Question

STATEMENT-1: Arg \((2+3 i)+\operatorname{Arg}(2-3 i)=0(\) Arg \(z\) stands for principal argument of \(z\) ) STATEMENT-2 : Argz \(+\operatorname{Arg} \bar{z}=0, z=x+i y, \forall x, y \in R\) (Arg \(z\) stands for principal argument of \(z\) ) (1) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (2) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (3) Statement-1 is True, Statement-2 is False (4) Statement-1 is False, Statement-2 is True

Short answer

Option (2). Both statements are true, but Statement-2 does not explain Statement-1.

Step-by-step solution

Analyze Statement-1

First, we need to find the principal arguments of the complex numbers \(2 + 3i\) and \(2 - 3i\). The principal argument of a complex number \(z = a + bi\) is given by \(\tan^{-1}(b/a)\).For \(2 + 3i\):\[ \text{Arg}(2 + 3i) = \tan^{-1}(3/2) \]For \(2 - 3i\):\[ \text{Arg}(2 - 3i) = \tan^{-1}(-3/2) = -\tan^{-1}(3/2) \]Adding these two:\[ \text{Arg}(2 + 3i) + \text{Arg}(2 - 3i) = \tan^{-1}(3/2) + (-\tan^{-1}(3/2)) = 0 \]Thus, Statement-1 is true.

Analyze Statement-2

Now, we evaluate Statement-2. For a complex number \(z = x + iy\) with its conjugate given by \(\bar{z} = x - iy\), the principal argument of \(z\) is \(\text{Arg}(z)\) and for \(\bar{z}\) is \(\text{Arg}(\bar{z}) = -\text{Arg}(z)\).Hence:\[ \text{Arg}(z) + \text{Arg}(\bar{z}) = \text{Arg}(z) + (-\text{Arg}(z)) = 0 \]So, Statement-2 is true.

Evaluate the Relationship Between the Statements

Although both Statement-1 and Statement-2 are true, Statement-2 does not directly explain why the sum of the principal arguments of \(2 + 3i\) and \(2 - 3i\) is zero. The arguments of these two specific complex numbers sum to zero due to their specific values rather than the general property of a complex number and its conjugate.

Final Answer

Based on the analysis, the correct option is (2) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

Key concepts

Principal Argument

The principal argument of a complex number is the angle formed between the positive real axis and the line representing the complex number in the complex plane. For any complex number in the form of \(z = a + bi\), the principal argument (\(\text{Arg}(z)\)) can be calculated using the formula \(\tan^{-1}(b/a)\). It is important to distinguish that the principal argument is usually taken in the interval \((-\pi, \pi]\).
For example, if you have the complex number \(2 + 3i\), its principal argument will be:
\[ \text{Arg}(2 + 3i) = \tan^{-1}\left(\frac{3}{2}\right) \] On the other hand, for \(2 - 3i\), the value will be:
\[ \text{Arg}(2 - 3i) = \tan^{-1}\left(\frac{-3}{2}\right) \] which simplifies to \(- \tan^{-1}(3/2)\).
Thus, when adding the principal arguments of \(2 + 3i\) and \(2 - 3i\), we get:
\[ \text{Arg}(2 + 3i) + \text{Arg}(2 - 3i) = \tan^{-1}\left(\frac{3}{2}\right) + \left(-\tan^{-1}\left(\frac{3}{2}\right)\right) = 0 \] This verifies the interpretation provided in Statement-1.

Complex Conjugate

The complex conjugate of a complex number \(z = x + iy\) is \(\bar{z} = x - iy\). In essence, it is the reflection of the complex number across the real axis.
One significant property of complex conjugates is that the principal argument of \(\bar{z}\) is the negative of the principal argument of \(z\). That is:
\[ \text{Arg}(\bar{z}) = -\text{Arg}(z) \] For example, for \(z = 2 + 3i\), \[ \text{Arg}(z) = \tan^{-1}\left(\frac{3}{2}\right) \] and the conjugate \(\bar{z} = 2 - 3i\) will have: \[ \text{Arg}(\bar{z}) = -\tan^{-1}\left(\frac{3}{2}\right) \] Hence: \[ \text{Arg}(z) + \text{Arg}(\bar{z}) = \tan^{-1}\left(\frac{3}{2}\right) + \left(-\tan^{-1}\left(\frac{3}{2}\right)\right) = 0 \] This explains the concept provided in Statement-2 and demonstrates this important property of complex conjugates.

Trigonometric Identities

Trigonometric identities are fundamental tools in understanding complex numbers, particularly when dealing with the principal arguments of complex numbers. One important identity is: \( \tan^{-1}(-x) = -\tan^{-1}(x) \).
For example, let's take \(\tan^{-1}(3/2)\) for the complex number \(2 + 3i\), and \(\tan^{-1}(-3/2)\) for its conjugate \(2 - 3i\). According to the identity: \[ \tan^{-1}(-3/2) = -\tan^{-1}(3/2) \] This directly influenced how the arguments in Statement-1 were added together to result in zero: \[ \text{Arg}(2 + 3i) + \text{Arg}(2 - 3i) = \tan^{-1}\left(\frac{3}{2}\right) + (-\tan^{-1}\left(\frac{3}{2}\right)) = 0 \] Understanding and applying these trigonometric identities is critical in solving complex number problems, especially in IIT JEE exams.