Solved by verified expert:see file Prove that there is no complex number such that Write each of the following complex numbers intrigonometric form: Prove that the sum of all the distinct th roots of unity isnzero. What geometric fact does this express?
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ENG 3420 Homework Assignment #1 Complex Numbers
Q1. Let k be an integer. Solve the following:
A. i 12k + i 24k
B.
(i
)(
21k + 31
2
i 5k +15
)
Q2. Calculate the following quantities:
25
A. ( 1 + i )
B.
C.
(1−i)5−1
5
(1+i) +1
1 + i tan( θ )
1 − i tan( θ )
( θ is a real number )
Q3. Two complex numbers z 1 and z 2 satisfy the condition
| z1 + z2 |
| z − z | = 1 . Show that the real part of z 1 z 2 is zero ( Re(z 1 z 1 ) = 0 ).
| 1 2|
Q4. Solve each of the following equations for z :
A. z 3 + i = 0
B. z 2 − iz + ( 1 + i ) = 0
2
C. 3iz − ( 3 + i ) z = 1 + i
Q5. Draw plots for each of the following equations or inequalities:
A. | z | + Re( z ) ≤ 1
B. Re( z 2 ) ≤
1
2
C. I m(z) = ||Re(z)| − 1|
D. || z 2 − 4 || = 9
E. || zz +− 11 || ≤ 1
Q6. Prove that there is no complex number such that |z | − z = i .
Q7. Prove the following identity and interpret it geometrically.
|| z 1 + z 2 ||
2
+ || z 1 − z 2 ||
2
= 2 (|| z 1 ||
2
+ || z 2 || 2 )
Q8. Write each of the following complex numbers in
trigonometric form:
A. 1 − i
B. 1 + i
C. − 1 − √3 i
D. √3 − i
E. 2 + √3 + i
F. √ 1 + i
Q9. Calculate the following quantities:
15
( )
B. (1 −
)
A.
C.
1 + i√3
√2 + i√2
√3 −i
2
(−1 + i√3) 15
(1 − i) 20
+
24
(−1 − i√3) 15
(1 + i) 20
Q10. Find all values of the following roots:
A. √ 1
3
B. √ i
3
C. √ − 4 + 3i
5
D.
√
6
1−i
√3 + i
Q11. Prove that the sum of all the distinct n th roots of unity is
zero. What geometric fact does this express?
Q12. Find all complex roots of z 5 = − 2 + 2i in the polar form.
Plot the answers on the complex plane.
Q13. Consider the following circuit with two components in
series:
Suppose that:
I. The source voltage is 8 mV at a frequency of 750 Hz
II. The resistance of the resistor is 50 Ω
III. The inductance of the inductor is 250 mH
IV. The capacitance of the capacitor is 20 μF
Determine:
A. The circular frequency ω
B. The total impedance of the circuit components
C. The current i in complex polar form (assuming a zero phase
for the source voltage)
D. The voltage across each component.
Q14. Consider the following circuit:
A. Find total impedance of circuit
B. Find voltage across inductor
C. Find current across resistor
…
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