Complex Numbers

If (1+i) (1+2i) (1+3i) ....... (1+ni) = a+ib, then 2 x 5 x 10 x ...... x (1+n²) is equal to

• $\sqrt{a^2+b^2}$

• $\sqrt{a^2-b^2}$

• $a^2+b^2$

• $a^2-b^2$

If  $\sqrt{\ a+ib}$  = x + iy, then possible value of  $\sqrt{\ a-ib}$ is 

•  $x^2+y^2$  

•  $\sqrt{x^2+y^2}$  

• x + iy

• x - iy

•  $\sqrt{x^2-y^2}$  

If  $\frac{\left(a^2+1\right)^2}{2a-i}$  = x + iy, then  $x^2+y^2$ is equal to 

•  $\frac{\left(a^2+1\right)^4}{4a^2+1}$  

•  $\frac{\left(a+1\right)^2}{4a^2+1}$  

•  $\frac{\left(a^2-1\right)^2}{\left(4a^2-1\right)^2}$  

• None of these

The least positive integer n such that  $\left(\frac{2i}{1+i}\right)^n$  is a positive integer, is

• 16

• 8

• 4

• 2

If $\left(x+iy\right)^{\frac{1}{3}}$   = a + ib, then  $\frac{x}{a}+\frac{y}{b}$  = 

• 0

• 1

• -1

• None of these

If  $z\ =\ \left(\frac{1+i}{1-i}\right)$  , then  $z^4$  equals

• 1

• -1

• 0

• None of these

 If  $z\ =\frac{1}{\left(2+3i\right)^2}\$ , then |z|

•  $\frac{1}{13}$  

•  $\frac{1}{5}$  

•  $\frac{1}{12}$  

• None of these

If x + iy = (1+i) (1+2i) (1+3i), then x² + y² =

• 0

• 1

• 100

• None of these

If i² = -1, then the sum i + i² + i³ + ...... upto 1000 terms is equal to

• 1

• -1

• i

• 0

If x + iy =  $\frac{3+5i}{7-6i}$  , then y = 

•  $\frac{9}{85}$  

•  $\frac{-9}{85}$  

•  $\frac{53}{85}$  

• None of these