Complex Number

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

Click on each question to see answer & solution.

Question No: #71

If `z = x+iy` (where `x,y \in \mathbb{R}`) and `|\frac{z-3}{z+3}|=2`, then find the locus of `z` in complex plane.

Question No: #72

If `z = x+iy` and `\frac{z-i}{z-1}=ia` (where `x,y,a` are real), then show that `(x-1/2)^2+(y-1/2)^2 = (1/\sqrt{2})^2`

Question No: #73

If `z = x+iy` (`x,y \in \mathbb{R}`) and `\frac{z+1}{z+i}` is purely imaginary, then show that locus of `z` is a circle with centre at `-1/2(1+i)` and radius `1/\sqrt{2}`

Question No: #74

If `z=x+iy` and `|z-2-i|=5`, then show that locus of `z` in complex plane is a circle. Also find its centre & radius.

Question No: #75

If `z = x+iy` then find the numerical value of area of circle determined by `z \bar{z} + (3-4i)z + (3+4i)\bar{z}=0`

Question No: #76

If `a,b,c` real number, `z` complex number and `a^2+b^2+c^2=1`, `b+ic = z(1+a)` then show that `\frac{a+ib}{1+c} = \frac{1+iz}{1-iz}`

Question No: #77

Find the value of --
i) `(1-\omega) (1-\omega^2) (1-\omega^4)(1-\omega^5)`
ii) `(1-\omega^2) (1-\omega^4) (1-\omega^8) (1-\omega^10)`
iii) `(3 + \omega + 3\omega^2)^4`
iv) `1 + \omega^28 + \omega^29`
v) `\omega^4 + \omega^8 + \omega^{-1}\cdot\omega^{-2}`
vi) `\omega^{3n} + \omega^{3n+1} + \omega^{3n+2}` where `n \in \mathbb{N}`
vii) `(\frac{-1+\sqrt{-3}}{2})^19 + (\frac{-1 - \sqrt{-3}}{2})^19`
viii) `(x + y\omega + z\omega^2)^2 + (x\omega + y\omega^2 + z)^2 + (x\omega^2 + y + z\omega)^2`
ix) `1\cdot(2-\omega)\cdot(2-\omega^2) + 2\cdot(3-\omega)\cdot(3-\omega^2)+ \cdots + (n-1)\cdot(n-\omega)\cdot(n-\omega^2)`

Question No: #78

Find the value of `\alpha^4 + \beta^4 + \alpha^{-1} \beta^{-1}` where `\alpha`, `\beta` are imaginary cube root of 1.

Question No: #79

Show that the value of `(a+\omega+\omega^2)(a + \omega^2 + \omega^4)(a + \omega^4 + \omega^8) \cdots` upto `2n` number of factors is `(a-1)^{2n}`

Question No: #80

Show that `\omega/9 [(1-\omega)(1-\omega^2)(1-\omega^4)(1-\omega^8) + 9(\frac{c+a\omega+b\omega^2}{a\omega^2+b+c\omega})] = -1`