If z=(2-i)/(3+i)+4 i (a) |z|=(5)/(√(2)) (b) |z|=(√(5))/(2) (c) z=tan^-1 7 (d) z=tan^-1((1)/(7))

Name: Problem 197, Chapter 1: IIT JEE Super Course in Mathematics ALGEBRA II
Uploaded: 2021-07-10T09:51:27-08:00
Duration: 3 min 15 s
Channel: Uma Kumari
Description: If z=(2-i)/(3+i)+4 i (a) |z|=(5)/(√(2)) (b) |z|=(√(5))/(2) (c) z=tan^-1 7 (d) z=tan^-1((1)/(7))

step 1 In this question, it is given that complex number z is equals to 2 minus iota upon 3 plus iota p

If z=(2-i)/(3+i)+4 i (a) |z|=(5)/(√(2)) (b) |z|=(√(5))/(2)...

Key Concepts

Complex Numbers

Complex numbers are entities in the form a + bi, where a is the real part and b is the imaginary part. They are fundamental in many areas of mathematics and engineering because they allow solutions to equations that have no real solutions.

Complex Arithmetic

Complex arithmetic involves performing operations such as addition, subtraction, multiplication, and division on complex numbers. These operations often require combining like terms and sometimes converting complex expressions into a standard form to simplify further analysis.

Rationalizing Complex Fractions

When dividing complex numbers, it is common to multiply the numerator and the denominator by the conjugate of the denominator. This process eliminates the imaginary unit from the denominator, thereby simplifying the expression into the standard form a + bi.

Modulus of a Complex Number

The modulus of a complex number, defined as the square root of the sum of the squares of its real and imaginary parts, represents the distance of the number from the origin of the complex plane. It is analogous to the absolute value for real numbers.

Argument of a Complex Number

The argument of a complex number is the angle formed by the line representing the number and the positive real axis in the complex plane. It is usually computed using the inverse tangent of the ratio of the imaginary part to the real part, and proper attention must be given to the quadrant in which the complex number lies.

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