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Modulus of complex number and its Properties - (Concept)
If z = x + iy is a complex number. Then, the modulus of z, denoted by | z |, is the distance of z from the origin in the Argand plane, and it is a non-negative real number equal to $\sqrt{\mathrm{x}^2+\mathrm{y}^2}$.
i.e. |z| =$\sqrt{x^2+y^2}$.
Every complex number can be represented as a point in the argand plane with the x-axis as the real axis and the y-axis as the imaginary axis.
$|z|=\sqrt{x^2+y^2}=r$ (length r from origin to point (x,y))
Properties of Modulus
i) $|z| \geq 0$
ii) $|z|=0$, iff $z=0$ and $|z|>0$, iff $z \neq 0$
iii) $-|z| \leq \operatorname{Re}(z) \leq|z|$ and $-|z| \leq \operatorname{Im}(z) \leq|z|$
iv) $|z|=|\bar{z}|=|-z|=|-\bar{z}|$
v) $z \bar{z}=|z|^2$
vi) $\left|z_1 z_2\right|=\left|z_1\right|\left|z_2\right| . \quad$ Thus, $\left|z^n\right|=|z|^n$
vii) $\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}$
viii) $\left|z_1 \pm z_2\right| \leq\left|z_1\right|+\left|z_2\right|_{\text {(Triangle inequality) this can be generalised for } \mathrm{n} \text { complex numbers. }}$
ix) $\left|z_1 \pm z_2\right| \geq\left|\left|z_1\right|-\left|z_2\right|\right|$
| Exam | Chapter |
| JEE MAIN | Complex numbers and quadratic equations |
The complex number satisfying
will be:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $z=-12-5 i$ and $w=2 \sqrt{11}+i y$ such that z and w both are equidistant from the origin then y equals $n\sqrt5$, the value of n is ______________(Where $y \epsilon R$ )
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be a complex number. Then the set of all complex numbers z satisfying the equation
for some real number k is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\mathrm{a}>0$ and $z=\frac{(1+i)^2}{a-i}$, has magnitude $\sqrt{\frac{2}{5}}$, then $\bar{z}$ is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The equation $\left | z-i \right |=\left | z-1 \right |,i=\sqrt{-1},$ represents:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let , if
then
equals
:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let , If
then
equals
:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let , where
is such that
then
equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let z is a complex number which is 3 units away from origin then :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
are equidistant from origin then:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let z is a complex number 5 units away from origin then can't be more than:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
z is a complex number 7 units away from origin then can't exceed:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
, then
equals ___________. Write up to 2 decimal places.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
are complex numbers such that
. If
and
then a.b equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
then
equals _____________. Write up to 3 decimal places.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
are two complex numbers such that
. if the distance
from the origin is 16 units then the distance (in Units) from
the origin equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be any two non-zero complex numbers such that
. If
then :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If then the maximum value of
is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If then the maximum value of
is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
z is a variable complex number which satisfies the maximum possible value of
equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
z & w are variable complex numbers satisfying and
then the maximum possible value of
equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
z is a complex number always satisfying , then minimum possible value of
equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If z is a complex number such that then the minimum value of
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are two non-zero complex numbers such that
then arg
is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
and
are two complex numbers such that
and
then
equal:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
and
are two complex numbers such that
,
and
, then
equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
&
are two complex numbers such that
,
and
, then
equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
A complex number $z$ lies above real axis at a distance 3 units from it and also on left of imaginary axis at a distance of 2 units from it then the complex number
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $z=2+i$ is reflected twice in order, firstly in real axis then in imaginary axis then the resulting complex number $z_1$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let z is a complex number such that then
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
&
are complex numbers such that
and
then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
A complex number z is said to be unimodular if . Suppose z1 and z2 are complex numbers such that
