane. Which is the principal value? (b) Use the definition of 2" to show that i' is always pure real. (c) Use the definition sin z = (e" -e-") to express sin z in real-imaginary form. (d) Express Im(e(z+tv)) in terms of z and y.
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