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Algebraic operation on Complex Numbers - (Concept)
1. Addition of Two Complex Numbers
$z_1=a+i b$ and $z_2=c+i d$ be any two complex numbers. Then, the sum $z_1+z_2$ is defined as
$
z_1+z_2=(a+i b)+(c+i d)=(a+c)+i(b+d)
$
For example, $z_1=(3-4 i)$ and $z_2=(2+5 i)$, then $z_1+z_2$ is
$
(3-4 i)+(2+5 i)=(3+2)+(-4+5) i=5+i
$
2. Difference of Two Complex Numbers
$z_1=a+i b$ and $z_2=c+i d$ be any two complex numbers. Then, the difference $z_1-z_2$ is defined as
$
z_1-z_2=(a+i b)-(c+i d)=(a-c)+i(b-d)
$
For example, $z_1=(-5+7 i)$ and $z_2=(-11+2 i)$, then $z_1-z_2$ is
$
\begin{aligned}
& (-5+7 i)-(-11+2 i)=-5+7 i+11-2 i \\
& =-5+11+7 i-2 i \\
& =(-5+11)+(7-2) i \\
& =6+5 i
\end{aligned}
$
3. Multiplication of Two Complex Numbers
$\mathrm{z}_1=\mathrm{a}+\mathrm{ib}$ and $\mathrm{z}_2=\mathrm{c}+\mathrm{id}$ be any two complex numbers. Then, the multiplication $\mathrm{z}_1 \cdot \mathrm{z}_2$ is defined as
$
\begin{aligned}
& z_1 \cdot z_2=(a+i b) \cdot(c+i d) \\
& =a c+i a d+i b c+i^2 b d \\
& =a c+i(a d+b c)-b d \\
& =(a c-b d)+i(a d+b c)
\end{aligned}
$
For example, $z_1=(4+3 i)$ and $z_2=(2-5 i)$, then $z_1 \cdot z_2$ is
$\begin{aligned} & (4+3 i)(2-5 i)=4(2)-4(5 i)+3 i(2)-(3 i)(5 i) \\ & =8-20 i+6 i-15\left(i^2\right) \\ & =(8+15)+(-20+6) i \\ & =23-14 i\end{aligned}$
4. Division of Two Complex Numbers
z1 = a + ib and z2 = c + id (and z2 is non-zero) be any two complex numbers. Then, the division $\frac{\mathrm{z}_1}{\mathrm{z}_2}$ is defined as
$\begin{aligned} & \frac{\mathrm{z}_1}{\mathrm{z}_2}=\frac{\mathrm{a}+\mathrm{ib}}{\mathrm{c}+\mathrm{id}} \cdot \frac{\mathrm{c}-\mathrm{id}}{\mathrm{c}-\mathrm{id}} \\ & \text { [multiplying numerator and denominator by } \mathrm{c}-\mathrm{id} \text { where one of } \mathrm{c} \text { and } \mathrm{d} \text { is non }-\mathrm{zero}] \\ & \frac{\mathrm{z}_1}{\mathrm{z}_2}=\frac{\mathrm{ac}-\mathrm{iad}+\mathrm{ibc}-\mathrm{i}^2 \mathrm{bd}}{\mathrm{c}^2-(\mathrm{id})^2}=\frac{\mathrm{ac}+\mathrm{i}(\mathrm{bc}-\mathrm{ad})+\mathrm{bd}}{\mathrm{c}^2-\mathrm{i}^2 \mathrm{~d}^2} \\ & \frac{\mathrm{z}_1}{\mathrm{z}_2}=\frac{\mathrm{ac}+\mathrm{bd}+\mathrm{i}(\mathrm{bc}-\mathrm{ad})}{\mathrm{c}^2+\mathrm{d}^2} \\ & \frac{\mathrm{z}_1}{\mathbf{z}_{\mathbf{2}}}=\frac{\mathbf{a c}+\mathbf{b d}}{\mathbf{c}^{\mathbf{2}}+\mathbf{d}^{\mathbf{2}}+\mathbf{i} \frac{\mathbf{b c}-\mathbf{a d}}{\mathbf{c}^{\mathbf{2}}+\mathbf{d}^{\mathbf{2}}}}\end{aligned}$
| Exam | Chapter |
| JEE MAIN | Complex numbers and quadratic equations |
Let then
equals:
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Let If
then
equals
:
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Let and
then
equals
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All the points in the set
lie on a:
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If $\mathrm{a}>0$ and $z=\frac{(1+i)^2}{a-i}$, has magnitude $\sqrt{\frac{2}{5}}$, then $\bar{z}$ is equal to :
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equals:
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If then the greatest common divisor of the least values of m and n is________
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Let and k>0. If the curve represented by
intersects the y-axis at the point P and Q where PQ=5, then the value of k is:
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If ,
then
equals:
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Let . If
then
equals
:
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If then
equals:
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If then
equals :
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equals:
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Let with
and it satisfies
for some natural number n. Then :
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If the real part of the complex number is
, then the value of the integral
is equal to:
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If the real part of the complex number is zero, then the value of
is equal to_____.