is unimodular and z2 is not unimodular.Then the point z1 lies on a :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Imaginary part of can be:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be a non-zero complex number such that
where
then z lies on the:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the four complex numbers $z, \bar{z}, \bar{z}-2 \operatorname{Re}(\bar{z})$ and $z-2 \operatorname{Re}(z)$ represent the vertices of a square of side 4 units in the Argand plane, the $|z|$ is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the least and largest real values of , for which the equation
has a solution, are p and q respectively; then
is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $z$ be those complex numbers which satisfy
$
|z+5| \leq 4 \text { and } z(1+i)+\bar{z}(1-i) \geqslant-10, i=\sqrt{-1}
$
If the maximum value of $|z+1|^2$ is $\alpha+\beta \sqrt{2}$, then the value of $(\alpha+\beta)$ is
$\qquad$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let a complex number be $w=1-\sqrt{3} i$. Let the another complex number $z$ be such that $|z w|=1_{\text {and }} \arg (z)-\arg (w)=\frac{\pi}{2}$. Then the area of the triangle with the vertices origin, $a$ and $w$ is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find |z| if :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
are two complex numbers such that sum of their distances from origin is 18 units then
satisfies always.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
then
equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let z and w are two complex numbers such that and
then
can't be less than :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
z is a complex number always satisfying then minimum possible value of
equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
z is a variable complex number satisfying then maximum possible value of
equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let z be a complex number such that (where
). The
is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If Where
is a complex number, then the value of
is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If then
lies on:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For , if the minimum value of
is
, then a value of
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\mathrm{S}_1=\left\{\mathrm{z}_1 \in \mathrm{C}:\left|\mathrm{z}_1-3\right|=\frac{1}{2}\right\}_{\text {and }}$
$\mathrm{S}_2=\left\{\mathrm{z}_2 \in \mathbf{C}:\left|\mathrm{z}_2-\left|\mathrm{z}_2+1\right|\right|=\left|\mathrm{z}_2+\left|\mathrm{z}_2-1\right|\right|\right\}$. Then, for $\mathrm{z}_1 \in \mathrm{~S}_1$ and $\mathrm{Z}_2 \in \mathrm{~S}_2$, the least value of $\left|z_2-z_1\right|$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If be a complex number such that
, then the maximum value of
is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
then
equals :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let , where z is any non-zero complex number.
The set is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If a complex number z satisfies the equation , then
is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Sum of squares of modulus of all the complex numbers satisfying
is equal to ___________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Determine the value of z if,
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let . Then
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $z_1, z_2, z_3$ are the vertices of an equilateral triangle such that $\left|z_1-3+4 i\right|=\left|i z_2-3 i-4\right|=\left|4+3 i-i z_3\right|$, then $\left|z_1+z_2+z_3\right|$ is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\mathrm{S}=\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}-\mathrm{i}|=|\mathrm{z}+\mathrm{i}|=|\mathrm{z}-1|\}$, then, $\mathrm{n}(\mathrm{S})$ is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $z_1$ and $z_2$ be two complex numbers such that $z_1+z_2=5$ and $z_1^3+z_2^3=20+15 i$. Then, $\left|z_1^4+z_2^4\right|$ equals-
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $S=\{z \in C:|z-1|=1$ and $(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2 \sqrt{2}\}$. Let $z_1, z_2 \in S$ be such that $\left|z_1\right|=\max _{Z \in S}|z|$ and $\left|z_2\right|=\min _{z \in S}|z|$. Then $\left|\sqrt{2} z_1-z_2\right|^2$ equals :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $P=\{z \in C:|z+2-3 i| \leq 1\}$ and $Q=\{z \in C: z(1+i)+\bar{z}(1-i) \leq-8\}$. Let in $P \cap Q,|z-3+2 i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $\left|z_1\right|^2+2\left|z_2\right|^2=\alpha+\beta \sqrt{2}$, where $\alpha, \beta$ are integers, then $\alpha+\beta$ equals_______.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $z$ is a complex number such that $|z| \geq 1$, then the minimum value of $\left|z+\frac{1}{2}(3+4 i)\right|$ is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let a, b be two real numbers such that IF the complex number
is of unit modulus and a
lies on the circle
,then a possible value of
