SATHEE: Complex Numbers 5 Question 16

Complex Numbers 5 Question 16

17.

Let a complex number $α, α \neq 1$ , be a root of the equation

$z^{p + q} - z^{p} - z^{q} + 1 = 0$

where, $p$ and $q$ are distinct primes. Show that either

$or \begin{aligned} 1 + α + α^{2} + \dots + α^{p - 1} = 0 \\ 1 + α + α^{2} + \dots + α^{q - 1} = 0 \end{aligned}$

but not both together.

(2002, 5M)

Show Answer

Solution:

Given, $z^{p + q} - z^{p} - z^{q} + 1 = 0$

$\Rightarrow (z^{p} - 1) (z^{q} - 1) = 0$

Since, $α$ is root of Eq. (i), either $α^{p} - 1 = 0$ or $α^{q} - 1 = 0$

$\Rightarrow$ Either $\frac{α^{p} - 1}{α - 1} = 0$ or $\frac{α^{q} - 1}{α - 1} = 0 [$ as $α \neq 1]$

$\Rightarrow$ Either $1 + α + α^{2} + \dots + α^{p - 1} = 0$

or $1 + α + \dots + α^{q - 1} = 0$

But $α^{p} - 1 = 0$ and $α^{q} - 1 = 0$ cannot occur simultaneously as $p$ and $q$ are distinct primes, so neither $p$ divides $q$ nor $q$ divides $p$ , which is the requirement for $1 = α^{p} = α^{q}$ .

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Complex Numbers 5 Question 16

17.

Solution:

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