Complex Number

Study Material (PDF)
Self Assesment
Study Material (PDF)
Self Assesment

Click on each question to see answer & solution.

Question No: #31

If `z_1 = 3i` and `z_2 = -1-i`, then find `arg(z_1/z_2)`
Ans: `-135°`
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Question No: #32

If `z_1 = \frac{1-i}{\sqrt{2}}`, `z_2 = \frac{1+i\sqrt{3}}{2}`, then find the principal value of argument of `z_1z_2`
Ans: `15°`
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Question No: #33

If `z_1, z_2` are two complex numbers and `|z_1 + z_2| = |z_1| + |z_2|`, then prove that `\text{arg}(z_1) = \text{arg}(z_2)`

Question No: #34

If `z_1, z_2` are two complex numbers and `|z_1 + z_2| = |z_1 - z_2|`, then find the value of `\text{amp}(z_1) - \text{amp}(z_2)`

Question No: #35

If `z_1, z_2` are two complex numbers and `|z_1| = |z_2| = 1` and `\text{arg}(z_1) + \text{arg}(z_2) = 0`. Then show that `z_1 = 1/z_2`

Question No: #36

If `z_1, z_2` are two complex numbers and `|z_1| = |z_2|` and `\text{arg}(z_1) ~ \text{arg}(z_2) = \pi`, then show that `z_1 + z_2 = 0`

Question No: #37

Prove that, `\text{amp}(z) - \text{amp}(-z) = \pm\pi` when `\text{amp}(z)` be positive or negetive.

Question No: #38

Find the value of --
i) `1+i+i^2+i^3+i^4`
ii) `1+i^2+i^4+i^6+\cdots + i^16`
iii) `i^n + i^{n+1} + i^{n+2} + i^{n+3}` where `n \in \mathbb{N}`
iv) `\frac{i+i^2+i^3+i^4}{1+i}`
v) `\frac{i^2+i^4+i^6+i^7}{1+i^2+i^3}`
vi) `\sum_{n=0}^225 i^n`
vii) `(1+i)^6(1+1/i)^6`
viii) `\sqrt{-3 + \sqrt{-3 + \sqrt{-3 + \cdots \ \infty}}}`
Ans: i) `1`
ii) `1`
iii) `0`
iv) `0`
v) `1 - i`
vi) `1 + i`
vii) `64`
viii) `\frac{1 \pm i \sqrt{11}}{2}`
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Question No: #39

i) If `a = \frac{1+i}{\sqrt{2}}` then show that `1+a^2+a^4+a^6=0`
ii) If `x = 2-i\sqrt{3}` then find the value of `2x^4 - 5x^3 - 3x^2 + 41x - 35`
iii) If `x = 2 + 3i` then find the value of `x^3 - 4x^2 + 13x + 5`
iv) If `x = -1 + i\sqrt{2}` then find the value of `x^4 + 4x^3 + 6x^2 + 4x + 9`
Ans: i) N.A.
ii) `0`
iii) `5`
iv) `12`
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Question No: #40

Find square root of following complex number --
i) `i`
ii) `-2i`
iii) `\frac{1+i}{1-i}`
iv) `\frac{-1+\sqrt{-3}}{2}`
v) `7-24i`
vi) `y+\sqrt{y^2-x^2}` where `y^2 \lt x^2`
vii) `1+i\sqrt{a^4-1}`
viii) `x-i\sqrt{x^4+x^2+1}`
ix) `1+i`
x) `1-i`
Ans: i) `\pm \frac{1}{\sqrt{2}}(1+i)`
ii) `\pm (1-i)`
iii) `\pm \frac{1}{\sqrt{2}} (1+i)`
iv) `\pm 1/2 (1+i\sqrt{3})`
v) `\pm (4-3i)`
vi) `\pm \frac{1}{\sqrt{2}} (\sqrt{y+x}+\sqrt{y-x})`
vii) `\pm \frac{1}{\sqrt{2}} (\sqrt{a^2+1}+i\sqrt{a^2-1})`
viii) `\pm \frac{1}{\sqrt{2}} (\sqrt{x^2+x+1}-i\sqrt{x^2-x+1})`
ix) `\pm (\sqrt{\frac{\sqrt{2}+1}{2}} + i \sqrt{\frac{\sqrt{2}-1}{2}})`
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