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4-1.Complex numbers
hard
જો $z$ એ સંકર સંખ્યા હોય તો સમીકરણ ${z^4} + z + 2 = 0$ ના બીજ શક્ય ન થવા માટે. . . .
A
$|z|\, < 1$
B
$|z|\, = 1$
C
$|z|\, > 1$
D
એકપણ નહીં.
Solution
(a)Suppose there exists a complex number $z$ which satisfies the given equation and is such that $|z|\, < 1$.
Then ${z^4} + z + 2 = 0$ ==> $ – 2 = {z^4} + z$==> $| – 2|\, = \,|{z^4} + z|$
==> $2 \le \,|{z^4}| + |z|$==> $2 < 2,$ because$|z|\, < 1$
But $2 < 2$ is not possible. Hence given equation cannot have a root $z$ such that $|z| < 1$
Standard 11
Mathematics
