We can solve problems that do not have any real solution. For instance, the non-negative equation x⁴=-3 has no real solution since the equation x²+8=0 has a real solution. The equation  D=v²−4xc<0 cannot be solved in the system of real numbers. Thus we must extend the real numbers to a wider approach and solve them.

Chapter 5 of the NCERT Class 11 Mathematics text deals with complex numbers and quadratic equations. With the previous Algebra chapters, this chapter has a total weightage of 37 marks, and one should know that exact Class 11 Maths Complex Numbers And Quadratic Equations Solutions to score well.

The term "complex number" refers to any number expressed as a+xb, for example, 9+9x, 7+8x. In this case, x is equal to -1. We may conclude that x² = 1. As a result, we may use x = -1 for any equation with no real solution.

 

It is a quadratic equation if the polynomial has two roots or has a degree 2. y=ax²+bx+d is the general form of a quadratic equation. The real numbers are a, b, and d.

Addition of Two Complex Numbers

Let there be two complex integers, z1 and z2.

 

For example, z1 = 3+4i and z2 = 4+3i

 

Assume a=3, b=4, c=4, and d=3 in this case.

z1 + z2 = (a + c) + (b + d) = i

= (3+4) + (4+3)i

⇒z1 + z2 = 7+7i

Complex number addition properties:

• • Closure law: The outcome of combining two complex numbers is likewise a complex number.

  • Commutative law: The complex numbers z1 and z2 can be commutated as z1+ z2 = z2+ z1.

  • Associative law: The associative law states that (z1+ z2) + z3 = z1 + (z2 + z3) when considering three complex numbers.

  • Additive identity: The additive identity number is a complex number having zeroes that go as 0+i0. For every complex number z, z+0 = z.

  • Additive inverse: Every complex number has an additive inverse as -z.

 

Difference Between Two Complex Numbers

 

It's analogous to adding complex numbers, as in z1 - z2 = z1 + ( -z2)

 

For example: (5+3i) - (2+1i) = (5-2) + (-2-1i) = 3 - 3i

 

Multiplication of Complex Numbers

 

When z1 and z2 have the same value, the product of the complex numbers is

(ac-bd) + (ad+bc)i  = z1 * z2

 

For example, z1 - z2 = z1 + ( -z2)

For example: (5+3i) - (2+1i) = (5-2) + (-2-1i) = 3 - 3i

 

Division of Complex Numbers

 

If you're asked for z1/z2 of a complex number,write it down as z1 (1/z2).

 

For example:   z1 = 4+2i and z2 = 2 - i z1 / z2 =(4+2i)×1/(2 - i) = (4+i2)(2/(2²+(-1)² ) + i (-1)/(2²+(-1)² )) =(4+i2) ((2+i)/5) = 1/5 [8+4i + 2(-1)+1] = 1/5 [8-2+1+41] = 1/5 [7+4i]

 

IOTA's Integral Power (i) =  √-1, i2 = -1, i3 = -i, i4 = 1

As a result, i4n+1 = i, i4n+2 = -1, i4n+3 = -i, i4n = 1

 

Note: When at least one of the supplied numbers, i.e. either zero or positive, the result a b: ab is true for any two real numbers a and b.

 

• • √-a × √-b ≠ √ab
    So, i2 = √-1 × √-1 ≠ 1

  • ‘i’ is neither positive, zero nor negative.

  • in + in+1 + in+2 + in+3 = 0

 

For example, Take √ − 25.

 

Notice that √ − 25 = √ ( − 1 ) ( 25 ) = √ − 1 ⋅ √ 25

But √ − 1 = i , is the imaginary number

Thus, √ − 25 = √ − 1 ⋅ √ 25

= i √ 25

= √ 25 i

= 5 i

 

Taking the square root of a negative real number yields I times the square root of the number if the number were positive. To put it another way, if you are looking for a unique way to express yourself

 

√ − a = i √ a , for a < 0

 

We can solve a larger range of quadratic equations now that we know how to get the square root of negative real values!