where [t] is greatest integer function, is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $z=\frac{1}{2}-2 i$ is such that $|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}$ and $\alpha, \beta \in \mathbb{R}$, then $\alpha+\beta$ is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$A\left ( z_{1} \right ),B\left ( z_{2} \right ),O\left (O \right )$ are vertices of right-angled isosceles triangle, right-angled at O then (z1 - i.z2) equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$z \bar{z}=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If z is origin, then $\left | z \right |=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $|z|=5 {\text { then }} \operatorname{Re}(z)$ can satisfy
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The modulus of 2 complex no's are 4 and 7 and the modulus of the sum of these complex numbers is 9, then $z_1 \overline{z_2}+\bar{z}_1 z_2=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
What is $|9-12 i|$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which value(s) of $\left|Z_1+Z_2\right|$ is/are acceptable, if $\left|Z_1\right|=3 \quad$ and $\quad\left|Z_2\right|=4$.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which value(s) of $\left|Z_1-Z_2\right|$ is/are acceptable, if $\left|Z_1\right|=7$ and $\left|Z_2\right|=41$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $(A+i B)=\frac{\sqrt{1+2 i}}{\sqrt{3+4 i}}$, then $\left(A^2+B^2\right)^2$ is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The magnitude and amplitude of $\frac{(1+i \sqrt 3 )(2+2i)}{(\sqrt 3 - i)}$ are respectively:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $z_1, z_2$ are two distinct complex number such that $\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2$, then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $z$ satisfies $10|z|^2+3 i(\bar{z})^2=3 i z^2+6$ and $\rho \& q$ are the respective maximum \& minimum value of $|z|$, then $p+q$ is equal to________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find modulus of $\frac{z_1}{z_2}$, if $\left|z_1\right|=3_{\&}\left|z_2\right|=2$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If z is a complex number, then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $f(z)=\frac{7-z}{1-z^2}$ where $z=1+2 i$, then $|f(z)|$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The complex number z which satisfies the condition $\left |\frac{i+z}{i-z} \right |=1$ lies on
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which of the following is correct for any two complex numbers $z_1\: \: and\: \: \: \: z_2$ ?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The inequality representing the following graph is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of the square of the modulus of the elements in the set
$
\{z=a+i b: a, b \in Z, z \in C,|z-1| \leq 1,|z-5| \leq|z-5 i|\} $ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
State true or false
$
\left|Z_1-Z_2\right|^2=\left|Z_1\right|^2+\left|Z_2\right|^2-2\left|Z_1 Z_2\right| \cos \left(\theta_1-\theta_2\right)
$
Where $\arg \left(Z_1\right)=\theta_1$ and $\arg \left(Z_2\right)=\theta_2$
| A. |
|
| B. |
|
State True or False
$\left | Z_1+Z_2 \right |^{2}=\left | Z_1 \right |^{2}+\left | Z_2 \right |^{2}+Z_1\overline{Z}_2+Z_2\overline{Z}_1$
| A. |
|
| B. |
|
w is a complex number such that $|w|=1$ and $\arg (w) \in(0,2 \pi)$, then number of solution of equation $\left|\frac{w}{\bar{w}}+\frac{\bar{w}}{w}\right|=1$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $z$ is a complex number satisfying $\left|z^3+z^{-3}\right| \leq 2$, then the maximum possible value of $\left|z+z^{-1}\right|$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $|z+4| \leq 3$, then the maximum value of $|z+1|_{\text {is }}$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\left | z_{1}z_{2} \right |=5\sqrt{10},$
$| z_{1} |=5\sqrt{2}$
then $\left | z_{2}\right |=?$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $a+i b=c+i d$, then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$|z_1 + z_2| = |z_1| + |z_2|$ is possible if
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the Modulus of the complex number $\small \frac{1-2i}{1-(1+i)^2}$.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Suppose $z$ is any root of $11 z^8+20 i z^7+10 i z-22=0$, where $i=\sqrt{-1}$. Then $S=|z|^2+|z|+1_{\text {satisfies }}$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$\left|(1+i) \frac{2+i}{(3+i)}\right|=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $z$ be a complex number such that $|z|=1$. If $\frac{2+\mathrm{k}^{2} \mathrm{z}}{\mathrm{k}+\overline{\mathrm{z}}}=\mathrm{kz}, \mathrm{k} \in \mathbf{R}$, then the maximum distance of $k+i k^{2}$ from the circle $|z-(1+2 i)|=1$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $A=\{z \in C:|z-2-i|=3\}$, $B=\{z \in C: \operatorname{Re}(z-i z)=2\}$ and $S=A \cap B$. Then $\sum_{z \in S}|z|^2$ is equal to ___________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Obviously, in the Argand plane, the modulus of the complex number x + iy = is the distance between the point P(x, y) and the origin O (0, 0)
