JEE Important Chapter - Complex Numbers and Quadratic Equations

Complex Numbers and Quadratic Equations is a fascinating and essential topic in mathematics. Every year, at least 1 to 3 problems from this chapter appear in IIT JEE and other exams. The concept of this chapter will be used in many other chapters, such as functions and coordinate geometry.

Begin by grasping fundamental ideas such as the definition of a complex number, integral powers of iota, and various representations of a complex number. Then go on to complex number algebra. The Argand plane, modulus, and argument of a complex number, as well as the triangle, are all fundamental concepts. Go over the concepts of solved problems again, and then do the same with the quadratic equation. Refer to the complex numbers and quadratic equations Class 11 solutions provided by Vedantu and download the complex numbers and quadratic equations Class 11 PDF to prepare for the exams.

JEE Main Maths Chapter-wise Solutions 2022-23 

Important Topics of Complex Numbers and Quadratic Equations

• Algebra of complex numbers

• Properties of complex numbers

• Modulus and conjugate of a complex numbers

• Argument of complex number

• Polar form of complex numbers

• Euler's Formula and De Moiver’s Theorem

• Geometry of Complex Numbers

• Cube root of unity

• Vector representation and rotation of complex numbers

• Nature of roots (in quadratic equations), the relation of coefficient and roots

• Transformation of quadratic equations and condition of common roots

• The discriminant of quadratic equations

Important Concepts of Complex Number and Quadratic Equation 

Complex Number 

A complex number is one that can be written as p + iq, where p and q are real values and i represents a solution to the x2-1 equation.

$\sqrt{i}=-1$ or, i2 = -1.

Some of the examples of complex numbers are: 8 – 2i, 2 +31i, etc, and the Complex numbers are denoted by ‘z’.

The general form of the complex number is z=p+iq

Here,

• p is known as the real part and is denoted by Re z.

• q is known as the imaginary part and it is denoted by Im z.

• If z = 12 + 35i, then the value Re z = 12 and Im z = 35. If z1 and z2 are two complex numbers such that z1 = p + iq and z2 = r + is , z1 and z2 are equal if p = r and q = s.

Modulus of a Complex Number

Consider the complex number z = x + iy. The modulus (absolute values) of z is thus defined as the positive square root of the sum of the squares of the real and imaginary parts, indicated by |z|  i.e. $|\text{z}|=\sqrt{{x^2}+{y^2}}$

It represents the distance of z from the origin in the set of complex number c, where the order relation is not defined

i.e. z1 > z2 or z1 < z2 has no meaning but when |z1| > |z2| or |z1|<|z2| has got its meaning since |z1| and |z2| are the real numbers.

Argand Plane

Any complex number z = x + iy can be represented geometrically by a point (x, y) in a plane, called the argand plane or gaussian plane. A purely number x, i.e. (x + 0i) is represented by the point (x, 0) on the x-axis. Therefore, the x-axis is called the real axis. A purely imaginary number iy i.e. (0 + iy) is represented by the point (0, y) on the y-axis. Therefore, the y-axis is known as the imaginary axis.

Argument of a Complex Number

The angle made by a line joining point z to the origin, with the positive direction of the x-axis in an anti-clockwise sense is called the argument or amplitude of a complex number. It is denoted by the symbol arg(z) or amp(z).

arg(z) = θ = $\tan^{-1}\left ( \dfrac{x}{y} \right )$

Image: Argument of complex number

The argument of z is not unique, and its general value of the argument of z is 2nπ + θ, but arg(0) is not defined. The unique value of θ such that -π < θ ≤ π is called the principal value of the amplitude or principal argument.

Principal Value of Argument

• if x > 0 and y > 0, then arg(z) = θ.

• if x < 0 and y > 0, then arg(z) = π – θ.

• if x < 0 and y < 0, then arg(z) = -(π – θ).

• if x > 0 and y < 0, then arg(z) = -θ.

Polar Form of a Complex Number

If z = x + iy is a complex number, then  z = |z| (cosθ + i sinθ), where θ = arg(z). This is called polar form. If the general value of the argument is θ, then the polar form of z is z = |z| [cos (2nπ + θ) + i sin(2nπ + θ)], where n is an integer.

What Is a Quadratic Equation?

A quadratic equation is a second-degree equation. The general form of the quadratic equation is ax2+bx+c = 0, where a, b, c are real numbers and a ≠ 0. For example, x2+2x+1 = 0.

An algebraic expression with several terms is called a polynomial. A polynomial of degree two of the form ax2+bx+c (a≠0) is called a quadratic expression in x. When a quadratic polynomial f(x) is equated to zero, we can term it a quadratic equation.

Solution of Quadratic Equation

The quadratic equation ax2 + bx + c = 0 with real coefficients has two roots given by $\dfrac{-b+\sqrt{D}}{2a}$ and $\dfrac{-b-\sqrt{D}}{2a}$, where D = b2 – 4ac, called the discriminant of the equation.

