Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding

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Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding Solutions Chapter-5. Step by step Solutions of ML Aggarwal ISC Class-11 Mathematics with Exe1.1,  Exe-1.2, Exe-1.3, Exe-1.4, Exe-1.5, Exe-1.6, Exe-1.7, and Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-11 Mathematics.

Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding Ch-5

Class: 11th
Subject: Mathematics
Chapter  : Ch-5 Complex Numbers of Section -A
Board ISC
Writer ML Aggarwal
Publications APC Arya Publications 2020-21

-: Select Topics :- 

Exe-5.1,

 Exe-5.2,

 Exe-5.3,

 Exe-5.4,

 Exe-5.5,

Exe-5.6,

Exe-5.7,

Chapter Test

Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding Ch-5

What are Complex Numbers?

The complex number is basically the combination of a real number and an imaginary number. The complex number is of the form a+ib. The real numbers are the numbers which we usually work on to do the mathematical calculations. But the imaginary numbers are not generally used for calculations but only in the case of imaginary numbers. Let us check the definitions for both the numbers.

What are Real Numbers?

Any number which is present in a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are real numbers. It is represented as Re(). For example: 12, -45, 0, 1/7, 2.8, √5, etc., are all real numbers.

What are Imaginary Numbers?

The numbers which are not real are imaginary numbers. When we square an imaginary number, it gives a negative result. It is represented as Im(). Example: √-2, √-7, √-11 are all imaginary numbers.

The complex numbers were introduced to solve the equation x2+1 = 0. The roots of the equation are of form x = ±√-1 and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots.

We denote √-1 with the symbol ‘i’, where i denotes Iota (Imaginary number).

An equation of the form z= a+ib, where a and b are real numbers, is defined to be a complex number. The real part is denoted by Re z = a and the imaginary part is denoted by Im z = ib.

See the table below to differentiate between a real number and an imaginary number.

Complex Number Real Number Imaginary Number
-1+2i -1 2i
7-9i 7 -9i
-6i 0 -6i (Purely Imaginary)
6 6 0i (Purely Real)

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

Complex Number Formulas

While performing the arithmetic operations of complex numbers such as addition and subtraction, combine similar terms. It means that combine the real number with the real number and imaginary number with the imaginary number.

Addition

(a + ib) + (c + id) = (a + c) + i(b + d)

Subtraction

(a + ib) – (c + id) = (a – c) + i(b – d)

Multiplication

When two complex numbers are multiplied by each other, the multiplication process should be similar to the multiplication of two binomials. It means that FOIL method (Distributive multiplication process) is used.

(a + ib). (c + id) = (ac – bd) + i(ad + bc)

How to divide the complex numbers?

To divide the complex number, multiply the numerator and the denominator by its conjugate. The conjugate of the complex number can be found by changing the sign between the two terms in the denominator value. Then apply the FOIL method to simplify the expression.

Exe-5.1

Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding Ch-5

Question 1:

Evaluate the following :

(i) ………….

……………….

Question 2:

If z = -3-i, find Re (z), Im (z), z¨ and |z|.

Question 3:

……………………

…………………….

……………………..

Question 27:

Find the conjugate of i7

Question 28:

………………….

……………………

…………………..

Question 44:

Find the real value of ……………. is a real number.

Question 45:

If …………….. such that …………

Exe-5.2

Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding Ch-5

Question 1:

Using distance formula, …………….. in complex number.

Question 2:

Using distance formula ……………. are collinear.

Question 3:

………………..

………………..

Question 4:

If ……………. and only if ………. .

Exe-5.3

Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding Ch-5

Question 1:

If z is a complex number, state ……………….  true or false :

(i)…………..

………………

Question 2:

Represent the following complex number …………………

(i)………….

……………..

Question 3:

………………..

………………..

………………….

Question 6:

Convert the following complex number into polar form :

(i)………….

……………….

Question 7:

Express sin …………….. in polar form.

Exe-5.4

Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding Ch-5

Question 1:

Write the equation or ……………. the following :

(i) A circle of radius ……………

(ii) all point ……………

(iii) …………..

…………….

Question 2:

Which regions ………… complex number.

…………….

Question 3:

……………..

………………

………………..

Question 8:

If  z= x + iy ………….. it on the Argand plane.

Exe-5.5

Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding Ch-5

Question 1:

Find square root of ……… your result.

Question 2:

Find the square roots  ………………… complex number .

……………..

Question 3:

Find the ……………………. complex number .

…………….

Question 4:

Find the square …………….. complex number.

(i)-1

(ii)…….

(iii) …………

Exe-5.6

Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding Ch-5

Question 1:

Find the cube roots of ………………… roots is equal to zero.

Question 2:

(i) If …………

(ii)………….

(iii) ………….

(iv) Prove that ………

Question 3:

…………………..

…………………..

………………….

Question 10:

Prove that :

………………

(i) 2 if n is a multiple of 3

(ii)-1ifnisnotamultiple of 3.

Exe-5.7

Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding Ch-5

Question 1:

If ………………. thewn find  |z1|.

Question 2:

If ……………..prove that.

…………………..

Question 3:

……………….

…………………

………………….

Question 6:

If n is an odd integer, prove that ……………. + 1 = 0.

Chapter Test

Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding Ch-5

Question 1:

Find the real values of x and y if

(i)……………

(ii)……………

(iii)………….

(iv)…………..

Question 2:

Show that ………….. all n ∈ N.

Question 3:

…………………..

……………………

…………………….

Question 15:

If x = a + b, y = aω + …………………… = 6ab.

-: End of Complex Numbers ML Aggarwal  ISC Class-11 Maths Chapter-5 Solution :-

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6 thoughts on “Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding”

    • Although we have try our best to do more and more resources for ICSE / ISC even if any error sorry for this . we will again analysis the solutions if such error occure

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