Inequality of 2 Complex Numbers

Q. The value of $\sum _{k=1}^{13}\frac{1}{\sin (\frac{\pi }{4}+\frac{(k-1)\pi }{6})\sin (\frac{\pi }{4}+\frac{k\pi }{6})}$ is equal to

1. $3-\sqrt{3}$
2. $2(3-\sqrt{3})$
3. $2(\sqrt{3}-1)$
4. $2(2+\sqrt{3})$

Q. If $\frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i$, then values of $x$ and $y$ are

1. $y=\frac{43}{63},x=-\frac{16}{7}$
2. $y=-\frac{43}{63},x=\frac{16}{7}$
3. $x=3,y=-1$
4. $x=1,y=0$

Q. If $z=x+iy$ is a complex number which satisfies $\frac{|z-5i|}{|z+5i|}=1$, then $z$ lies on

1. the real axis
2. the straight line, $y=5$
3. a circle passing through the origin
4. the imaginary axis

Q. Let $z=x+iy$ and $\omega =\frac{1-iz}{z-i}$. If $|\omega |=1$, then $z$ lies on

1. the imaginary axis
2. on the real axis
3. a unit circle
4. a parabola

Q. If at least one value of the complex number z = x + iy satisfy the condition $|z+\sqrt{2}|=a^2-3a+2$ and the inequality $|z+i\sqrt{2}|=a^2,$ then

1. a > 2
2. a = 12
3. a < 2
4. None of these

Q. If $b_1+b_2+b_3+b_4=0$ where $b_i\in R$ such that the sum of no two values being zero and $b_1{z}_1+b_2{z}_2+b_3{z}_3+b_4{z}_4=0$ where $z_1,z_2,z_3,z_4$ are arbitary complex numbers such that no three of them are collinear, so the required condition for complex numbers to be concylic is

1. $|b_1{b}_4||z_1-z_4{|}^2=|b_2{b}_3||z_2-z_3{|}^2$
2. $|b_1{b}_3||z_1-z_3{|}^2=|b_2{b}_4||z_2-z_4{|}^2$
3. $|b_1{b}_2||z_1-z_2{|}^2=|b_3{b}_4||z_3-z_4{|}^2$
4. None of these