Question

Statement 1: The product of all values of $(cos\alpha +isin\alpha {)}^{\frac{3}{5}}$ is $cosn3\alpha +isin3\alpha$.
Statement 2: The product of fifth roots of unity is 1.

• A. Both the statements are true, and Statement 2 is the correct explanation for Statement 1.
• B. Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.
• C. Statement 1 is true and Statement 2 is false.
• D. Statement 1 is false and Statement 2 is true.

Answer 1

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

Let
$z=(cos\theta +isin\theta {)}^3$
Taking the fifth root, we get
$z^{\frac{1}{5}}=(cos\theta +isin\theta {)}^{\frac{3}{5}}=x$
Now there will be $5$, corresponding values of $x$.
Product if all the $5$ values will be
$x^5$
$=(cos\theta +isin\theta {)}^3$
$=cos3\theta +isin3\theta$ ... using De-Moivre's rule.
Now consider $x^n=1$
Hence if $n$ is odd, the nth roots of unity will be
$1,a_1,\overline{a_1},a_2,\overline{a_2}....$
Now $a_1.\overline{a_1}=|a_1{|}^2=1$
$a_2.\overline{a_2}=|a_2{|}^2=1$
:
:
Hence one root will be one, and the rest $n-1$ roots will occur in pair with its conjugate.
Hence product will be $1$.
Substituting, $n=5$, we get the roots as
$1,a_1,\overline{a_1},a_2,\overline{a_2}$
Hence product if all the roots will be $1$. And sum of all the roots will be $0$.
Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.