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Quiz 2: Polar form and roots of complex numbers

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
What are the modulus and the principal argument of -5 - 5i?
a) 5 and 5π4   b) 5 and -3π4
c) √ - 5 2 and 5π4   d) √ - 5 2 and -3π- 4
e) 5√2- and 3π 4

 

Not correct. Choice (a) is false.
Remember that if z = x + iy then |z| = ∘------- x2 + y2 and the principal argument of z is greater than -π and less than or equal to π.
Not correct. Choice (b) is false.
Remember that if z = x + iy then |z| = ∘ ------- x2 + y2.
Not correct. Choice (c) is false.
Remember that the principal argument of z is greater than -π and less than or equal to π.
Your answer is correct.
|- 5- 5i| = ∘ (- 5)2 +-(- 5)2 = √50-= 5√2 .
Recall that the principal argument is greater than -π and less than or equal to π.
Not correct. Choice (e) is false.
Remember that the principal argument of z is greater than -π and less than or equal to π. Check which quadrant -5 - 5i lies in, using a diagram.
 

Question 2

 
 
Check all options corresponding to the polar form of 2 - 2i.
a) ( π) 2cis - 4-   b) ( ) 7π- 2 cis 4
c) √ - ( ) 2 2cis - π 4   d) √ - (π) 2 2cis 4
e) ( ) 2√2-cis 7π- 4

 

There is at least one mistake.
For example, choice (a) should be false.
There is at least one mistake.
For example, choice (b) should be false.
There is at least one mistake.
For example, choice (c) should be true.
There is at least one mistake.
For example, choice (d) should be false.
There is at least one mistake.
For example, choice (e) should be true.
Your answers are correct
  1. False.
  2. False.
  3. True.
  4. False.
  5. True.
 

Question 3

 
 
The cartesian form of 8cis(π) is
a) 8   b) 8π
c) 8 - i   d) 8i
e) -8

 

Not correct. Choice (a) is false.
Recall that if z = r cisθ then the cartesian form of z is r cosθ + ir sinθ.
Not correct. Choice (b) is false.
Recall that if z = r cisθ then the cartesian form of z is r cosθ + ir sinθ.
Not correct. Choice (c) is false.
Recall that if z = r cisθ then the cartesian form of z is r cosθ + ir sinθ.
Not correct. Choice (d) is false.
Recall that if z = r cisθ then the cartesian form of z is r cosθ + ir sinθ.
Your answer is correct.
8cisπ = 8cosπ + i8sinπ = -8 + 0i = -8.
 

Question 4

 
 
If z = -1 + i then z expressed in polar form is
a) √- 2 cis π 4   b) √ - 3π 2cis 4--
c) 1 π √-- cis(- 4-) 2   d) cis π 4
e) π cis(- 4)

 

Not correct. Choice (a) is false.
Suppose z = x + iy⁄=0 has polar form r cisθ. Then r = ∘ -2---2- x + y, x- r = cosθ and y- r = sinθ.
Your answer is correct.
√2-(cos 3π+ isin 3π) 4 4
√ - = 2(-√1-+ i√1-) = - 1+ i. 2 2
Not correct. Choice (c) is false.
Suppose z = x + iy⁄=0 has polar form r cisθ. Then r = ∘x2-+-y2-, x- r = cosθ and y- r = sinθ.
Not correct. Choice (d) is false.
Suppose z = x + iy⁄=0 has polar form r cisθ. Then r = ∘ ------- x2 + y2, x r- = cosθ and y- r = sinθ.
Not correct. Choice (e) is false.
Suppose z = x + iy⁄=0 has polar form r cisθ. Then r = ∘ ------- x2 +y2, x- r = cosθ and y- r = sinθ.
 

Question 5

 
 
An equivalent form of the complex number 1√--+ i 1√- 2 2 is
a) cos iπ- 4   b) √1- cis iπ 2 4
c) 1 π √-- cis-- 2 4   d) π- cis4
e) None of the above

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
π π π cis--= cos--+ isin -- 4 4 4
1 1 = √2- + i√2- .
Not correct. Choice (e) is false.
 

Question 6

 
 
If z = -1 + i and 1 1 w = √--+ i√-- 2 2 then zw equals
a) -√- 2 + √- 2i   b) -√- 2
c) 0   d) √ - 2i
e) None of the above

 

Not correct. Choice (a) is false.
Recall that if z = a + ib and w = c + id then zw = ac - bd + i(ad + bc).
Your answer is correct.
Not correct. Choice (c) is false.
Recall that if z = a + ib and w = c + id then zw = ac - bd + i(ad + bc).
Not correct. Choice (d) is false.
Recall that if z = a + ib and w = c + id then zw = ac - bd + i(ad + bc).
Not correct. Choice (e) is false.
Recall that if z = a+ib and w = c+id then zw = ac-bd+i(ad+bc).
 

