Properties of Modulus

Q.

If $\omega$ is a complex number stisfying $|\omega +\frac{1}{\omega }|$=2,

then maximum distance of $\omega$ from origin is

1. 

2. 

3. 

4. None of these

Q. If $|Z_1+Z_2|$ = $\left|Z_1\right|+\left|Z_2\right|$, then find the value of arg $\left(\frac{Z_1}{Z_2}\right)$

1. 

2. 

3. 

4. 0

Q. If Z is a complex number such that $\left|z\right|$ greater than or equal to 2, then the minimum value of $|z+\frac{1}{2}|$.

1. 

2. 

3. 

4. Less than

Q.

Let $Z_1$=3+4i and |$Z_2$|=1, then

1. maximum value of |+| is 6

2. Maximum value of |+| is 7

3. Minimum value of |+| is 4

4. Minimum value of |+| is 3

Q. For all complex numbers $z_1,z_2$ satisfying $\left|z_1\right|$= 12 and $|z_2-3-4i|=5$ the minimum value of $|z_1-z_2|$ is

1. 2
2. 7
3. 17
4. 0

Q.

If z is a complex number, then the minimum value of |z|+|z-1| is

1. 0

2. 1

3. None of these

4. 1/2

Q. If $\left|z\right|=1$ and $\frac{z_1-z_3}{z_2-z_3}=\frac{z-i}{z+i}$ then $z_1,z_2,z_3$ will be vertices of a

1. None of these
2. equilateral triangle
3. acute angled triangle
4. obtuse angled triangle

Q.

If $z_1,z_{2}and{z}_3$ are complex numbers such that

|$z_1$|=|$z_2$|=|$z_3$|=$|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|=1$,

then |$z_1$+$z_2$+$z_3$|

1. Equal to 3

2. Less than 1

3. Equal to 1

4. Greater than 3

Q. A complex number z is said to be unimodular, if $|z|=1$. If and $z_1$ and $z_2$ are complex numbers such that $\frac{z_1-2z_2}{2-\left(z_1\overline{z_2}\right)}$ is unimodular and $z_2$ is not unimodular.
Then, the point $z_1$ lies on a

1. Straight line parallel to X-axis
2. Straight line parallel to X-axis
3. Circle of radius 2
4. Circle of radius $\sqrt{2}$

Q. Let complex numbers $\alpha$ and $\frac{1}{\overline{\alpha }}$ lie on circles $(x-x_0{)}^2+(y-y_0{)}^2=r^{2}and(x-x_0{)}^2+(y-y_0{)}^2=4r^2$, respectively.
If $z_0=x_0+i{y}_0$ satisfies the equation $2|z_0{|}^2=r^2+2$, then $|\alpha |$ is equal to

1. $\frac{1}{\sqrt{2}}$
2. $\frac{1}{2}$
3. $\frac{1}{\sqrt{7}}$
4. $\frac{1}{3}$