 

Consider x2 + 2. Let's find the roots of this function.

 x2 + 2 = 0

x2 = − 2

√ x2 = ± √ − 2

 x = ± √ − 2

Before, we could not have evaluated ± √ − 2 , because we did not know about imaginary numbers. However, now we know that √ − 2 = i √ 2 .

 

Thus,

x = ± √ − 2

x = ± i √ 2

So, x2+ 2 has two imaginary roots: i √ 2 and − i √ 2.

A complex number's modulus is defined as |z|, where z = a + bi is a complex number. The complex number's modulus will be defined as follows: |z| = x + yi

 

1. 1. |z|= 0 then it shows that x=y=0 

   2. |-z| = |z|

 

Assume that z1 and z2 are two complex numbers.

 

     3.  |z1.z2| = | z1 | |z2|

     4. |z1 + z2| ≤ |z1| + |z2|

     5. |z1/ z2| = |z1| / |z2|

 

Modulus of a Complex Number

 

We multiply the conjugate complex by the complex number supplied in the. As a result, the product zz is defined as the square of the Absolute value or modulus of a complex number. Let's write zz = |z|2 as an example.

 

As per the explanation, zz provides a calculation of the complex number's absolute value or magnitude.

 

Therefore, |z|2 = (a2 + b2)

Hence, |z| =  √(a2 + b2)

 

The modulus, or absolute value, of the complex number z is given by the equation above.

 

Conjugate of a Complex Number

 

A complex number's complex conjugate is a number whose real and imaginary parts are of identical size but have opposite signs.

 

The discovery of polynomial roots is a task for complex conjugates. The complex conjugate root theorem states that if a complex number in one variable with real coefficients is a root of a polynomial, then its conjugate is also a root.

 

The Cartesian or XY-plane or Z-plane used to depict any given pair of points graphically is presumably something you're already familiar with. The x-axis and y-axis are two axes that are perpendicular to each other. As a result, the XY plane can contain any ordered pair (x, y). You may use this Cartesian plane to locate any couple of real-number locations.

 

The Argand plane is identical to the XY-plane or the Cartesian plane, except that the x-axis is the real axis, and the y-axis is the imaginary axis. As a result, the argand plane is used to find complex numbers visually.

 

1. Find the modulus and the amplitude for the given complex number z = -1-i.

 

The modulus is given by |z| = √(x²+y²)

 

For z=c+ib, the value of the acute angle of which θ = tan−1|y/x|

 

Then look for the values of (c, b).

 

If (c, b) lies in the first quadrant of the plane, Argument = θ.

If (c, b) lies in the second quadrant of the plane, Argument = π – θ.

If (c, b) lies in the third quadrant of the plane, Argument = -π + θ.

If (c, b) lies in the fourth quadrant of the plane, Argument = −θ.

 

Hence, |z| = 2–√

 

The acute angle is given by θ=tan−1|yx|

 

Hence, θ = π/4

 

Hence, arg =  –3π/4

 

2. Represent the equation z = √3 + i in the polar form.

 

√3 = r sin  θ

= 1 = r sin θ

 

r = |z| =  √3+1 = 2

sin θ =  12 and cos θ =3–√2

This gives θ = π/6

Hence, the polar form of z is given by

Z = 2 (cos (π/6) + i sin (π/6))

 

What are the topics covered under NCERT Complex Number Class 11 Maths Chapter 5?

An introduction to complex numbers, algebra of complex numbers, modulus, and conjugate of a complex number are some of the essential subjects covered in complex number and quadratic equation class 11.

 

What is the concept of complex numbers and quadratic equations?

Any quadratic equation has the generic form y = ax²+ bx + c, where a, b, and c can be any real (or complex) integer. The coefficients are a, b, and c in this case. Because of its degree 2, a quadratic equation has two roots or solutions.

 

What is the marks distribution for Class 11 Maths?