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Properties of nth Root of a Complex Number

Let the four ...

Question

Let the four roots of unity be $z_{1}, z_{2}, z_{3}$ , and $z_{4}$ , respectively.
Statement 1: $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2} = 0$
Statement 2: $z_{1} + z_{2} + z_{3} + z_{4} = 0$ .

Both the statements are true, and Statement 2 is the correct explanation for Statement 1.

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Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.

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Statement 1 is true and Statement 2 is false.

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Statement 1 is false and Statement 2 is true.

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Solution

The correct option is B Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.
$x^{4} = 1$
$x^{2} = \pm 1$
$x^{2} = 1$ and $x^{2} = - 1$
$x = \pm 1$ and $x = \pm i$
Hence the four roots are
$1, - 1, i, - i$ .
Now
$z_{1} = 1 = - z_{2}$
$z_{3} = i = - z_{4}$
Hence
$z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2}$
$= 1 + 1 + (i)^{2} + (- i)^{2}$
$= 2 - 2$
$= 0$ ...(i)
And also
$z_{1} + z_{2} + z_{3} + z_{4}$
$= 1 - 1 + i - i$
$= 0$ ...(ii)
Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.

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(p \lor q) \land \sim p

\sim p \land q

Choose the correct option.

Statement $1 : (p \land \sim q) \land (\sim p \land q)$ is a fallacy.

Statement $2 : (p \Rightarrow q) \Leftrightarrow (\sim q \Rightarrow\sim p)$ is a tautology.

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