Inequality of 2 Complex Numbers

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 $|z-\frac{4}{z}|=2$, then the greatest value of $|z|$ is

1. $1+\sqrt{2}$

2. $2+\sqrt{2}$

3. $\sqrt{3}+1$

4. $\sqrt{5}+1$

Q. If |z - 25i| $\le$ 15, then |max amp(z) - min.amp(z)| =

1. $co{s}^{-1}\left(\frac{3}{5}\right)$
2. $\pi -2co{s}^{-1}\left(\frac{3}{5}\right)$
3. $\frac{\pi }{2}+co{s}^{-1}\left(\frac{3}{5}\right)$
4. $si{n}^{-1}\left(\frac{3}{5}\right)-co{s}^{-1}\left(\frac{3}{5}\right)$

Q. The locus of z satisfying the inequality $lo{g}_{1/3}|z+1|>lo{g}_{1/3}|z-1|$ is

1. I (z) < 0
2. R (z) < 0
3. R (z) > 0
4. None of these

Q. If |z - 25i| $\le$ 15, then |max amp(z) - min.amp(z)| =

1. $\pi -2co{s}^{-1}\left(\frac{3}{5}\right)$
2. $\frac{\pi }{2}+co{s}^{-1}\left(\frac{3}{5}\right)$
3. $si{n}^{-1}\left(\frac{3}{5}\right)-co{s}^{-1}\left(\frac{3}{5}\right)$
4. $co{s}^{-1}\left(\frac{3}{5}\right)$

Q. The locus of z satisfying the inequality $lo{g}_{1/3}|z+1|>lo{g}_{1/3}|z-1|$ is

1. R (z) < 0
2. R (z) > 0
3. I (z) < 0
4. None of these

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

Q.

If $|z-\frac{4}{z}|=2$, then the greatest value of $|z|$ is

1. $1+\sqrt{2}$

2. $2+\sqrt{2}$

3. $\sqrt{3}+1$

4. $\sqrt{5}+1$

Q. If $z_1,z_2,z_3$ are the vertices of a triangle in argand plane such that $|z_1-z_2|=|z_1-z_3|,$ then $\arg (\frac{2z_1-z_2-z_3}{z_3-z_2})$ is

1. $\pm \frac{\pi }{3}$
2. $0$
3. $\pm \frac{\pi }{2}$
4. $\pm \frac{\pi }{6}$

Q. Let $z=a+ib$ (where $a,b\in R$ and $i=\sqrt{-1})$ such that $|2z+3i|=|z^2|.$ Identify the correct statement(s)?

1. $|z{|}_{\text{maximum}}$ is equal to $3$.
2. $|z{|}_{\text{minimum}}$ is equal to $1$.
3. If $|z|\text{is maximum}$, then $a^3+b^3$ is equal to $27$.
4. $|z|\text{is minimum}$, then $(a^2+2b^2)$ is $2$.