First slide

Algebra of complex numbers

Question

Dividing f(z) by z - i, we obtain the remainder i and dividing it by z + i, we get the remainder 1 + i, then, remainder upon the division of f(z) by z² + 1 is

Moderate

A

$\frac{1}{2} (z + 1) + i$
B

$\frac{1}{2} (iz + 1) + i$
C

$\frac{1}{2} (iz - 1) + i$
D

$\frac{1}{2} (z + i) + 1$

Solution

$f (z) = g (z) (z - i) (z + i) + az + b; a, b \in C$
Given, f(i) = i
$\Rightarrow ai + b = i$ (1)
and $f (- i) = 1 + i$
$\Rightarrow a (- i) + b = 1 + i$ (2)
From (l) and (2), we have
$a = \frac{i}{2}, b = \frac{1}{2} + i$
Hence, the required remainder is $a z + b = \frac{1}{2} i z + \frac{1}{2} + i$

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