Complex Number

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

Click on each question to see answer & solution.

Question No: #51

If a complex number `z` is such that that `|z|=1`, then show that `\frac{z-1}{z+1}` is purely imaginary.
Ans: N.A.
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Question No: #52

If `x+iy = \sqrt{\frac{a+ib}{c+i\d}}`, then show that `(x^2+y^2)^2 = \frac{a^2+b^2}{c^2+d^2}`
Ans: N.A.
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Question No: #53

If `\sqrt{x-iy} = a-ib` (where `x,y,a,b \in \mathbb{R}`) then show that `\sqrt{x+iy} = a+ib`

Question No: #54

i) If `(a-ib)^{1/3} = p-iq` (where `a,b,p,q \in \mathbb{R}`) then show that `(a+ib)^{1/3}=p+iq`
ii) If `(x-iy)^{1/3} = a-ib` (where `x,y,a,b \in \mathbb{R}`) then show that `x/a + y/b = 4(a^2-b^2)`

Question No: #55

If `x = cos\theta + isin\theta` (where `\theta` real), then show that `x^2 + 1/x^2` is a real number.

Question No: #56

If `x - 1/x = 2isin\theta`, then show that `x^4 - 1/x^4 = 2isin4\theta`

Question No: #57

If `a = cos\alpha + isin\alpha` and `b = cos\beta + isin\beta` then show that `a/b + b/a = 2cos(\alpha-\beta)`

Question No: #58

If `x = cos\theta + isin\theta` (where `\theta` real) and `1 + \sqrt{1-a^2} = na`, then show that `a/{2n}(1+nx)(1+n/x) = 1+acos\theta`
Ans: N.A.
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Question No: #59

The three vertices of an equilateral triangle are represented by three complex numbers `z_1, z_2, z_3`. Then show that --
i) `\frac{1}{z_1 - z_2} + \frac{1}{z_2 - z_3} + \frac{1}{z_3 - z_1} = 0`
ii) `z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1`

Question No: #60

The three points in a complex plane are represented by `z_1, z_2, z_3` such that `|z_1| = |z_2| = |z_3|` and they form an equilateral triangle in complex plane. Prove that `z_1 + z_2 + z_3 = 0`