Note:

(i) When D = 0, roots are real and equal. When D > 0 roots are real and unequal. Further, if a,b, c ∈ Q and D is a perfect square, then the roots of the quadratic equation are real and unequal and if a, b, c ∈ Q and D is not a perfect square, then the roots are irrational and occur in pairs. When D < 0, the roots of the equation are non-real (or complex).

(ii) Let α, β be the roots of quadratic equation ax2 + bx + c = 0, then the sum of roots α + β =$\dfrac{-b}{a}$ and the product of roots αβ = $\dfrac{c}{a}$. 

Important Formulae

Solved Examples 

1. Express the question in the form of a + bi: $\left ( -5i \right )\left ( \dfrac{1}{8}i \right )$

Solution: $\left ( -5i \right )\left ( \dfrac{1}{8}i \right )$

Solving the above equation, we get:

 = $\left ( \dfrac{-5}{8} i^2 \right )$ = $\left ( \dfrac{-5}{8} (-1) \right )$ = $\left ( \dfrac{5}{8}\right )$ 

Now, write the final answer in the form of a+ib

=$\left ( \dfrac{5}{8}+0i\right )$

Question 2: (cos θ + i sin θ)4 / (sin θ + i cos θ)5 is equal to ____________.

Solution:

(cos θ + i sin θ)4 / (sin θ + i cos θ)5

= (cos θ + i sin θ)4 / i5 ([1 / i] sin θ + cos θ)5

= (cosθ + i sin θ)4 / i (cos θ − i sin θ)5

= (cos θ + i sin θ)4 / i (cos θ + i sin θ)−5 (By property) = 1 / i (cos θ + i sin θ)9

= sin(9θ) − i cos (9θ).

Solved Questions of Previous Years Question Papers

Question 1: If z is a complex number, then the minimum value of |z| + |z − 1| is ______.

Solution:

First, note that |−z|=|z| and |z1 + z2| ≤ |z1| + |z2|

Now |z| + |z − 1| = |z| + |1 − z| ≥ |z + (1 − z)|

= |1|

= 1

Hence, minimum value of |z| + |z − 1| is 1.

Question 2: Find the complex number z satisfying the equations $\dfrac{\left | z-12 \right |}{\left | z-8i \right |}=\dfrac{8}{3}$, $\dfrac{\left | z-4 \right |}{\left | z-8 \right |}=1$

Solution:

We have

$\dfrac{\left | z-12 \right |}{\left | z-8i \right |}=\dfrac{8}{3}$,

$\dfrac{\left | z-4 \right |}{\left | z-8 \right |}=1$

Let z = x + iy, then

$\dfrac{\left | z-12 \right |}{\left | z-8i \right |}=\dfrac{8}{3}$

⇒ 3|z − 12| = 5 |z − 8i|

3 |(x − 12) + iy| = 5 |x + (y − 8) i|

9 (x − 12)2 + 9y2 = 25x2 + 25 (y − 8)2 ….(i) and

$\dfrac{\left | z-4 \right |}{\left | z-8 \right |}=1$

⇒ |z − 4| = |z − 8|

|x − 4 + iy| = |x − 8 + iy|

(x − 4)2 + y2 = (x − 8)2 + y2

⇒ x = 6

Putting x = 6 in (i), we get y2 − 25y + 136 = 0

y = 17, 8

Hence, z = 6 + 17i or z = 6 + 8i

Question 3: If the cube roots of unity are 1, ω, ω2, find the roots of the equation (x − 1)3 + 8 = 0.

Solution:

(x − 1)3 = −8 ⇒ x − 1 = (−8)1/3

x − 1 = −2, −2ω, −2ω2

x = −1, 1 − 2ω, 1 − 2ω2

Practise Questions

1. The area of the triangle with vertices A(z), b(iz) and, c(z+iz) is

a. 1

b. $\dfrac{1}{2}$|z|2

c. $\dfrac{1}{2}$

d. $\dfrac{1}{2}$ |z+iz|2

Answer: 1-b

2. Let a complex number be $W=1-\sqrt{3}i$. Let another complex number z be |zw|=1 such that and arg(z)-arg(w) = $\dfrac{\pi}{2}$. The area of the triangle with vertices origin, and is equal to:

a. 4

b. $\dfrac{1}{2}$

c. $\dfrac{1}{4}$

d. 2 

Answer: 2-b

Conclusion

Students will learn about JEE Mathematics complex number and quadratic equations by reading the articles. They will know about its wide range of applications in real-life problems. For example, in physics, when dealing with a circuit involving capacitors and inductance, we use complex numbers to find the circuit's impedance. To do so, we use complex numbers to represent the capacitor and inductance quantities responsible for the contribution of impedance. Students can also gain a thorough understanding of the concept by working through the solved problems and examples. Visit Vedantu’s website and refer to Class 11 Maths Chapter 5 miscellaneous exercise solutions, NCERT solutions for class 11 Maths Chapter 5, complex numbers and quadratic equations Class 11 solutions and download complex numbers and quadratic equations Class 11 PDF to be better prepared.

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