Q 11 :

Let $S=\{z\in \mathrm{ℂ}:|\frac{{\displaystyle z-6i}}{{\displaystyle z-2i}}|=1\text{ and }|\frac{{\displaystyle z-8+2i}}{{\displaystyle z+2i}}|=\frac{{\displaystyle 3}}{{\displaystyle 5}}\}.$

Then $\sum _{z\in S}|z{|}^2$ is equal to                      [2026]

• 423

   

• 385

   

• 413

   

• 398

   

Q 12 :

Let $\alpha =\frac{-1+i\sqrt{3}}{2}\text{ and }\beta =\frac{-1-i\sqrt{3}}{2},i=\sqrt{-1}.$ 

If ${(7-7\alpha +9\beta )}^{20}+{(9+7\alpha -7\beta )}^{20}+{(-7+9\alpha +7\beta )}^{20}+{(14+7\alpha +7\beta )}^{20}=m^{10},$ then $m$ is __________ .         [2026]

Q 13 :

$\text{Let }A=\{z\in \mathrm{ℂ}: |z-2|\le 4\} \text{ and }$

$B=\{z\in \mathrm{ℂ}: |z-2|+|z+2|=5\}.$

Then the max  $\left\{\right|z_1-z_2:z_1\in A and z_2\in B\}$ is:   [2026]

• 9

   

• 8

   

• 15/2

   

• 17/2

   

Q 14 :

Let $z=(1+i)(1+2i)(1+3i)$...$(1+ni),$ where $i=\sqrt{-1}$. If $|z{|}^2=44200$, then $n$ is equal to _________.                [2026]

Q 15 :

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}, i=\sqrt{-1}$ then ${(z^{201}-i)}^8$ is equal to [2026]

• 0

   

• 1

   

• 256

   

• -1

   

Q 16 :

Let z be the complex number satisfying |z − 5| ≤ 3 and having maximum positive principal argument.

Then $34 {\left|\left|\frac{{\displaystyle 5z-12}}{{\displaystyle 5iz+16}}\right|\right|}^2$ is equal to:       [2026]

• 16

   

• 12

   

• 20

   

• 26

   

Q 17 :

Let $S=\{z\in \mathrm{ℂ}:4z^2+\overline{z}=0\}.$ Then $\sum _{z\in S}|z{|}^2$ is equal to:  [2026]

• 7/64

   

• 1/16

   

• 5/64

   

• 3/16

   

Q 18 :

Let z be a complex number such that $|z-6|=5 and |z+2-6i|=5$. Then the value of $z^3+3z^2-15z+141$ is equal to    [2026]

• 42

   

• 50

   

• 37

   

• 61