Question

If $\frac{2+3i}{(7-i)(4+2i)}=a+ib$, then $a=$

• A. $\frac{9}{100}$
• B. $\frac{7}{100}$
• C. $\frac{-9}{100}$
• D. $\frac{-7}{100}$

Answer 1

The correct option is A $\frac{9}{100}$

$\frac{2+3i}{(7-i)(4+2i)}$ $=\frac{2+3i}{28+14i-4i-2i^2}$

$=\frac{2+3i}{28+10i-2(-1)}$

$=\frac{2+3i}{30+10i}$

$=\frac{1}{10}\left(\frac{2+3i}{3+i}\right)$

$=\frac{1}{10}(\frac{2+3i}{3+i}\times \frac{3-i}{3-i})$

$=\frac{1}{10}\left(\frac{6-2i+9i-3i^2}{9-i^2}\right)$

$=\frac{1}{10}\left(\frac{6+7i-3(-1)}{9-(-1)}\right)$

$=\frac{9+7i}{100}$

$=a+ib$

Comparing the coefficients we get, $a=\frac{9}{100}$