Question 7

 
 
Suppose that w = 2 cis π 4 . Check all options which equal w4.
a) -16i   b) -16
c) 2cisπ   d) 8cisπ
e) 16cisπ

 

There is at least one mistake.
For example, choice (a) should be false.
Recall that if w = r cisθ then w4 = r4(cisθ)4 = r4 cis(4θ).
There is at least one mistake.
For example, choice (b) should be true.
π π w4 = 24(cos --+isin--)4 = 16(cosπ + isinπ ) = - 16. 4 4
There is at least one mistake.
For example, choice (c) should be false.
Recall that if w = r cisθ then w4 = r4(cisθ)4 = r4 cis(4θ).
There is at least one mistake.
For example, choice (d) should be false.
Recall that if w = r cisθ then w4 = r4(cisθ)4 = r4 cis(4θ).
There is at least one mistake.
For example, choice (e) should be true.
w4 = 24(cis π)4 = 16cisπ. 4
Your answers are correct
  1. False. Recall that if w = r cisθ then w4 = r4(cisθ)4 = r4 cis(4θ).
  2. True. π π w4 = 24(cos --+isin--)4 = 16(cosπ + isinπ ) = - 16. 4 4
  3. False. Recall that if w = r cisθ then w4 = r4(cisθ)4 = r4 cis(4θ).
  4. False. Recall that if w = r cisθ then w4 = r4(cisθ)4 = r4 cis(4θ).
  5. True. w4 = 24(cis π)4 = 16cisπ. 4
 

Question 8

 
 
It is known that the polynomial equation
z4 - 4z3 + 14z2 - 36z + 45 = 0
has 3i and 2 -i as two of its roots. What are the other two roots?
a) 3i, 2 + i   b) 2 - 3i, i
c) -3i, 2 + i   d) 1 - 3i, 2 + i
e) There is not enough information to be able to work this out.

 

Not correct. Choice (a) is false.
Recall that if z = a + ib is a root of a polynomial with real coefficients then so is ¯z = a- ib .
Not correct. Choice (b) is false.
Recall that if z = a + ib is a root of a polynomial with real coefficients then so is ¯z = a- ib .
Your answer is correct.
Any polynomial equation with real coefficients has
non-real roots in complex conjugate pairs.
Not correct. Choice (d) is false.
Recall that if z = a + ib is a root of a polynomial with real coefficients then so is ¯z = a - ib .
Not correct. Choice (e) is false.
Recall that if z = a + ib is a root of a polynomial with real coefficients then so is ¯z = a - ib .
 

Question 9

 
 
The 5th roots of -1 are
a) (π-) cis 5 , ( ) 3π- cis 5 , ( ) 3π- cis - 5 and ( π) cis - 5
b) ( ) cis ± π- 5 , ( ) cis ± 3π- 5 , -1
c) cis(± π-) 10 , ( ) cis ± 3π-- 10 and -1
d) -1, i, i + 1, -i and -i - 1
e) (π) cis 5 , ( 3π) cis 5 , ( 7π) cis 5 and ( 9π) cis 5

 

Not correct. Choice (a) is false.
How many fifth roots of a number are there?
Your answer is correct.
Observe that the fifth roots of -1 are spaced at equal intervals of 2π- 5 around the unit circle.
Not correct. Choice (c) is false.
Suppose z = r cisθ, so that z5 = r5 cis(5θ). Then r5 cis(5θ) = -1 = 1cis(π + 2kπ) where k is any integer. Solve for r and find all θ which satisfy this equation.
Not correct. Choice (d) is false.
Suppose z = r cisθ, so that z5 = r5 cis(5θ). Then r5 cis(5θ) = -1 = 1cis(π + 2kπ) where k is any integer. Solve for r and find all θ which satisfy this equation.
Not correct. Choice (e) is false.
How many fifth roots of a number are there?
 

Question 10

 
 
Find the 8th roots of
1 1 - √--+ √--i. 2 2
a) ( ) √1-cis ±π- 2 4 , ( ) √1-cis ± 3π- 2 4 , ±1 and ±i
b) ( ) √2 cis ± π- 4 , ( ) √2cis ± 3π- 4 , ±√2 and ±i√2-
c) π cis(±--) 4 , 3π cis(±---) 4 , π cis(± -) 2 and cis(± π)
d) ( 29π) cis - 32-- , ( 21π) cis - -32- , ( 13π ) cis --32- , ( 5π) cis - 32- , ( 3π) cis 32- , (11π ) cis -32- , (19π) cis -32- and (27π) cis -32-
e) ( π ) cis ± 4- , ( 3π ) cis ±-4- , ( 5π) cis ± 4-- and ( 7π ) cis ±-4-

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.