Complex Numbers and Quadratic Equations Class 11 Notes CBSE Maths Chapter 5 [PDF] (2024)

Section–A (1 Mark Questions)

1. If a+2i and 3+bi are conjugate of each other then find value of a and b.

Ans. Let $z=a+2 i$

$\bar{z}=a-2 i$

But we are given that $\bar{z}=3+b i$

$$ \therefore a-2 i=3+b i $$

Comparing the real and imaginary parts, we get $a=3$ and $b=-2$.

2. The conjugate of the complex number $\dfrac{1-i}{1+i}$ is ______.

Ans. $$\text { 2. } \begin{aligned} & \frac{1-i}{1+i}=\frac{1-i}{1+i} \times \frac{1-i}{1-i} \\ & =\frac{1+i^2-2 i}{1-i^2} \\ & =\frac{1-1-2 i}{1+1}=-i \end{aligned} $$

Hence, conjugate of $\frac{1-i}{1+i}$ is $0+i$.

3. If z=2+3i , then value of $\left| z-1+4i \right |$ is______.

Ans. Given that: $z=2+3 i$

Then, $|z-1+4 i|=|2+3 i-1+4 i|$

$$ \begin{aligned} & =|1+7 i|=\sqrt{1^2+7^2} \\ & =\sqrt{50}=5 \sqrt{2} \end{aligned} $$

4. The sum of the series $i+i^{2}+i^{3}+\cdots$ up to 1000 terms is equal to ……….

Ans. $i+i^2+i^3+\ldots .$. upto 1000 terms

$$=i+i^2+i^3+\ldots+i^{1000}=0$$

Hence, the value of the filler is 0 .

5. Find the amplitude of $\sqrt{3}+i$ .

Ans. Let $z=\sqrt{3}+i$

Here, $z$ lies in first quadrant.

Therefore

$\tan\theta=\left|\frac{\operatorname{Im}(z)}{\operatorname{Re}(z)}\right|=\left|\frac{1}{\sqrt{3}}\right|=\frac{1}{\sqrt{3}}$

$$ \therefore \theta=\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right) \text { i.e., } \theta=\frac{\pi}{6} \text {. } $$

Section–B (2 Marks Questions)

6. Show that $(z+3)(\vec{z}+3)$ is equivalent to $\left| z+3 \right |^{2}$ .

Ans. Let $z=x+i y$

$$ \begin{aligned} & \operatorname{So}(z+3)(\bar{z}+3)=(x+i y+3)(x-i y+3) \\ & =[(x+3)+i y][(x+3)-i y] \\ & =(x+3)^2-i^2 y^2 \\ & =(x+3)^2+y^2 \\ & =|x+3+i y|^2=|z+3|^2 . \end{aligned} $$

7. If $\left ( \dfrac{1+i}{1-i} \right )^{x}=1$ , then show that $x=4n$ , where $n\epsilon N $.

Ans. Given that: $\left(\frac{1+i}{1-i}\right)^x=1$

$$ \begin{aligned} & \Rightarrow\left(\frac{(1+i)(1+i)}{(1-i)(1+i)}\right)^x=1 \\ & \Rightarrow\left(\frac{1+i^2+2 i}{1-i^2}\right)^x=1 \\ & \Rightarrow\left(\frac{1-1+2 i}{1+1}\right)^x=1 \\ & \Rightarrow\left(\frac{2 i}{2}\right)^x=1 \\ & \Rightarrow(i)^x=(i)^{4 n} \\ & \Rightarrow x=4 n, n \in N \end{aligned} $$

8. Find the value of p such that the difference of the rootsof the equation $x^{2}-px+8=0$ is 2.

Ans. Let $\alpha, \beta$ be the roots of the equation

$$ x^2-p x+8=0 $$

Therefore $\alpha+\beta=p$ and $\alpha \beta=8$

Now, $\alpha-\beta= \pm \sqrt{(\alpha+\beta)^2-4 \alpha \beta}$

Therefore, $2= \pm \sqrt{p^2-32}$

$$ \begin{aligned} & \Rightarrow p^2-32=4 \\ & \Rightarrow p= \pm 6 . \end{aligned} $$

9. If $x+iy=\dfrac{a+ib}{a-ib}$, prove that $x^{2}+y^{2}=1$.

Ans. We have,

$$ \begin{aligned} & x+i y=\frac{(a+i b)(a+i b)}{(a-i b)(a+i b)} \\ & =\frac{a^2-b^2+2 a b i}{a^2+b^2}=\frac{a^2-b^2}{a^2+b^2}+\frac{2 a b}{a^2+b^2} i \end{aligned} $$

So that, $x-i y=\frac{a^2-b^2}{a^2+b^2}-\frac{2 a b}{a^2+b^2} i$

Therefore,

$$ \begin{aligned} & x^2+y^2=(x+i y)(x-i y) \\ & =\frac{\left(a^2-b^2\right)^2}{\left(a^2+b^2\right)^2}+\frac{4 a^2 b^2}{\left(a^2+b^2\right)^2}=\frac{\left(a^2+b^2\right)^2}{\left(a^2+b^2\right)^2}=1 \end{aligned} $$

10. Express $\frac{5+\sqrt{2} i}{1-\sqrt{2} i}$ in the form of.

Ans. $$ \text { 10. } \begin{aligned} & \frac{5+\sqrt{2} i}{1-\sqrt{2} i} \\ & =\frac{5+\sqrt{2} i}{1-\sqrt{2} i} \times \frac{1+\sqrt{2} i}{1+\sqrt{2} i} \\ & =\frac{5+5 \sqrt{2} i+\sqrt{2} i-2}{1-(\sqrt{2} i)^2}=\frac{3+6 \sqrt{2} i}{1+2} \\ & =\frac{3(1+2 \sqrt{2} i)}{3}=1+2 \sqrt{2} i . \end{aligned} $$

11. What is the conjugate of $\dfrac{\sqrt{5+12 i}+\sqrt{5-12 i}}{\sqrt{5+12 i}-\sqrt{5-12 i}}$

Ans. Let, $z=\frac{\sqrt{5+12 i}+\sqrt{5-12 i}}{\sqrt{5+12 i}-\sqrt{5-12 i}}$

$=\frac{\sqrt{5+12 i}+\sqrt{5-12 i}}{\sqrt{5+12 i}-\sqrt{5-12 i}} \times \frac{\sqrt{5+12 i}+\sqrt{5-12 i}}{\sqrt{5+12 i}+\sqrt{5-12 i}}$

$=\frac{5+12 i+5-12 i+2 \sqrt{25+144}}{5+12 i-5+12 i}$

$=\frac{10+2 \times 13}{24 i}=\frac{36}{24 i}$

$=-\frac{3 i}{2}$

$=0-\frac{3}{2} i$

Therefore, the conjugate of $z=0+\frac{3}{2} i$.

12. Express $z=\frac{1}{(2+i)^2}$ in standard form.

Ans. $$ \begin{aligned} & \text { 12. } \frac{1}{(2+i)^2}=\frac{1}{4+i^2+4 i} \\ & =\frac{1}{4-1+4 i}=\frac{1}{3+4 i} \\ & =\frac{1}{3+4 i} \times \frac{3-4 i}{3-4 i} \\ & =\frac{3-4 i}{9-16 i^2}=\frac{3-4 i}{25} \end{aligned} $$

13. If $\frac{(1+i)^2}{2-i}=x+i y$, find x+y.

Ans. Given: $\frac{(1+i)^2}{2-i}=x+i y$

Upon expansion we get,

$$ \begin{aligned} & \frac{1^2+i^2+2(1)(i)}{2-i}=x+i y \\ & \frac{1+(-1)+2 i}{2-i}=x+i y \\ & \frac{2 i}{2-i}=x+i y \end{aligned} $$

Now, let us multiply and divide by $(2+i)$, we get

$$ \begin{aligned} & \frac{2 i}{2-i} \times \frac{2+i}{2+i}=x+i y \\ & \frac{4 i+2 i^2}{2^2-i^2}=x+i y \\ & \frac{2(-1)+4 i}{4-(-1)}=x+i y \\ & \frac{-2+4 i}{5}=x+i y \end{aligned} $$

Let us equate real and imaginary parts on both sides we get,

$x=-\frac{2}{5}$ and $y=\frac{4}{5}$

So, $x+y=-\frac{2}{5}+\frac{4}{5}=\frac{2}{5}$.

PDF Summary - Class 11 Maths Complex Numbers and Quadratic Equations Notes (Chapter 5)

1. Definition

When a given number is in the form of $a + ib$, where $a,b \in R$ and $i = \sqrt { - 1} $ it is called a complex number and such number is denoted by ‘$z$’.

$z = a + ib$

Where,

$a$= real part of complex number and,

$b$= imaginary part of complex number.

1.1 Conjugate of a Complex Number

Consider a complex number $z = a + ib$,

Then its conjugate is written as '$\bar z$'.

Whose value is defined as $\bar z = a - ib$.

2. Algebra of Complex Numbers

Let ${z_1} = a + ib$ and ${z_2} = c + id$ be two complex numbers where ${\text{a}},{\text{b}},{\text{c}},{\text{d}} \in {\text{R}}$ and ${\text{i}} = \sqrt { - 1} $.

1. Addition :

$ z_{1}+z_{2} =(a+b i)+(c+d i) $

$ =(a+c)+(b+d) i $

2. Subtraction :

$ z_{1}-z_{2} =(a+b i)-(c+d i) $

$ =(a-c)+(b-d) $

3. Multiplication :

$ z_{1} \cdot z_{2} =(a+b i)(c+d i) $

$ =a(c+d i)+b i(c+d i) $

$ =a c+a d i+b c i+b d i^{2} $

$ =a c-b d+(a d+b c) i $ $\left(\because i^{2}=-1\right)$

Note…

1. $\begin{array}{r} a+i b=c+i d \\ \Leftrightarrow a=c \& b=d \end{array}$

2. $i^{4 k+r}=\left\{\begin{array}{cl}1 ; & r=0 \\ i ; & r=1 \\ -1 ; & r=2 \\ -i ; & r=3\end{array}\right.$

3. $\sqrt b \sqrt a = \sqrt {ba} $ is only possible if atleast one of either a or b is non-negative.

3. Argand Plane

Any complex number $z = a + ib$ can be represented by a unique point $P(a,\,\,b)$ in the argand plane.

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$P(a,\,\,b)$ represents the complex number $z = a + ib$.

3.1 Modulus and Argument of Complex Number

Consider a complex number $z = a + ib$.

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(i) Distance of $z$ from the origin is referred to as the modulus of complex number $z$.

It is represented by ${\text{r}} = |{\text{z}}| = \sqrt {{{\text{a}}^2} + {{\text{b}}^2}} $

(ii) Here, $\theta $ i.e.The angle made by ray OP with positive direction of the real axis is called the argument of ${\mathbf{z}}$.

Note.

${{\text{z}}_1} > {{\text{z}}_2}$ or ${{\text{z}}_1} < {{\text{z}}_2}$ has no meaning but $\left| {{{\text{z}}_1}} \right| > \left| {{{\text{z}}_2}} \right|$ or $\left| {{{\text{z}}_1}} \right| < \left| {{{\text{z}}_2}} \right|$ holds meaning.

3.2 Principal Argument

The argument $'\theta '$ of complex numbers $z = a + ib$ is called the principal argument of $z$if $ - \pi < \theta \leqslant \pi $.

Consider$\tan \alpha = \left| {\dfrac{b}{a}} \right|$, and $\theta $ be the $\arg \left( z \right)$.

i.

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ii.

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iii.

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iv.

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In (iii) and (iv) the principal argument is given by $ - \pi + \alpha $ and $ - \alpha $respectively.

4. Polar Form

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$\begin{array}{l} a=r \cos \theta \quad \& b=r \sin \theta \\ \text { where } r=|z| \text { and } \theta=\arg (z) \\ \therefore \quad z \quad=a+i b \\ \quad=r(\cos \theta+\operatorname{isin} \theta) \end{array}$

Note…

A complex number $z$ can also be represented as $z = r{e^{i\theta }}$, it is known as Euler’s form.

Where,

$\mathrm{r}=|\mathrm{Z}| \& \theta=\arg (\mathrm{Z})$

5. Some Important Properties

1. $\overline {(\bar z)} = z$

2. $z + \bar z = 2\operatorname{Re} (z)$

3. $z - \bar z = 2i\operatorname{Im} (z)$

4. $\overline {{{\text{z}}_1} + {{\text{z}}_2}} = {\overline {\text{z}} _1} + {\overline {\text{z}} _2}$

5. $\overline {{{\text{z}}_1}{{\text{z}}_2}} = {\overline {\text{z}} _1}.{\overline {\text{z}} _2}$

6. $|{\text{z}}| = 0 \Rightarrow {\text{z}} = 0$

7. $z.\,\bar z = |z{|^2}$

8. $\left| {{z_1}{z_2}} \right| = \left| {{z_1}} \right|\left| {{z_2}} \right|;\left| {\dfrac{{{z_1}}}{{{z_2}}}} \right| = \dfrac{{\left| {{z_1}} \right|}}{{\left| {{z_2}} \right|}}$

9. $|\overline {\text{z}} | = |{\text{z}}| = | - {\text{z}}|$

10. ${\left| {{z_1} \pm {z_2}} \right|^2} = {\left| {{z_1}} \right|^2} + {\left| {{z_2}} \right|^2} \pm 2\operatorname{Re} \left( {{z_1}{{\bar z}_2}} \right)$

11. $\left| {{z_1} + {z_2}} \right| \leqslant \left| {{z_1}} \right| + \left| {{z_2}} \right|$ (Triangle Inequality)

12. $\left| {\,{{\text{z}}_1} - {{\text{z}}_2}} \right|\,\, \geqslant \,\,\,\left| {\left| {\,{{\text{z}}_1}} \right| - \left| {\,{{\text{z}}_2}} \right|} \right|$

13. ${\left| {{\text{a}}{{\text{z}}_1} - {\text{b}}{{\text{z}}_2}} \right|^2} + {\left| {{\text{b}}{{\text{z}}_1} + {\text{a}}{{\text{z}}_2}} \right|^2} = \left( {{{\text{a}}^2} + {{\text{b}}^2}} \right)\left( {{{\left| {{{\text{z}}_1}} \right|}^2} + {{\left| {{{\text{z}}_2}} \right|}^2}} \right)$

14. $\operatorname{amp} \left( {{{\text{z}}_1} \cdot {{\text{z}}_2}} \right) = \operatorname{amp} {{\text{z}}_1} + \operatorname{amp} {{\text{z}}_2} + 2{\text{k}}\pi ;\,\,\,\,{\text{k}} \in {\text{I}}$

15. $\operatorname{amp} \left( {\dfrac{{{{\text{y}}_0}}}{{{{\text{y}}_1}}}} \right) = \operatorname{amp} {{\text{z}}_1} - \operatorname{amp} {{\text{z}}_2} + 2{\text{k}}\pi ;\,\,\,\,\,{\text{k}} \in {\text{I}}$

16. $\operatorname{amp} \left( {{z^n}} \right) = n\operatorname{amp} (z) + 2k\pi ;\,\,\,k \in I$

6. De-Moivre's Theorem

Statement: $\cos n\theta + i\sin n\theta $ is the value or one of the values of ${(\cos \theta + i\sin \theta )^n}$ according as if 'n' is integer or a rational number. The theorem is very useful in determining the roots of any complex quantity.

7. Cube Root of Unity

Roots of the equation ${{\text{x}}^3} = 1$ are called cube roots of unity.$

Roots of the equation $x^{3}=1$ are called cube roots of unity.

$\begin{array}{l} \quad x^{3}-1=0 \\ (x-1)\left(x^{2}+x+1\right)=0 \\ x=1 \quad \text { or } \quad x^{2}+x+1=0 \end{array}$

i.e $x=\underbrace{\dfrac{-1+\sqrt{3} \mathrm{i}}{2}}_{\mathrm{w}}$ or $\mathrm{x}=\underbrace{\dfrac{-1-\sqrt{3} \mathrm{i}}{2}}_{w^{2}}$

(i) The cube roots of unity are $1,\dfrac{{ - 1 + i\sqrt 3 }}{2},\dfrac{{ - 1 - i\sqrt 3 }}{2}$.

(ii) ${\omega ^3} = 1$

(iii) If ${\text{w}}$ is one of the imaginary cube roots of unity then $1 + \omega + {\omega ^2} = 0$.

(iv) In general $1 + {\omega ^{\text{r}}} + {\omega ^{2{\text{r}}}} = 0$; where ${\text{r}} \in {\text{I}}$ but is not the multiple of 3 .

(v) In polar form the cube roots of unity are:

$\cos 0 + i\sin 0;\cos \dfrac{{2\pi }}{3} + i\sin \dfrac{{2\pi }}{3},\cos \dfrac{{4\pi }}{3} + i\sin \dfrac{{4\pi }}{3}$

(vi) The three cube roots of unity when plotted on the argand plane constitute the vertices of an equilateral triangle.

(vii) The following factorisation should be remembered:

$\begin{array}{l} a^{3}-b^{3}=(a-b)(a-\omega b)\left(a-\omega^{2} b\right) \\ x^{2}+x+1=(x-\omega)\left(x-\omega^{2}\right) \\ a^{3}+b^{3}=(a+b)(a+\omega b)\left(a+\omega^{2} b\right) \\ a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)\left(a+\omega b+\omega^{2} c\right)\left(a+\omega^{2} b+\omega c\right) \end{array}$

8. ‘n’ nth Roots of Unity

Solution of equation ${x^n} = 1$ is given by,

$\begin{array}{*{20}{l}} {{\text{x}} = \cos \dfrac{{2{\text{k}}\pi }}{{\text{n}}} + {\text{i}}\sin \dfrac{{2{\text{k}}\pi }}{{\text{n}}}}&{;{\text{k}} = 0,1,2, \ldots ,{\text{n}} - 1} \\ { = {{\text{e}}^{{\text{i}}\left( {\dfrac{{2{\text{k}}\pi }}{{\text{n}}}} \right)}}}&{;{\text{k}} = 0,1, \ldots .,{\text{n}} - 1} \end{array}$

Note.

1. We may take any n consecutive integral values of ${\text{k}}$ to get ' ${\text{n}}$ ' ${{\text{n}}^{{\text{th }}}}$ roots of unity.

2. Sum of ' $'n'\,\,{n^{{\text{th }}}}$ roots of unity is zero, $n \in N$

3. The points represented by '${\text{n}}$', nth roots of unity are located at the vertices of a regular polygon of $n$ sides inscribed in a unit circle, centered at the origin and one vertex being one positive real axis.

Properties:

If $1,{\alpha _1},{\alpha _2},{\alpha _3} \ldots .{\alpha _{n - 1}}$ are the ${\text{n}},{{\text{n}}^{{\text{th }}}}$ root of unity then:

(i) They are in G.P. with common ratio ${{\text{e}}^{{\text{i}}(2\pi /{\text{n}})}}$

(ii) ${1^p} + \alpha _0^0 + \alpha _1^\circ + \ldots . + \alpha _{m - 0}^\circ = \left[ {\begin{array}{*{20}{l}} {0,{\text{ if }}p \ne {k^n}} \\ {n,{\text{ if }}p = kn} \end{array}} \right.$ where $k \in Z$

(iii) $\left( {1 - {\alpha _1}} \right)\left( {1 - {\alpha _2}} \right) \ldots \ldots \left( {1 - {\alpha _{n - 1}}} \right) = {\text{n}}$

(iv) $\left( {1 + {\alpha _1}} \right)\left( {1 + {\alpha _2}} \right) \ldots \ldots \left( {1 + {\alpha _{n - 1}}} \right) = \left[ {\begin{array}{*{20}{l}} {0,{\text{ if n is even }}} \\ {1,{\text{ if n is odd }}} \end{array}} \right.$

(v) $1.{\alpha _1} \cdot {\alpha _2} \cdot {\alpha _3} \ldots \ldots \ldots {\alpha _{n - 1}} = \left[ {\begin{array}{*{20}{c}} { - 1,{\text{ if }}n{\text{ is even }}} \\ {1,{\text{ if }}n{\text{ is odd }}} \end{array}} \right.$

Note...

(i) $\cos \theta + \cos 2\theta + \cos 3\theta + \ldots . + \cos n\theta = \dfrac{{\sin (n\theta /2)}}{{\sin (\theta /2)}}\cos \left( {\dfrac{{n + 1}}{2}} \right)\theta $

(ii) $\sin \theta + \sin 2\theta + \sin 3\theta + \ldots . + \sin n\theta = \dfrac{{\sin (n\theta /2)}}{{\sin (\theta /2)}}\sin \left( {\dfrac{{n + 1}}{2}} \right)\theta $.

9. Square Root of Complex Number

Let ${\text{x}} + {\text{iy}} = \sqrt {{\text{a}} + {\text{ib}}} $, Squaring both sides, we get

${(x + iy)^2} = a + ib$

i.e., ${x^2} - {y^2} = a,{\mkern 1mu} {\mkern 1mu} 2xy = b$

Solving these equations, we get square roots of $z$.

10. LOCI in Complex Plane

(i) $\left| {z - {z_0}} \right| = $ a represents the circumference of a circle, centred at ${{\text{z}}_{\text{o}}}$, radius a.

(ii) $\left| {{\text{z}} - {{\text{z}}_{\text{0}}}} \right| < $ a represents the interior of the circle.

(iii) $\left| {z - {z_0}} \right| > $ a represents the exterior of this circle.

(iv) $\left| {z - {z_1}} \right| = \left| {z - {z_2}} \right|$ represents $ \bot $ bisector of segment with endpoints ${z_1}{\text{ and }}{z_2}$.

(v) $\left| {\dfrac{{ - {z_1}}}{{ - {z_2}}}} \right| = k$ represents: $\left\{ {\begin{array}{*{20}{l}} {{\text{ circle, k}} \ne 1} \\ { \bot {\text{ bisector, k}} = 1} \end{array}} \right\}$

(vi) $\arg ({\text{z}}) = \theta $ is a ray starting from the origin (excluded) inclined at an $\angle \theta $ with a real axis.

(vii) Circle described on line segment joining ${z_1}{\text{ and }}{z_2}$ as diameter is:

$\left( {z - {z_1}} \right)\left( {\bar z - {{\bar z}_2}} \right) + \left( {z - {z_2}} \right)\left( {\bar z - {{\bar z}_1}} \right) = 0$

(viii) If $\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}$ are the vertices of an equilateral triangle where $\mathrm{z}_{0}$ is its circumcentre then

(a) $\dfrac{1}{\mathrm{z}_{2}-\mathrm{z}_{3}}+\dfrac{1}{\mathrm{z}_{3}-\mathrm{z}_{1}}+\dfrac{1}{\mathrm{z}_{1}-\mathrm{z}_{2}}=0$

(b) $z_{0}^{1}+z_{1}^{1}+z_{2}^{1}-z_{1} z_{2}-z_{2} z_{3}-z_{3} z_{1}=0$

(c) $z_{0}^{1}+z_{1}^{1}+z_{2}^{1}=3 z_{/}^{1}$

(ix) If $\mathrm{A}, \mathrm{B}, \mathrm{C} \& \mathrm{D}$ are four points representing the complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3} \& \mathrm{z}_{4}$ then

$\mathrm{AB}|| \mathrm{CD}$ if $\dfrac{\mathrm{Z}_{4}-\mathrm{Z}_{3}}{\mathrm{z}_{2}-\mathrm{z}_{1}}$ is purely real ;

$\mathrm{AB} \perp \mathrm{CD}$ if $\dfrac{\mathrm{z}_{4}-\mathrm{z}_{3}}{\mathrm{z}_{2}-\mathrm{z}_{1}}$ is purely imaginary $]$

11. Vectorial Representation of a Complex

Every complex number can be considered as if it is the position vector of that point. If the point P represents the complex number z then,

$\overrightarrow {OP} \, = \,\,z{\text{ and }}\left| {\overrightarrow {OP} } \right| = \left| {\,z\,} \right|.$

(“Image will be updated soon)

Note.

(i) If $\overrightarrow {{\text{OP}}} = {\text{z}} = {\text{r}}{{\text{e}}^{{\text{i}}\theta }}$ then $\overrightarrow {{\text{OQ}}} = {{\text{z}}_1} = {\text{r}}{{\text{e}}^{{\text{i}}(\theta + \,\phi )}} = {\text{z}}{\text{.}}{{\text{e}}^{{\text{i}}\phi }}$.

If $\overrightarrow {{\text{OP}}} $ and $\overrightarrow {{\text{OQ}}} $ are of unequal magnitude, then

$\widehat {{\text{OQ}}} = \widehat {{\text{OP}}}{{\text{e}}^{{\text{i}}\phi }}$

(ii) If ${z_1},\,\,{z_2}{\text{ and }}{{\text{z}}_3}$ are three vertices of a triangle ABC described in the counterclockwise sense, then

$\dfrac{{{z_3} - z}}{{{z_2} - z}} = \dfrac{{AC}}{{AB}}(\cos \alpha + i\sin \alpha ) = \dfrac{{AC}}{{AB}} \cdot {e^{i\alpha }} = \dfrac{{\left| {{z_3} - {z_1}} \right|}}{{\left| {{z_2} - {z_1}} \right|}} \cdot {e^{i\alpha }}$

12. Some Important Results

(i) If ${{\text{z}}_1}$ and ${{\text{z}}_2}$ are two complex numbers, then the distance between ${z_1}$ and ${z_2}$ is $\left| {{z_2} - {z_1}} \right|$.

(ii) Segment Joining points ${\text{A}}\left( {{{\text{z}}_1}} \right)$ and ${\text{B}}\left( {{{\text{z}}_2}} \right)$ is divided by point ${\text{P}}({\text{z}})$ in the ratio ${{\text{m}}_1}:{{\text{m}}_2}$ then ${\text{z}} = \dfrac{{{{\text{m}}_1}{{\text{z}}_2} + {{\text{m}}_2}{\text{z}}}}{{{{\text{m}}_1} + {{\text{m}}_2}}},\;{{\text{m}}_1}$ and ${{\text{m}}_2}$ are real.

(iii) The equation of the line joining ${z_1}$ and ${z_2}$ is given by

$\left| {\begin{array}{*{20}{c}} z&{\bar z} \\ z&{\bar z} \\ {{z_2}}&{{{\bar z}_2}} \end{array}} \right| = 0{\text{ (non parametric form) }}$

Or

$\dfrac{{z - z}}{{\bar z - \bar z}} = \dfrac{{z - {z_2}}}{{\bar z - {{\bar z}_2}}}$

(iv) $\bar az + a\bar z\, + b = 0$ represents a general form of line.

(v) The general eqn. of circle is:

$z\bar z + a\bar z + \bar az + b = 0$ (where ${\text{b}}$ is real no.).

Centre : $( - a)$ and radius,

$\sqrt {|a{|^2} - b} = \sqrt {a\bar a - b} $.

(vi) Circle described on line segment joining ${z_1}$ and ${z_2}$ as diameter is:

$\left( {z - {z_1}} \right)\left( {\bar z - {{\bar z}_2}} \right) + \left( {z - {z_2}} \right)\left( {\bar z - {{\bar z}_1}} \right) = 0$

(vii) Four pts. ${z_1},{z_2},{z_3},{z_4}$ in anticlockwise order will be concyclic, if and only if

$\theta = \arg \cdot \left( {\dfrac{{{z_2} - {z_4}}}{{{z_1} - {z_4}}}} \right) = \arg \left( {\dfrac{{{z_2} - {z_3}}}{{{z_1} - {z_3}}}} \right)$

$\Rightarrow \arg \left( {\dfrac{{{z_2} - {z_4}}}{{{z_1} - {z_4}}}} \right) - \arg \cdot \left( {\dfrac{{{z_2} - {z_3}}}{{{z_1} - {z_3}}}} \right) = 2n\pi ;(n \in I)$

$\Rightarrow \arg \left[ {\left( {\dfrac{{{z_2} - {z_4}}}{{{z_1} - {z_4}}}} \right)\left( {\dfrac{{{z_1} - {z_3}}}{{{z_2} - {z_3}}}} \right)} \right] = 2n\pi$

$ \Rightarrow \quad \left( {\dfrac{{{z_2} - {z_4}}}{{{z_1} - {z_4}}}} \right) \times \left( {\dfrac{{{z_1} - {z_3}}}{{{z_2} - {z_3}}}} \right)$ is real and positive.

(viii) If ${z_1},{z_2},{z_3}$ are the vertices of an equilateral triangle where ${z_0}$ is its circumcentre then

(a) $\dfrac{1}{{{z_2} - {z_3}}} + \dfrac{1}{{{z_3} - {z_1}}} + \dfrac{1}{{{z_1} - {z_2}}} = 0$

(b) $z_0^1 + z_1^1 + z_2^1 - {z_1}{z_2} - {z_2}{z_3} - {z_3}{z_1} = 0$

(c) ${\text{z}}_0^1 + {\text{z}}_1^1 + {\text{z}}_2^1 = 3{\text{z}}_{}^1$

(ix) If $A,B,C{\text{ and }}D$ are four points representing the complex numbers ${z_1},{z_2},{z_3}{\text{ and }}{z_4}$ then

${\text{AB}}||{\text{CD}}$ if $\dfrac{{{z_4} - {z_3}}}{{{z_2} - {z_1}}}$ is purely real;

$AB \bot CD{\text{ if }}\dfrac{{{z_4} - {z_3}}}{{{z_2} - {z_1}}}$ is purely imaginary.

(x) Two points $P\left( {{z_1}} \right)$ and $Q\left( {{z_2}} \right)$ lie on the same side or opposite side of the line $\bar az\, + \,a\bar z\, + b$ accordingly as $\bar a{z_1}\, + \,a{\bar z_1}\, + b{\text{ and }}\bar a{z_2}\, + \,a{\bar z_2}\, + b$ have same sign or opposite sign.

Important Identities

(i) ${{\text{x}}^2} + {\text{x}} + 1 = ({\text{x}} - \omega )\left( {{\text{x}} - {\omega ^2}} \right)$

(ii) ${{\text{x}}^2} - {\text{x}} + 1 = ({\text{x}} + \omega )\left( {{\text{x}} + {\omega ^2}} \right)$

(iii) ${x^2} + xy + {y^2} = (x - y\omega )\left( {x - y{\omega ^2}} \right)$

(iv) ${{\text{x}}^2} - {\text{xy}} + {{\text{y}}^2} = ({\text{x}} + \omega {\text{y}})\left( {{\text{x}} + {\text{y}}{\omega ^2}} \right)$

(v) ${x^2} + {y^2} = (x + iy)(x - iy)$

(vi) ${x^3} + {y^3} = (x + y)(x + y\omega )\left( {x + y{\omega ^2}} \right)$

(vii) ${{\text{x}}^3} - {{\text{y}}^3} = ({\text{x}} - {\text{y}})({\text{x}} - {\text{y}}\omega )\left( {{\text{x}} - {\text{y}}{\omega ^2}} \right)$

(viii) ${x^2} + {y^2} + {z^2} - xy - yz - zx = \left( {x + y\omega + z{\omega ^2}} \right)\left( {x + y{\omega ^2} + z\omega } \right)$

or $\quad \left( {{\text{x}}\omega + {\text{y}}{\omega ^2} + {\text{z}}} \right)\left( {{\text{x}}{\omega ^2} + {\text{y}}\omega + {\text{z}}} \right)$

or $\quad \left( {{\text{x}}\omega + {\text{y}} + {\text{z}}{\omega ^2}} \right)\left( {{\text{x}}{\omega ^2} + {\text{y}} + {\text{z}}\omega } \right)$

(ix) ${x^3} + {y^3} + {z^3} - 3xyz = (x + y + z)\left( {x + \omega y + {\omega ^2}z} \right)$$\left( {{\text{x}} + {\omega ^2}{\text{y}} + \omega {\text{z}}} \right)$

1. Quadratic Expression

The standard form of a quadratic expression in ${\text{x}}$ is, $f({\text{x}}) = a{x^2} + bx + c$, where $a,b,c \in R{\text{ and }}a \ne 0$. General form of a quadratic equation in ${\text{x}}$ is, $a{x^2} + bx + c = 0$, where $a,b,c \in R{\text{ and }}a \ne 0$.

2. Roots of Quadratic Equation

(a) The solution of the quadratic equation,

$a{x^2} + bx + c = 0$ is given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

The expression ${\text{D}} = {{\text{b}}^2} - 4{\text{ac}}$ is called the discriminant of the quadratic equation.

(b) If $\alpha {\text{ and }}\beta $ are the roots of the quadratic equation $a{x^2} + bx + c = 0$, then

(i) $\alpha + \beta = \dfrac{{ - {\text{b}}}}{a}$

(ii) $\alpha \beta = \dfrac{{\text{c}}}{a}$

(iii) $|\alpha - \beta | = \dfrac{{\sqrt {\text{D}} }}{{|{\text{a}}|}}$

(c) A quadratic equation whose roots are $\alpha {\text{ and }}\beta $ is $(x - \alpha )(x - \beta ) = 0$ i.e.,

${{\text{x}}^2} - (\alpha + \beta ){\text{x}} + \alpha \beta = 0\quad $ i.e.,

${{\text{x}}^2} - ($ sum of roots $){\text{x}} + $ product of roots $ = 0$

Note.

$ y=\left(a x^{2}+b x+c\right) \equiv a(x-\alpha)(x-\beta) $

$ =a\left(x+\dfrac{b}{2 a}\right)^{2}-\dfrac{D}{4 a} $

3. Nature of Roots

(a) Consider the quadratic equation $a{x^2} + bx + c = 0$ where a, $a,b,c \in R{\text{ and }}a \ne 0$ then;

(i) ${\text{D}} > 0 \Leftrightarrow $ roots are real and distinct (unequal).

(ii) ${\text{D}} = 0 \Leftrightarrow $ roots are real and coincident (equal).

(iii) ${\text{D}} < 0 \Leftrightarrow $ roots are imaginary.

(iv) If ${\text{p}} + {\text{iq}}$ is one root of a quadratic equation, then the other must be the conjugate $p - iq$ and vice versa. $({\text{p}},{\text{q}} \in {\text{R and i}} = \sqrt { - 1} )$.

(b) Consider the quadratic equation $a{x^2} + bx + c = 0$ where $a,b,c \in Q{\text{ and }}a \ne 0$ then;

(i) If ${\text{D}} > 0$ and is a perfect square, then roots are rational and unequal.

(ii) If $\alpha = p + \sqrt q $ is one root in this case, (where $p$ is rational and $\sqrt q $ is a surd) then the other root must be the conjugate of it i.e., $\beta = p - \sqrt q $ and vice versa.

Note.

Remember that a quadratic equation cannot have three different roots and if it has, it becomes an identity.

4. Graph of Quadratic Equation

Consider the quadratic expression, ${\text{y}} = {\text{a}}{{\text{x}}^2} + {\text{bx}} + {\text{c}}$, $a \ne 0{\text{ and }}a,b,c \in R$ then;

(i) The graph between ${\text{x}},{\text{y}}$ is always a parabola. If ${\text{a}} > 0$ then the shape of the parabola is concave upwards and if ${\text{a}} < 0$ then the shape of the parabola is concave downwards.

(ii) ${\text{y}} > 0\forall {\text{x}} \in {\text{R}}$, only if ${\text{a}} > 0{\text{ and D}} < 0$

(iii) ${\text{y}} < 0\forall {\text{x}} \in {\text{R}}$, only if ${\text{a}} < 0{\text{ and D}} < 0$

5. Solution of Quadratic Inequalities

$a x^{2}+b x+c>0(a \neq 0)$

(i) If $D>0$, then the equation $a x^{2}+b x+c=0$ has two different roots $\left(\mathrm{x}_{1}<\mathrm{x}_{2}\right)$

Then $\mathrm{a}>0 \quad \Rightarrow \quad \mathrm{x} \in\left(-\infty, \mathrm{x}_{1}\right) \cup\left(\mathrm{x}_{2}, \infty\right)$

$\mathrm{a}<0 \quad \Rightarrow \quad \mathrm{x} \in\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)$

(“Image will be updated soon)

(ii) Inequalities of the form $\dfrac{\mathrm{P}(\mathrm{x})}{\mathrm{Q}(\mathrm{x})} \gtrless 0$ can be quickly solved using the method of intervals (wavy curve).

6. Maximum and Minimum Value of Quadratic Equation

Maximum and minimum value of $y = a{x^2} + bx + c$ occurs at $x = - \left( {\dfrac{b}{{2a}}} \right)$ according as:

For $a > 0,$ we have:

$y \in \left[ {\dfrac{{4ac - {b^2}}}{{4a}},\,\infty } \right)$

(Image will be updated soon)

${y_{\min }} = \dfrac{{ - D}}{{4a}}{\text{ at }}x = \dfrac{{ - b}}{{2a}},{\text{ and }}{{\text{y}}_{\max }} \to \infty $

For $a < 0,$ we have:

$y \in \left( { - \infty ,\,\,\dfrac{{4ac - {b^2}}}{{4a}}} \right]$

(Image will be updated soon)

${y_{\max }} = \dfrac{{ - D}}{{4a}}{\text{ at }}x = \dfrac{{ - b}}{{2a}},{\text{ and }}{{\text{y}}_{\min }} \to \infty $

7. Theory of Equations

If ${\alpha _1},{\alpha _2},{\alpha _3}, \ldots \ldots ,{\alpha _n}$ are the roots of the ${{\text{n}}^{{\text{th }}}}$ degree polynomial equation:

$f({\text{x}}) = {{\text{a}}_0}{{\text{x}}^{\text{n}}} + {{\text{a}}_1}{{\text{x}}^{{\text{n}} - 1}} + {{\text{a}}_2}{{\text{x}}^{{\text{n}} - 2}} + \ldots \ldots + {{\text{a}}_{{\text{n}} - 1}}{\text{x}} + {{\text{a}}_{\text{n}}} = 0$

where ${a_0},{a_1}, \ldots \ldots .{a_n}$ are all real and ${a_0} \ne 0$

Then,

$\sum {{\alpha _1}} = - \dfrac{{{{\text{a}}_1}}}{{{{\text{a}}_0}}}$

$\sum {{\alpha _1}} {\alpha _2} = \dfrac{{{{\text{a}}_2}}}{{{{\text{a}}_0}}}$

$\sum {{\alpha _1}} {\alpha _2}{\alpha _3} = - \dfrac{{{a_3}}}{{{a_0}}}$;

${\alpha _1}{\alpha _2}{\alpha _3} \ldots .{\alpha _{\text{n}}} = {( - 1)^{\text{n}}}\dfrac{{{{\text{a}}_{\text{n}}}}}{{{{\text{a}}_0}}}$

8. Location of Roots

Let $f(x) = a{x^2} + bx + c$, where $a > 0{\text{ and }}a,b,c \in R$

(i) Conditions for both the roots of $f(x) = 0$ to be greater than a specified number ‘k’ are:

${\text{D}} \geqslant 0{\text{ and }}f({\text{k}}) > 0{\text{ and }}( - {\text{b}}/2{\text{a}}) > {\text{k}}$

(ii) Conditions for both roots of $f({\text{x}}) = 0$ to lie on either side of the number ‘k’ (in other words the number ‘k’ lies between the roots of $f({\text{x}}) = 0$ is:

$a\,f({\text{k}}) < 0$

(iii) Conditions for exactly one root of $f\left( x \right) = 0$ to lie in the interval $\left( {{k_1},\,\,{k_2}} \right){\text{ i}}{\text{.e}}{\text{., }}{k_1} < x < {k_2}$ are:

$D > 0{\text{ and }}f\left( {{k_1}} \right).f\left( {{k_2}} \right) < 0$

(iv) Conditions that both the roots of $f\left( x \right) = 0$ to be confined between the numbers ${k_1}{\text{ and }}{{\text{k}}_2}$ are $\left( {{k_1} < {k_2}} \right)$:

$D \geqslant 0{\text{ and }}f\left( {{k_1}} \right) > 0{\text{ and }}f\left( {{k_2}} \right) > 0{\text{ and }}{k_1} < \left( {\dfrac{{ - b}}{{2a}}} \right) < {k_2}$

9. Maximum and Minimum Values of Rational Numbers

Here we shall find the values attained by a rational expresion of the form $\dfrac{{{a_1}{x^2} + {b_1}x + {c_1}}}{{{a_2}{x^2} + {b_2}x + {c_2}}}$ for real values of $x$.

10. Common Factors

(a) Only One Common Root

Let $\alpha $ be the common root of $a{x^2} + bx + c = 0$ and ${a^\prime }{x^2} + {b^\prime }x + {c^\prime } = 0$, such that $a,{a^\prime } \ne 0$ and a ${b^\prime } \ne {a^\prime }b$. Then, the condition for one common root is:

${\left( {c{a^\prime } - {c^\prime }a} \right)^2} = \left( {a{b^\prime } - {a^\prime }b} \right)\left( {b{c^\prime } - {b^\prime }c} \right)$

(b) Two common roots

Let $\alpha ,\,\beta $ be the two common roots of

$a{x^2} + bx + c = 0{\text{ and }}{a^\prime }{x^2} + {b^\prime }x + {c^\prime } = 0$ such that ${\text{a}},{{\text{a}}^\prime } \ne 0$.

Then, the condition for two common roots is: $\dfrac{a}{{{a^\prime }}} = \dfrac{b}{{{b^\prime }}} = \dfrac{c}{{{c^\prime }}}$

11. Resolution Into Two Linear Factors

The condition that a quadratic function $f(x,y) = a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c$ may be resolved into two linear factors is that:

$abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0$ or,

$\left| {\begin{array}{*{20}{l}} a&h&g \\ h&b&f \\ g&f&c \end{array}} \right| = 0$

12. Formation of a Polynomial Equations

If ${\alpha _1},{\alpha _2},{\alpha _3}, \ldots \ldots ,{\alpha _n}$ are the roots of the ${n^{{\text{th }}}}$ degree polynomial equation, then the equation is

${{\text{x}}^{\text{n}}} - {{\text{S}}_1}{{\text{x}}^{{\text{n}} - 1}} + {{\text{S}}_2}{{\text{x}}^{{\text{n}} - 2}} + {{\text{S}}_3}{{\text{x}}^{{\text{n}} - 3}} + \ldots \ldots + {( - 1)^{\text{n}}}{{\text{S}}_{\text{n}}} = 0$

where ${{\text{S}}_{\text{k}}}$ denotes the sum of the products of roots taken ${\text{k}}$ at a time.

