Complex Number

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

Click on each question to see answer & solution.

Question No: #41

Find the fourth root of `7-24i`
Ans: `\pm 1/sqrt{2}(3-i)`, `\pm 1/sqrt{2}(1+3i)`
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Question No: #42

Show that one of the values of `\sqrt{i} + \sqrt{-i}` is `\sqrt{2}`

Question No: #43

Show that one of the values of `\sqrt{1+i} - \sqrt{1-i}` is `i\sqrt{2(\sqrt{2}-1)}`
Ans: N.A.
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Question No: #44

If `y = \sqrt{x^2+6x+8}` where `x \gt 0`, then show that one of the values of `\sqrt{1+iy} + \sqrt{1-iy}` is `\sqrt{2x+8}`

Question No: #45

i) If `z = x+iy` where `x,y \in \mathbb{R}` and `i=\sqrt{-1}` and `|z-2| = |2z-1|`, then prove that `x^2+y^2 = 1`
ii) If `z = x+iy` where `x,y \in \mathbb{R}` and `i=\sqrt{-1}` and `|z+6| = |2z+3|`, then prove that `x^2+y^2 = 9`

Question No: #46

If `z = x+iy` (`x,y \in \mathbb{R}`) and `i=\sqrt{-1}` and `|2z+1| = |z-2i|` then show that `3(x^2+y^2)+4(x+y)=3`
Ans: N.A.
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Question No: #47

If `z = 3+2i` and `\frac{2z-1}{z-2} = x+iy` (where `x, y` are real), then find the value of `x, y`.
Ans: `x=13/5`, `y= - 6/5`
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Question No: #48

If `a, b, c, d, x, y` are real number and `(a+ib)(c+i``d) = (x+iy)`, then show that `(ac-bd)^2+(ad+bc)^2 = x^2+y^2`

Question No: #49

If `a^2+b^2=1` (where `a,b \in \mathbb{R}`), then show that a real value of `x` satisfies the equation `\frac{1-ix}{1+ix}=a-ib`

Question No: #50

If a complex number `z` is such that `\frac{z-1}{z+1}` is purely imaginary, then show that `|z| = 1`
Ans: N.A.
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