(i) For any two complex numbers $z_1,z_2$ and any real numbers $a,b$ , $?a^2{z}_1+b^2{z}_2?-?b^2{z}_1+a^2{z}_2?=.....$

(ii) The value of $\sqrt[]{-25}\times \sqrt[]{-9}$ is .....................

(iii) The number $\frac{{\left(1-i\right)}^3}{1-i}$ is equal to ...............

(iv) The sum of the series $i+i^2+i^3+\dots$ up to 1000 terms is ..........

(v) Multiplicative inverse of $1+i$ is ................

(vi) If $z_1$ and $z_2$ are complex numbers such that $z_1+z_2$ is a real number, then $z_2=....$

(vii)  $\left(z\right)+z\left(z\ne 0\right)$ is ...............

(viii) If  $?z+4?\le 3$ , then the greatest and least values of $?z+1?$ are ..... and .....

(ix) If $\frac{z-2}{z+6}=\frac{\pi }{6},$ then the locus of $z$ is ............

(x) If $?z?=4$ and $\left(z\right)=\frac{5\pi }{6}$ , then $z=\dots \dots \dots \dots$

  $\left|\frac{1}{2}-2i+1\right|=\alpha \left(\frac{1}{2}-2i\right)+\beta \left(1+i\right)$  

$\sqrt[]{\frac{9}{4}+4}=\alpha \left(\frac{1}{2}-2i\right)+\beta \left(1+i\right)$

$\frac{5}{2}=\alpha \left(\frac{1}{2}\right)+\beta +i\left(-2\alpha +\beta \right)$             

$\frac{\alpha }{2}+\beta =\frac{5}{2}$      ...(1)

 –2α + β = 0                    …(2)

Solving (1) and (2)

$\frac{\alpha }{2}+2\alpha =\frac{5}{2}$

$\frac{5}{2}\alpha =\frac{5}{2}$            

a = 1

b = 2

-> a + b = 3

$z^2=-\overline{iz}$

$\left|z^2\right|=\left|-\overline{iz}\right|$

$\left|z^2\right|=\left|z\right|$

$\left|z^2\right|-\left|z\right|=0$

$\left|z\right|\left(\left|z\right|-1\right)=0$

|z| = 0 (not acceptable)

|z| = 1

|z|² = 1

Given : x² – 70x + l = 0

->Let roots be a and b

->b = 70 – a

->= a (70 – a)

l is not divisible by 2 and 3

->a = 5, b = 65

-> $\frac{\sqrt[]{5-1}+\sqrt[]{65-1}}{\left|60\right|}=\left|\frac{4+8}{60}\right|=\frac{1}{5}$

z₁ + z₂ = 5

$z_1^3+z_2^3=20+15i$            

  $z_1^3+z_2^3={\left(z_1+z_2\right)}^3-3z_1{z}_2\left(z_1+z_2\right)$           

$z_1^3+z_2^3=125-3z_1\cdot z_2\left(5\right)$            

 ⇒ 20 + 15i = 125 – 15z₁z₂

⇒ 3z₁z₂ = 25 – 4 – 3i

3z₁z₂ = 21– 3i

z₁⋅z₂ = 7 – i

(z₁ + z₂)² = 25

$z_1^2+z_2^2=25-27\left(7-i\right)$     

= 11 + 2i

  ${\left(z_{\text{?}1}^2+z_2^2\right)}^2$         = 121 − 4 + 44i

⇒   $z_1^4+z_2^4+2{\left(7-i\right)}^2=117+44i$

⇒   $z_1^4+z_2^4$ = 117 + 44i − 2(49 −1−14i )

= 21 + 72i

⇒   $\left|Z_1^4+Z_2^4\right|=75$

$x^2-2\left(3k-1\right)x+8k^2-7>0\Rightarrow a>0,\&D<0,$

a = 1 > 0 and D < 0

4 (3k – 1)² – 4 (8k² – 7) < 0

  $\left(9k^2+1-6k-8k^2+7<0\right)$

$k(k-4)-2(k-4)<0$

$k\in \left(2,4\right)$

K = 3