Question

Match the equations in Column I with their number of real roots in Column II
$\begin{array}{cccc}& \text{column I}& & \text{Colum II}\\ \text{A}& \text{If}z=x+iy,\text{and if}& \text{p}& i\\ & \left|\begin{array}{c}z-1\end{array}\right|=\left|\begin{array}{c}z-2\end{array}\right|=\left|\begin{array}{c}z-i\end{array}\right|\text{then z many be}& & \\ \text{B}& \text{If}{z}^{13}{w}^{19}=1& \text{q}& -i\\ & z^5{w}^7=1& & \\ & z^2+w^2=-2& & \\ & \text{z and w are complex numbers the z}& & \\ & \text{may be}& & \\ \text{C}& \text{In part 'B', w may be}& \text{r}& 0\\ \text{D}& R\left(\frac{z-2}{z+4i}\right)& \text{s}& \frac{3}{2}+\frac{3}{2}i\\ & & \text{t}& (1,-2\pm \sqrt{3})\end{array}$

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$R\left(\frac{z-2}{z+4i}\right)=0$
$\Rightarrow (x-1{)}^2-(y+2{)}^2=-3$
$\Rightarrow (y+2{)}^2-(x-1{)}^2=3$
Clearly for $x=1,y=-2\pm \sqrt{3}$