First slide

Algebra of complex numbers

Question

$Let the complex number z be root of the equation 11 z^{10} + 10 i z^{9} + 10 i z - 11 = 0, then$ which of the following statement(s) is/are correct?

Difficult

A

${|z|}^{2} + |z| + 1 = 3$
B

${|z|}^{2} - |z| + 1 = 1$
C

${|z|}^{2} + |z| + 1 = 7$
D

${|z|}^{2} + |z| + 1 = 8$

Solution

   $11 z^{10} + 10 i z^{9} + 10 i z - 11 = 0$
   $\Rightarrow z^{9} (11 z^{} + 10 i) = 11 - 10 i z \Rightarrow z^{9} = \frac{11 - 10 i z}{11 z + 10 i} \Rightarrow |z^{9}| = |\frac{11 i + 10 z}{11 z + 10 i}|$
$\Rightarrow {|11 i + 10 z|}^{2} - {|11 z + 10 i|}^{2} = (11^{2} - 10^{2}) (1 - {|z|}^{2}) = 21 (1 - {|z|}^{2})$
$f o r |z| < 1 \Rightarrow$ ${|11 i + 10 z|}^{2} - {|11 z + 10 i|}^{2}$ $> 0$
$\Rightarrow |z^{9}| =$ $= |\frac{11 i + 10 z}{11 z + 10 i}|$ $> 1$ $\Rightarrow |z^{9}| > 1 w h i c h c o n t r a d i c t s w i t h |z| < 1$
$f o r |z| > 1 w e g e t |z^{9}| < 1 t h e r e f o r e |z| = 1$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App