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The least positive integer n such that is a positive integer is ______.
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The equation represents a part of a circle having radius equal to :
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Let z=1 + ai be a complex number, a > 0, such that z3 is a real number. Then the sum 1+z+z2+.....+z11 is equal to :
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if is real, then the point represented by the complex number
lies
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If are two complex numbers such that
is a purely imaginary number, then
is equal to:
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The value of is.
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If be the roots of the equation
, then the least value of n for which
is:
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Let A=Then the sum of the elements in A is.
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Let the complex number z = x + iy be such tha t is purely imaginary. If
then
is equal to :
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Let , then
is equal to_______.
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$\frac{1-2i}{2+i}+\frac{4-i}{3+2i}=$
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The reciprocal of $3+\sqrt7i$ is
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What should be added to complex no $2+3 i$ to get $1-i$
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Find the difference. $z_1=2+i_{\&} z_2=-1+2 i$
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If $a, a^{\prime}, b \ $ and $b^{\prime}$ are real numbers, then, $\left(\frac{a+i b}{a^{\prime}+i b^{\prime}}\right)$ will also be real number if:
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If $z \times (3+4i)=2+3i,$ then the value of $z$ is:
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Let $\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4$ be the solution of the equation $4 x^4+8 x^3-17 x^2-12 x+9=0$ and $\left(4+x_1^2\right)\left(4+x_2^2\right)\left(4+x_3^2\right)\left(4+x_4^2\right)=\frac{125}{16} m$. Then the value of $m$ is____________.
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For what values of a and b, $a-2 i=b+(a-4) i$
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The real value of θ for which the expression $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is a real number is:
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The value of $(z + 3) (\bar{z}+3)$ is equivalent to
| A. |
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$\left(\mathrm{x}_1, \mathrm{y}_1\right),\left(\mathrm{x}_2, \mathrm{y}_2\right)$ and $\left(\mathrm{x}_3, \mathrm{y}_3\right)$ are co-ordinates which satisfy both $x^3-3 x y^2=4, y^3-3 x^2 y=5$.
Then the value of $\left|\frac{y_1 y_1 y_3}{\left(y_1-x_1\right)\left(y_2-x_2\right)\left(y_3-x_3\right)}\right|$ is
| A. |
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${ }_{\text {If }} z=\frac{6-3 i}{2+i}$ then $\operatorname{Re}(z)-\operatorname{Im}(z)$ equals
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$\left(\frac{1}{1-2 i}+\frac{3}{1+i}\right)\left(\frac{3+4 i}{2-4 i}\right)$
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Let $\mathrm{z} \in \mathrm{C}$ be such that $\frac{\mathrm{z}^2+3 \mathrm{i}}{\mathrm{z}-2+\mathrm{i}}=2+3 \mathrm{i}$. Then the sum of all possible values of $z^2$ is
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If $z_1, z_2, z_3 \in C$ are the vertices of an equilateral triangle, whose centroid is $z_0$, then $\sum_{k=1}^3\left(z_k-z_0\right)^2$ is equal to
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Let z1 = a + ib and z2 = c + id be any two complex numbers. Then, the sum z1 + z2 is defined as follows:
z1 + z2 = (a + c) + i (b + d), which is again a complex number.
