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MATH1001 Quizzes

Quiz 2: Polar form and roots of complex numbers
Question 1 Questions
What are the modulus and the principal argument of 5 5i? Exactly one option must be correct)
a)
5 and 5π 4
b)
5 and 3π 4
c)
52 and 5π 4
d)
52 and 3π 4
e)
52 and 3π 4

Choice (a) is incorrect
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.
Choice (b) is incorrect
Remember that if z = x + iy then |z| = x2 + y2.
Choice (c) is incorrect
Remember that the principal argument of z is greater than π and less than or equal to π.
Choice (d) is correct!
| 5 5i| = (5)2 + (5)2 = 50 = 52.
Recall that the principal argument is greater than π and less than or equal to π.
Choice (e) is incorrect
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.
Check all options giving a polar form of 2 2i. (Zero or more options can be correct)
a)
2cis π 4
b)
2cis 7π 4
c)
22cis π 4
d)
22cis π 4
e)
22cis 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.
Correct!
  1. False
  2. False
  3. True
  4. False
  5. True
The cartesian form of 8cis(π) is Exactly one option must be correct)
a)
8
b)
8π
c)
8 i
d)
8i
e)
8

Choice (a) is incorrect
Recall that if z = rcisθ then the cartesian form of z is rcosθ + irsinθ.
Choice (b) is incorrect
Recall that if z = rcisθ then the cartesian form of z is rcosθ + irsinθ.
Choice (c) is incorrect
Recall that if z = rcisθ then the cartesian form of z is rcosθ + irsinθ.
Choice (d) is incorrect
Recall that if z = rcisθ then the cartesian form of z is rcosθ + irsinθ.
Choice (e) is correct!
8cisπ = 8cosπ + i8sinπ = 8 + 0i = 8.
If z = 1 + i then z expressed in polar form is Exactly one option must be correct)
a)
2cis π 4
b)
2cis 3π 4
c)
1 2cis(π 4 )
d)
cis π 4
e)
cis(π 4 )

Choice (a) is incorrect
Suppose z = x + iy0 has polar form rcisθ. Then r = x2 + y2, x r = cosθ and y r = sinθ.
Choice (b) is correct!
2(cos 3π 4 + isin 3π 4 )
= 2( 1 2 + i 1 2) = 1 + i.
Choice (c) is incorrect
Suppose z = x + iy0 has polar form rcisθ. Then r = x2 + y2, x r = cosθ and y r = sinθ.
Choice (d) is incorrect
Suppose z = x + iy0 has polar form rcisθ. Then r = x2 + y2, x r = cosθ and y r = sinθ.
Choice (e) is incorrect
Suppose z = x + iy0 has polar form rcisθ. Then r = x2 + y2, x r = cosθ and y r = sinθ.
An equivalent form of the complex number 1 2 + i 1 2 is Exactly one option must be correct)
a)
cos iπ 4
b)
1 2cis iπ 4
c)
1 2cis π 4
d)
cis π 4
e)
None of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
cis π 4 = cos π 4 + isin π 4
= 1 2 + i 1 2.
Choice (e) is incorrect
If z = 1 + i and w = 1 2 + i 1 2 then zw equals Exactly one option must be correct)
a)
2 + 2i
b)
2
c)
0
d)
2i
e)
None of the above

Choice (a) is incorrect
Recall that if z = a + ib and w = c + id then zw = ac bd + i(ad + bc).
Choice (b) is correct!
Choice (c) is incorrect
Recall that if z = a + ib and w = c + id then zw = ac bd + i(ad + bc).
Choice (d) is incorrect
Recall that if z = a + ib and w = c + id then zw = ac bd + i(ad + bc).
Choice (e) is incorrect
Recall that if z = a + ib and w = c + id then zw = ac bd + i(ad + bc).
Suppose that w = 2cis π 4 . Check all options which equal w4. (Zero or more options can be correct)
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 = rcisθ then w4 = r4(cisθ)4 = r4 cis(4θ).
There is at least one mistake.
For example, choice (b) should be True.
w4 = 24(cos π 4 + isin π 4 )4 = 16(cosπ + isinπ) = 16.
There is at least one mistake.
For example, choice (c) should be False.
Recall that if w = rcisθ 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 = rcisθ 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 )4 = 16cisπ.
Correct!
  1. False Recall that if w = rcisθ then w4 = r4(cisθ)4 = r4 cis(4θ).
  2. True w4 = 24(cos π 4 + isin π 4 )4 = 16(cosπ + isinπ) = 16.
  3. False Recall that if w = rcisθ then w4 = r4(cisθ)4 = r4 cis(4θ).
  4. False Recall that if w = rcisθ then w4 = r4(cisθ)4 = r4 cis(4θ).
  5. True w4 = 24(cis π 4 )4 = 16cisπ.
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? Exactly one option must be correct)
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.

Choice (a) is incorrect
Recall that if z = a + ib is a root of a polynomial with real coefficients then so is z̄ = a ib.
Choice (b) is incorrect
Recall that if z = a + ib is a root of a polynomial with real coefficients then so is z̄ = a ib.
Choice (c) is correct!
Any polynomial equation with real coefficients has
non-real roots in complex conjugate pairs.
Choice (d) is incorrect
Recall that if z = a + ib is a root of a polynomial with real coefficients then so is z̄ = a ib.
Choice (e) is incorrect
Recall that if z = a + ib is a root of a polynomial with real coefficients then so is z̄ = a ib.
The 5th roots of 1 are Exactly one option must be correct)
a)
cis π 5 , cis 3π 5 , cis 3π 5 and cis π 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)
cis π 5 , cis 3π 5 , cis 7π 5 and cis 9π 5

Choice (a) is incorrect
How many fifth roots of a number are there?
Choice (b) is correct!
Observe that the fifth roots of 1 are spaced at equal intervals of 2π 5 around the unit circle.
Choice (c) is incorrect
Suppose z = rcisθ, 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.
Choice (d) is incorrect
Suppose z = rcisθ, 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.
Choice (e) is incorrect
How many fifth roots of a number are there?
Find the 8th roots of
1 2 + 1 2i.
Exactly one option must be correct)
a)
1 2cis ±π 4 , 1 2cis ±3π 4 , ±1 and ±i
b)
2cis ±π 4 , 2cis ±3π 4 , ±2 and ±i2
c)
cis(±π 4 ), cis(±3π 4 ), cis(±π 2 ) and cis(±π)
d)
cis 29π 32 , cis 21π 32 , cis 13π 32 , cis 5π 32 , cis 3π 32 , cis 11π 32 , cis 19π 32 and cis 27π 32
e)
cis ±π 4 , cis ±3π 4 , cis ±5π 4 and cis ±7π 4

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect