Complex Numbers and Quadratic Equations - Topics, Books, Preparation Tips FAQs

Complex Numbers: Definition

A complex number is a number that can be expressed in the form $x + iy$, where $x$ and $y$ are real numbers and $i$ is the imaginary unit, also called iota. Complex numbers are used to represent quantities that cannot be expressed on the real number line alone.

Example:
$z = 5 + 2i$

• Real part: $\operatorname{Re}(z) = 5$

• Imaginary part: $\operatorname{Im}(z) = 2$

The imaginary unit $i$ (iota) is defined as:
$i = \sqrt{-1}$

Powers of $i$

The first few powers of $i$ are important for calculations:
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$

Special Cases

1. Purely real number: If $\operatorname{Im}(z) = 0$, the complex number becomes a real number.
   Example: $2 + 0i = 2$

2. Purely imaginary number: If $\operatorname{Re}(z) = 0$, the complex number becomes purely imaginary.
   Example: $0 + 2i = 2i$

Quadratic Equations: Definition

The highest power of the variable in a polynomial expression is called the degree of the polynomial.

A polynomial equation in which the highest degree of a variable term is $2$ is called a quadratic equation.

The standard form of a quadratic equation is: $a x^2 + b x + c = 0, \quad a \neq 0$
Here, $a$, $b$, and $c$ are constants (they may be real or complex) and are called the coefficients of the equation. $a$ is also called the leading coefficient.

Examples: $-5x^2 - 3x + 2 = 0$
$x^2 = 0$
$(1+i)x^2 - 3x + 2i = 0$

Since the degree of a quadratic polynomial is $2$, it always has 2 roots (number of real roots + number of imaginary roots = $2$).

Roots of Quadratic Equations

If $f(x)$ is a polynomial, then the equation $f(x) = 0$ is called a polynomial equation.

The roots of the polynomial equation are the values of $x$ for which $f(x) = 0$.

If $x = \alpha$ is a root of $f(x) = 0$, then:
$f(\alpha) = 0$

Example: For $x^2 - 3x + 2 = 0$, $x = 2$ is a root because: $f(2) = 2^2 - 3(2) + 2 = 4 - 6 + 2 = 0$

A polynomial of degree $n$ has $n$ roots (real or complex). Therefore, a quadratic equation always has exactly $2$ roots.

Complex Numbers Formulae

This section covers the essential formulae related to complex numbers, including conjugates, modulus, square roots, and algebraic operations. Understanding these formulae helps simplify calculations and solve problems involving complex numbers efficiently.

Conjugate of Complex Numbers

The conjugate of a complex number $z$ is represented by $\bar{z}$. The pair $(z, \bar{z})$ is called a complex-conjugate pair.

If $z = a + ib$ (where $a, b$ are real numbers), the conjugate is: $ \bar{z} = a - ib $

Conjugates are obtained by changing the sign of the imaginary part, while the real part remains unchanged.

Modulus of a Complex Number

For $z = x + iy$, the modulus of complex zumber $z$, denoted by $|z|$, is the distance from the origin in the Argand plane: $|z| = \sqrt{x^2 + y^2}$

It is always a non-negative real number.

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: $|z| = \sqrt{x^2 + y^2} = r$ where $r$ is the distance from the origin to the point $(x, y)$.

Square Root of a Complex Number

If $z = a + ib$, the square root of complex number $z$ is another complex number whose square equals $z$. If $\sqrt{z} = x + iy$, then: $(x + iy)^2 = a + ib$

The formula to find the square root is: $\sqrt{z} = \pm \left( \sqrt{\frac{|z| + \operatorname{Re}(z)}{2}} + i \sqrt{\frac{|z| - \operatorname{Re}(z)}{2}} \right)$
If $\operatorname{Im}(z) < 0$, the sign of the imaginary part is negative.

Algebraic Operations on Complex Numbers

This section explains the fundamental algebraic operations on complex numbers, including addition, subtraction, multiplication, and division.

Addition

Let $z_1 = a + ib$ and $z_2 = c + id$, then: $z_1 + z_2 = (a + c) + i(b + d)$

Example: $(2 + 3i) + (2 + 2i) = (2 + 2) + i(3 + 2) = 4 + 5i$

Subtraction

Let $z_1 = a + ib$ and $z_2 = c + id$, then: $z_1 - z_2 = (a - c) + i(b - d)$

Example: $(6 + 10i) - (10 + 3i) = (6 - 10) + i(10 - 3) = -4 + 7i$

Multiplication

Let $z_1 = a + ib$ and $z_2 = c + id$, then: $(a + ib)(c + id) = (ac - bd) + i(ad + bc)$

Example: $2i(2 + 6i) = (0 + 2i)(2 + 6i) = 0 + 12i^2 + 4i = -12 + 2i$

Division

Let $z_1 = c_1 + id_1$ and $z_2 = c_2 + id_2$, then the quotient is:
$\frac{z_1}{z_2} = z_1 \cdot \frac{1}{z_2}$
To divide, multiply $z_1$ by the reciprocal of $z_2$:
$\frac{z_1}{z_2} = z_1 \cdot \frac{\bar{z_2}}{|z_2|^2}$

Quadratic Equations

Solving quadratic equations means finding the value(s) of the variable that satisfy the equation. These values are called roots, solutions, or zeros. Since the degree of a quadratic equation is $2$, it can have a maximum of $2$ roots.

Example: For the equation $x^2 - 3x + 2 = 0$, we can see that $x = 1$ and $x = 2$ satisfy the equation, so these are its roots. But how do we find roots if they are not obvious?

There are several methods to solve quadratic equations:

• Solving by factoring

• Solving by completing the square

• Solving by graphing

• Solving using the quadratic formula

Nature of Roots

The nature of the roots of a quadratic equation depends on the discriminant $D = b^2 - 4ac$.

1. Real and Distinct Roots

   • $D > 0$

   • The curve intersects the $x$-axis at two distinct points.

2. Real and Equal Roots

   • $D = 0$

   • The curve intersects the $x$-axis at only one point.

3. Complex Roots

   • $D < 0$

   • The curve does not intersect the $x$-axis.

How to Find the Nature of Roots

Step 1: Compare the given quadratic equation with the standard form $ax^2 + bx + c = 0$ to find the coefficients $a$, $b$, and $c$.

Step 2: Substitute the coefficients into the discriminant: $D = b^2 - 4ac$ and solve.

Step 3: Determine the nature of the roots based on $D$:

• If $D < 0$, the roots are complex

• If $D = 0$, the roots are real and equal

• If $D > 0$, the roots are real and distinct

Sum and Product of Roots of a Quadratic Equation

Assume $\alpha$ and $\beta$ are the roots of a quadratic equation $ax^2 + bx + c = 0$.

Sum of Roots:

$\alpha + \beta = \frac{-b + \sqrt{D}}{2a} + \frac{-b - \sqrt{D}}{2a} $

$= \frac{-b}{2a} + \frac{\sqrt{D}}{2a} + \frac{-b}{2a} - \frac{\sqrt{D}}{2a} $

$= \frac{-2b}{2a} $

$= \frac{-b}{a} $

Product of roots:

$\alpha \cdot \beta = \frac{(-b + \sqrt{D})}{2a} \cdot \frac{(-b - \sqrt{D})}{2a} $

$= \frac{(-b)^2 - (\sqrt{D})^2}{(2a)^2} $

$= \frac{b^2 - D}{4a^2} $

$= \frac{b^2 - b^2 + 4ac}{4a^2} \quad [D = b^2 - 4ac] $

$= \frac{c}{a} $

A quadratic equation can be formed using the sum and product of roots:

$x^2 - (\alpha + \beta)x + (\alpha \cdot \beta) = 0 $

Location of Roots

Now let us look into the location of roots of quadratic equations.

Let $f(x) = ax^2 + bx + c$ where $a, b, c$ are real numbers and $a \neq 0$. Let $x_1$ and $x_2$ be the roots of $f(x)$ and let $k$ be any real number.

1. If both roots of $f(x)$ are less than $k$:

i) $D \ge 0$ (the roots may be distinct or equal)

ii) $a f(k) > 0$ (in both cases, $a f(k)$ is positive; if $a < 0$, then $f(k) < 0$, and multiplying two negative values gives a positive value)

iii) $k > -\frac{b}{2a}$ (since $-\frac{b}{2a}$ lies between $x_1$ and $x_2$, and $x_1, x_2 < k$, so $-\frac{b}{2a} < k$)

2. If both roots of $f(x)$ are greater than $k$:

i) $D \ge 0$

ii) $a f(k) > 0$

iii) $k < -\frac{b}{2a}$ (since $-\frac{b}{2a}$ lies between $x_1$ and $x_2$, and $x_1, x_2 > k$, so $-\frac{b}{2a} > k$)

Condition for a number $k$ with respect to the roots:

• Let $f(x) = ax^2 + bx + c$ with real coefficients and $a \neq 0$.

• Let $x_1$ and $x_2$ be the real roots of the quadratic equation.

Case (i): $k$ lies between the roots $x_1$ and $x_2$

i) $D \ge 0$

ii) $k_1 < -\frac{b}{2a} < k_2$, where $k_1 < k_2$

Case (ii): $k_1$ and $k_2$ lie between the roots

i) $a f(k_1) < 0$ and $a f(k_2) < 0$

Solved examples based on Complex Numbers and Quadratic Equations

Example 1: If z is a non-real complex number, then the minimum value of $\frac{\operatorname{Im} z^5}{(\operatorname{Im} z)^3}$.

Solution:

As we have learned

Polar Form of a Complex Number -

$z=r(\cos \theta+i \sin \theta)$

- wherein

$\mathrm{z}=$ modulus of z and $\theta$ is the argument of Z

Euler's Form of a Complex Number -

$z=r e^{i \theta}$

- wherein

r denotes the modulus of z and $\theta$ denotes the argument of z.

$z=x+i y=r(\cos \theta+i \sin \theta)$

$=r e^{i \theta}$

So, $\operatorname{Im} z^5=\operatorname{Im}\left(r e^{i \theta}\right)^5$

$=\operatorname{Im}\left(r^5 e^{i \theta 5}\right)$

$=r^5 \sin 50$

$(\operatorname{Im} z)^5=(r \sin \theta)^5$

$=\left(r^5 \sin ^5 \theta\right)$

So, $\frac{\operatorname{Im} z^5}{(\operatorname{Im} z)^5}=\frac{\sin 5 \theta}{\sin ^5 \theta}$

for minimum value, differentiating w.r.t $\theta$

So, $\frac{\sin ^5 \theta \cdot 5 \cos \theta-5 \sin 5 \theta \sin ^4 \theta \cos \theta}{\sin ^{10} \theta}$

$\Rightarrow \sin \theta \cdot \cos 5 \theta-\sin 5 \theta \cos \theta=0$

$
\begin{aligned}
& \Rightarrow \sin 4 \theta \cdot=0 \\
& 4 \theta=n \pi \\
& \theta=n \pi / 4
\end{aligned}
$

for $\mathrm{n}=1$

$\frac{\sin 5 \theta}{\sin ^5 \theta}=\frac{-1 / \sqrt{2}}{(1 / \sqrt{2})^5}=-4$

Hence, the answer is -4.

Example 2: Let $z_0$ be a root of the quadratic equation, $x^2+x+1=0$. If $z=3+6 i z_0^{81}-3 i z_0^{93}$ then arg z is equal to :

Solution:

Definition of Argument/Amplitude of z in Complex Numbers -

$\theta=\tan ^{-1}\left|\frac{y}{x}\right|, z \neq 0$

$\theta, \pi-\theta,-\pi+\theta,-\theta$ are Principal Arguments if z lies in the first, second, third, or fourth quadrant respectively.

Cube roots of unity -

$z=(1)^{\frac{1}{3}} \Rightarrow z=\cos \frac{2 k \pi}{3}+i \sin \frac{2 k \pi}{3}$

k=0,1,2 so z gives three roots

$\Rightarrow 1, \frac{-1}{2}+i \frac{\sqrt{3}}{2}(\omega), \frac{-1}{2}-i \frac{\sqrt{3}}{2}\left(\omega^2\right)$

- wherein

$\omega=\frac{-1}{2}+\frac{i \sqrt{3}}{2}, \omega^2=\frac{-1}{2}-\frac{i \sqrt{3}}{2}, \omega^3=1,1+\omega+\omega^2=0$

$1, \omega, \omega^2$ are cube roots of unity.

Quadratic Equation

$x^2+x+1=0$, roots are, $\omega$ and $\omega^2$ where $\omega$ is the cube root of unity.

$
z=3+6 i\left(z_0\right)^{81}-3 i\left(z_0\right)^{90}
$

$z_0=\omega$ and $\omega^2$

$
\begin{aligned}
& z=3+6 i(\omega)^{81}-3 i(\omega)^{93} \\
& z=3+3 i \quad \because \omega^3=1 \\
& \arg (z)=\frac{1}{4}
\end{aligned}
$
Hence, the required answer is $\frac{\pi}{4}$.

Example 3: z is a complex number such that $z+\bar{z}=5$ and $z-\bar{z}=7 i$ then z equals

Solution:

We know the properties of the conjugate of a complex number:

$\operatorname{Im}(z) = \frac{z - \bar{z}}{2i}$, where

$\operatorname{Im}(z)$ denotes the imaginary part of $z$

$\bar{z}$ denotes the conjugate of $z$

$z + \bar{z} = 5$

$z - \bar{z} = 7i$

Adding both: $2z = 5 + 7i \Rightarrow z = \frac{5}{2} + \frac{7}{2}i$

Or, using formulas: $\operatorname{Re}(z) = \frac{z + \bar{z}}{2} = \frac{5}{2}$ and $\operatorname{Im}(z) = \frac{z - \bar{z}}{2i} = \frac{7}{2}$

Hence, the answer is $z = \frac{5}{2} + \frac{7}{2}i$

Example 4: z is a complex number such that $z-\bar{z}=4 i$ , then $\operatorname{Im}(z)$ equals

Solution:

As we learned in properties of the conjugate of a complex number:

$\operatorname{Im}(z) = \frac{z - \bar{z}}{2i}$ where $\operatorname{Im}(z)$ denotes the imaginary part of $z$

$\bar{z}$ denotes the conjugate of $z$

$\operatorname{Im}(z) = \frac{z - \bar{z}}{2i} = \frac{4i}{2i} = 2$

Hence, the answer is $2$.

Example 5: The multiplicative inverse of the complex number $z=\frac{5+i \sqrt{2}}{1-i \sqrt{2}}$ is:

Solution:

As we have learnt, the multiplicative inverse of $z$ is $\frac{1}{z}$

Now, $z = \frac{5+i \sqrt{2}}{1-i \sqrt{2}}$

$\frac{1}{z} = \frac{1-i \sqrt{2}}{5+i \sqrt{2}}$

$= \frac{1-i \sqrt{2}}{5+i \sqrt{2}} \cdot \frac{5-i \sqrt{2}}{5-i \sqrt{2}}$

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

$= \frac{3-6 \sqrt{2} i}{27}$

$= \frac{3(1-2 \sqrt{2} i)}{27}$

$= \frac{1-2 \sqrt{2} i}{9}$

So, the multiplicative inverse is $\frac{1-2 i \sqrt{2}}{9}$.

Hence, the answer is option 2.

List of Topics related to Complex Numbers according to NCERT/JEE MAIN

This section covers all essential topics on complex numbers as per NCERT and JEE Main syllabus. Understanding these topics thoroughly helps in solving a wide variety of problems in algebra, coordinate geometry, and higher-level mathematics.

NCERT Resources

Below provided are the NCERT Solutions, Exemplar problems, and concise notes for Class 11 Maths Chapter 5 – Complex Numbers and Quadratic Equations. These resources will help in understanding concepts and practising exam-relevant questions.

NCERT Solutions for Class 11 Maths Chapter 5 -Complex Numbers and Quadratic Equations

NCERT Exemplar Class 11 Maths Solutions for Chapter 5 - Complex Numbers and Quadratic Equations

NCERT Class 11 Maths Notes for Chapter 5 - Complex Numbers and Quadratic Equations

Recommended Books for Complex Numbers and Quadratic Equations

Here is a list of standard books that provide clear explanations, solved examples, and practice problems on complex numbers and quadratic equations, ideal for NCERT, JEE Main, and other competitive exams.

NCERT Subjectwise Resources

These NCERT resources cover each subject in a structured manner, offering theory, examples, and exercises that are essential for building a strong foundation in topics like complex numbers, quadratic equations, and other key concepts.

Practice Questions based on Complex Numbers and Quadratic Equations

These practice questions helps in boosting your understanding of complex numbers, including operations, conjugates, modulus, and quadratic equations, roots, allowing you to apply formulas and solve problems efficiently.