(A) First,simplify the expression inside the parenthesis: $\frac{i - 1}{i + 1} = \frac{(i - 1)(i - 1)}{(i + 1)(i - 1)} = \frac{i^2 - 2i + 1}{i^2 - 1}

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(A) First,simplify the expression inside the parenthesis: $\frac{i - 1}{i + 1} = \frac{(i - 1)(i - 1)}{(i + 1)(i - 1)} = \frac{i^2 - 2i + 1}{i^2 - 1} = \frac{-1 - 2i + 1}{-1 - 1} = \frac{-2i}{-2} = i$. Thus,the expression becomes $i^n$. For $i^n$ to be a real number,$n$ must be a multiple of $2$ (since $i^2 = -1$ and $i^4 = 1$). The least positive integer $n$ is $2$.

Class 11 Mathematics 4-1.Complex numbers Integral power of iota, Algebraic operations and Equality of complex numbers

Easy

The least positive integer $n$ which will reduce $\left( \frac{i - 1}{i + 1} \right)^n$ to a real number is

A
$2$
B
$3$
C
$4$
D
$5$

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4-1.Complex numbers

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Integral power of iota, Algebraic operations and Equality of complex numbers

Which of the following statements is true?

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For any two complex numbers $z_{1}$ and $z_{2},$ prove that $\operatorname{Re}(z_{1} z_{2})=\operatorname{Re} z_{1} \operatorname{Re} z_{2}-\operatorname{Im} z_{1} \operatorname{Im} z_{2}.$

Medium

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If one root of the quadratic equation $ix^2 - 2(i + 1)x + (2 - i) = 0$ is $2 - i$,then the other root is

Easy

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Evaluate: $\left[i^{18}+\left(\frac{1}{i}\right)^{25}\right]^{3}$.

Medium

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The value of $\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}-1=$

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The least positive integer $n$ which will reduce $\left( \frac{i - 1}{i + 1} \right)^n$ to a real number is

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Which of the following statements is true?

For any two complex numbers $z_{1}$ and $z_{2},$ prove that $\operatorname{Re}(z_{1} z_{2})=\operatorname{Re} z_{1} \operatorname{Re} z_{2}-\operatorname{Im} z_{1} \operatorname{Im} z_{2}.$

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