JAMIA 2021 — Mathematics PYQ

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Question

JAMIA 2021 — Mathematics PYQ

JAMIA | Mathematics | 2021

If $(\frac{1 + i}{1 - i})^{x} = 1$ , then

Choose the correct answer:

Explanation

Step 1: Simplify the expression inside the bracket

Multiply the numerator and the denominator by the conjugate of the denominator $(1+i)$ :

\frac{1+i}{1-i} = \frac{1+i}{1-i} \times \frac{1+i}{1+i}

\frac{1+i}{1-i} = \frac{(1+i)^2}{1^2 - i^2}

\frac{1+i}{1-i} = \frac{1 + 2i + i^2}{1 - (-1)}

Since $i^2 = -1$ :

\frac{1+i}{1-i} = \frac{1 + 2i - 1}{2}

\frac{1+i}{1-i} = \frac{2i}{2} = i

Step 2: Substitute back into the original equation

The original equation $\left(\dfrac{1+i}{1-i}\right)^x = 1$ becomes:

i^x = 1

Step 3: Solve for $x$

We know the powers of $i$ :

$i^1 = i$
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$

The value of $i^x$ is $1$ whenever $x$ is a multiple of $4$ .

x = 4n, \quad \text{where } n \in \mathbb{Z}