Particular Cases

(a) Quadratic Equation: If $\alpha, \beta$ be the roots the quadratic equation, then the equation is :

$\mathrm{x}^{2}-\mathrm{S}_{1} \mathrm{x}+\mathrm{S}_{2}=0 \quad \text { i.e. } \quad \mathrm{x}^{2}-(\alpha+\beta) \mathrm{x}+\alpha \beta=0$

(b) Cubic Equation: If $\alpha, \beta, \gamma$ be the roots the cubic equation, then the equation is :

$\mathrm{x}^{3}-\mathrm{S}_{1} \mathrm{x}^{2}+\mathrm{S}_{2} \mathrm{x}-\mathrm{S}_{3}=0 \quad \text { i.e }$

$x^{3}-(\alpha+\beta+\gamma) x^{2}+(\alpha \beta+\beta \gamma+\gamma \alpha) x-\alpha \beta \gamma=0$

(i) If $\alpha$ is a root of equation $\mathrm{f}(\mathrm{x})=0$, the polynomial $\mathrm{f}(\mathrm{x})$ is exactly divisible by $(\mathrm{x}-\alpha)$. In other words, $(x-\alpha)$ is a factor of $f(x)$ and conversely.

(ii) Every equation of nth degree $(n \geq 1)$ has exactly n roots $\&$ if the equation has more than $\mathrm{n}$ roots, it is an identity.

13. Transformation of Equations

(i) To obtain an equation whose roots are reciprocals of the roots of a given equation, it is obtained by replacing ${\text{x}}$ by $1/{\text{x}}$ in the given equation.

(ii) Transformation of an equation to another equation whose roots are negative of the roots of a given equation-replace ${\text{x}}$ by $ - {\text{x}}$.

(iii) Transformation of an equation to another equation whose roots are square of the roots of a given equation-replace ${\text{x}}$ by $\sqrt {\text{x}} $.

(iv) Transformation of an equation to another equation whose roots are cubes of the roots of a given equation-replace $x$ by ${x^{1/3}}$.

Class 11 Maths Notes of Complex Numbers

What is a Complex Number?

A complex number can be noted as a + ib, here “a” is a real number and “b” is an imaginary number. The complex number comprised of the symbol “i” which assures the condition i2 = −1. Complex numbers are mentioned as the addition of one-dimensional number lines. In a complex plane, a complex number can be denoted by a + bi and is usually represented in the form of the point (a, b).

We have to see that a complex number with no real part, such as – i, -5i, etc, is called as entirely imaginary. Also, a complex number with absolutely no imaginary part is also known as a real number.

A number of form x + iy, where x and y are real numbers, is called a complex number, x is called the real part, and y is called the imaginary part of the complex number i.e. Re(Z) = x and Im(Z) = y.

Properties of Complex Numbers

The properties of a complex number are listed below:

1. a and b are two real numbers and if, a + ib = 0 then a = 0, b = 0

2. When the real numbers are a, b and c; and a + ib = c + id ,then a = c and b = d.

3. A set of three complex numbers namely z1, z2, and z3 satisfy the commutative, associative, as well as distributive laws.

4. If both the sum as well as the product of any two given complex numbers are real then the complex numbers are conjugate to each other.

5. For two complex numbers z1 and z2 : |z1 + z2| ≤ |z1| + |z2|

6. The sum of any two conjugate complex numbers is always real.

7. The product of any two conjugate complex numbers is always real.

Importance of Ch 5 Class 11 Maths Revision Notes for Complex Numbers and Quadratic Equations

All the class 11 maths chapter 5 revision notes have been made by Vedantu in accordance with the latest CBSE syllabus so that there is no mistake if changes have been created by the CBSE board. The ch 5 class 11 maths revision notes for Complex Numbers and Quadratic Equations have been made by teachers who have vast experience and know exactly what is needed.

Further, students will be able to see chapter-wise Mathematics Revision Notes with short keynotes that could raise their preparations. The CBSE revision notes class 11 maths chapter 5 are prepared in a concise manner from the latest edition of the books.

The CBSE class 11 Maths Chapter 5 revision notes for Complex Numbers and Quadratic Equations are available in a PDF format so that students can simply refer to it whenever required thorough Vedantu. The teachers who prepare these class 11 maths chapter 5 revision notes have done so after rigorously going through the last ten year's question papers and then taking them down.

Because of the large amount of content present in most of the books, it becomes very difficult for the students to keep up with all of it. The class 11 revision notes maths ch 5 can play a role in helping easier studying methods. Vedantu is one of India’s leading education platforms for students all over the country.

Algebra of Complex Numbers - In this section of Notes of Class 11 Revision Notes Chapter 5, you would learn about different operations that can be performed on complex numbers. Let us say there are two complex numbers:

C1 = x + iy and C2 = a + ib

• Addition of Complex Numbers - Two complex numbers are added by adding each part (real and imaginary) separately. Therefore C1 + C2 = (x + iy) + (a + ib) = (x + a) + (y + b)i.

• Subtraction of Complex Numbers - Two complex numbers are subtracted by subtracting each part (real and imaginary) separately. Therefore C1 - C2 = (x + iy) - (a + ib) = (x - a) + (y - b)i.

• Multiplication of Complex Numbers - Two complex numbers are multiplied by multiplying each part (real and imaginary) of the first complex number with each part of the second complex number. Hence:

C1 * C2 = (x + iy) * (a + ib) = (x * a) + (a * ib) + (y * a * i) + (y * b * i2.)

• Division of Complex Numbers - We use the concept of conjugates to divide one complex number by another. Both the numerator and denominator are multiplied by the conjugate of the denominator in the division process. Hence:

C1/C2 = (x + iy) / (a + ib) = (x + iy) * (a - ib)/(a + ib) * (a - ib) = ((xa + yb)/ (a2 + b2)) + ((ya - xb)/(a2 + b2)) i

Modulus and Argument of a Complex Number - A complex number C = x + iy can be represented on an argand plane by a point P(x, y). In that plane:

• The distance of point P from the origin is called the modulus of the complex number C and is denoted by r = |C| = √(x2 + y2).

• The angle which a straight line joining the origin and point P makes with the positive direction of the X-axis is known as the argument of the complex number C.

• If the argument (θ) of C lies between -π and +π i.e. -π < θ < +π, then θ is called the principal argument of C.

Why Should Students go Through Vedantu Revision Notes?

The studying process is usually student-specific, that is some students prefer kinesthetic studying, some are auditory students, and others may find visual studying to be more efficient. But these different processes are just a part of the studying experience of every student, the other important aspect is the revision. Revision is very essential. Students have to strengthen their studying, hence revisions are a way to begin about this phase as it will help them grasp the concept easily.

• Enables the student to strengthen their studying.

• Students become more confident during examinations.

• Saves valuable time during examinations.

• The correctness of answers are higher.