Properties of Modulus

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 $\sqrt{2}$

4. Circle of radius 2

Q. If $|z_1|=1,|z_2|=2,|z_3|=3$ and $|9z_1{z}_2+4z_1{z}_3+z_2{z}_3|=12,$ then the value of $|z_1+z_2+z_3|$ is

1. $1$
2. $2$
3. $3$
4. $4$

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

Q.

( z + a ) ( $\overline{z}+a)$ , where a is real , is equivalent to :

1. 

2. 

3. 

4. 

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.

Let f = {(1, 1), (2, 3), (0, -1), (-1, -3)} be a function from Z to Z defined by f(x) = ax + b for some integer a, b. Determine a, b.

Q. If $z_1,z_2,z_3$ are three distinct complex numbers such that $\frac{3}{|z_2-z_3|}=\frac{4}{|z_3-z_1|}=\frac{5}{|z_1-z_2|},$ then $\frac{9}{z_2-z_3}+\frac{16}{z_3-z_1}+\frac{25}{z_1-z_2}$ equals

Q.

If $|z|\le 4$, then maximum value of $|iz+3-4i|$ is equal to:

1. 2

2. 4

3. 3

4. 9

Q. For any two complex numbers $z_1,z_2$ we have $|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2.$ Then

1. $Re\left(\frac{z_1}{z_2}\right)=0$
2. $Im\left(\frac{z_1}{z_2}\right)=0$
3. $Re\left(z_1{z}_2\right)=0$
4. $Im\left(z_1{z}_2\right)=0$

Q. If the mth, nth and pth terms of an AP and GP be equal and be respectively x, y, z, then

1. 
2. None of the these
3. 
4. 

Q. If $z_1$ and $z_2$ are complex numbers and $u=\sqrt{z_1{z}_2},$ then $|\frac{z_1+z_2}{2}+u|+|\frac{z_1+z_2}{2}-u|$ is equal to

1. $|z_1|-|z_2|$
2. $|z_1{|}^2+|z_2{|}^2$
3. $0$
4. $|z_1|+|z_2|$

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. Less than 1

2. Equal to 1

3. Greater than 3

4. Equal to 3

Q. If $z_1,z_2,z_3$ 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 $|z_1+z_2+z_3|$ is

1. equal to 1
2. less than 1
3. greater than 3
4. equal to 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. 0
2. 2
3. 7
4. 17

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. Equal to 1

3. Less than 1

4. Greater than 3

Q.

If |$z_1$| = 1, |$z_2$| = 2, |$z_3$| = 3 and |$9z_1{z}_2+4z_1{z}_3$| = 12, then the value of |$z_1$ + $z_2$ + $z_3$| is equal to:

1. 3

2. 4

3. 6

4. 2

Q.

If |z+7| ≤ 9, for z ∈ C, the greatest value of |Z+2| is __

|$Z_1$ + $Z_2$| ≤ |$Z_1$| + |$Z_2$|

Q. The multiplicative inverse of a non-zero complex number $z$ is

1. $\frac{z}{|z{|}^2}$
2. $\frac{\overline{z}}{|z{|}^2}$
3. $\frac{\overline{z}}{|z|}$
4. $\frac{z}{|z|}$

Q.

If x = 1 + a + $a^2$ .......... to $\infty$ (|a|<1), y = 1 + b + $b^2$ ......... to $\infty$ (|b| < 1), then Z = 1 + ab + $a^2$ $b^2$ + $a^3$ $b^3$..... to ∞ is

1. 

2. 

3. 

4. 

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

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.

If $\left|z\right|=1$ prove that $\frac{1+z}{1+z}=z.$

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

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

Q. If $|z|\le 1,|w|\le 1,$ then

1. $|z-w{|}^2=(|z|-|w|{)}^2+(\arg z-\arg w{)}^2$
2. $|z-w{|}^2\le (|z|-|w|{)}^2+(\arg z-\arg w{)}^2$
3. $|z-w{|}^2\ge (|z|-|w|{)}^2+(\arg z-\arg w{)}^2$
4. None of the above

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. $\frac{2}{5}$

2. $\frac{3}{2}$

3. $\frac{5}{2}$

4. Less than $\frac{2}{3}$

Q. Let $z_1$ be a complex root of the equation $a_n{z}^n+a_{n-1}{z}^{n-1}+\cdots +a_{1}z+a_0=3(|z|<1)$, where $|a_i|<2$ for $i=0,1,2,\dots ,n$. Then

1. $\frac{1}{3}<|z_1|<1$
2. $|z_1|<\frac{1}{3}$
3. $\frac{1}{2}<|z_1|<1$
4. $|z_1|<\frac{1}{2}$

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. 0

4. 

Q. Let $f\left(x\right)=\{\begin{array}{cc}{x}^2,& x\ge 0\\ ax,& x<0\end{array}$
The set of real values of $a$ such that $f\left(x\right)$ will have local minima at $x=0,$ is:

1. $a\in R^{+}$
2. $a\in R$
3. $a\in \varphi$
4. $a\in R^{-}$

Q. If $z$ is a complex number such that $|z|\ge 2,$ then the minimum value of $|z+\frac{1}{2}|$:

1. is equal to $\frac{5}{2}$
2. lies in the interval $(1,2)$
3. is strictly greater than $\frac{5}{2}$
4. is strictly greater than $\frac{3}{2}$ but less than $\frac{5}{2}$

Q. The value of $Im\left(z\overline{z}\right)$

1. is $1$
2. is $0$
3. is $-1$
4. depends on $z$