Complex Number

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

Click on each question to see answer & solution.

Question No: #1

Covert the following complex number into `A + iB` form --
i) `(\frac{1+i}{1-i})^3`
ii) `(\frac{1-i}{1+i})^100`
iii) `\frac{(1+i)^2}{3-i}`
iv) `\frac{\sqrt{3}+i\sqrt{2}}{2\sqrt{3}+i\sqrt{2}}`
v) `\frac{1}{1-\cos\theta-i\sin\theta}`
Ans: i) `0+(-1)i`
ii) `1+0.i`
iv) `4/7 + i \frac{\sqrt{6}}{14}`
v) `1/2 + i/2 cot \frac{\theta}{2}`
SEE SOLUTION

Question No: #2

If `\frac{2+i}{2-3i} = A+iB` then find the value of `A^2+B^2`
Ans: `65/169`
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Question No: #3

If `x+iy = \frac{2}{3+cos\theta+i\sin\theta}` then show that `2x^2+2y^2 = 3x-1`
Ans: N.A.
SEE SOLUTION

Question No: #4

If `(x+iy)^5 = a+ib` then find the value of `(y+ix)^5`
Ans: `b+ia`
SEE SOLUTION

Question No: #5

Find conjugate of the following complex numbers --
i) `-\sqrt{5}+7i`
ii) `\frac{1}{1+i}`
iii) `\frac{2-i}{(1-3i)^2}`
Ans: iii) `\frac{-1-2i}{10}`
SEE SOLUTION

Question No: #6

Which one is correct? i) `2+3i \gt 1+4i`, ii) `3+3i \gt 6+2i`, iii) `5+9i \gt 5+6i`, iv) none of these.

Question No: #7

For any complex number `z`, show that `|z| \gt \frac{|\text{Re}(z)|+|\text{Im}(z)|}{\sqrt{2}}`
Ans: N.A.
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Question No: #8

If `z_1` and `z_2` are two complex numbers, then show that--
i) `|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2[ |z_1|^2 + |z_2|^2 ]`

Question No: #9

Find the least positive whole number of `n`, that satisfies the equation `(\frac{1+i}{1-i})^n=1`

Question No: #10

If `(1+i)(2+i)(3+i)\cdots(n+i)=a+ib`, then show that `2 \cdot 5 \cdot 10 \ \cdots (n^2+1) = a^2+b^2`