Question

sammishra sankhya (1/(2-i)+2/(1-i)(3+4i/(2-3i)) ko maanak roop mein likhiye​

Answer 1

Answer:

$\huge \orange{ - \frac{ 89}{65} + \frac{6}{5} i}$

Step-by-step explanation:

$\frac{1}{2 - i} + \frac{2}{1 - i} ( \frac{3 + 4i}{2 - 3i} ) \\ \\ = \frac{1}{2 - i} \times \frac{2 + i}{2 + i} + \\ \frac{2}{1 - i} \times \frac{1 + i}{1 + i} ( \frac{3 + 4i}{2 - 3i} \times \frac{2 + 3i}{2 + 3i} ) \\ \\ = \frac{2 + i}{2 {}^{2} - {i}^{2} } + \frac{2(1 + i)}{1 - {i}^{2} } ( \frac{(3 + 4i)(2 + 3i)}{2 {}^{2} - (3i) {}^{2} } ) \\ \\ = \frac{2 + i}{4 + 1} + \frac{2(1 + i)}{2} \{ \frac{6 - 12 + 17i}{4 + 9} \} \\ \\ = \frac{2 + i}{5} + \frac{(1 + i)( - 6 + 17i)}{13} \\ \\ = \frac{2 + i}{5} + \frac{ - 6 + 17i - 6i - 17}{13} \\ \\ = \frac{2 + i}{5} + \frac{ - 23 + 13i}{13} \\ \\ = \frac{13(2 + i) + 5( - 23 + 13i)}{65} \\ \\ = \frac{26 - 115 + 13i + 65i}{65} \\ \\ = \frac{ - 89 + 78i}{65} \\ \\ = \frac{ - 89}{65} + \frac{78}{65}i \\ \\ = \frac{ - 89}{65} + \frac{6}{5} i$

Answer 2

Answer:

Given:-

$\longrightarrow\sf\frac{1}{2 - i} + \frac{2}{1 - i} \bigg( \frac{3 + 4i}{2 - 3i} \bigg)$

$\longrightarrow\sf\frac{1}{2 - i} + \frac{2}{1 - i} \bigg( \frac{3 + 4i}{2 - 3i} \bigg) \\ \\ \\ \longrightarrow \sf \frac{1}{2 - i} \times \frac{2 + i}{2 + i} + \frac{2}{1 - i} \times \frac{1 + i}{1 + i} \bigg( \frac{3 + 4i}{2 - 3i} \times \frac{2 + 3i}{2 + 3i} \bigg) \\ \\ \\ \longrightarrow \: \sf \frac{2 + i}{2 {}^{2} - {i}^{2} } + \frac{2(1 + i)}{1 - {i}^{2} } \bigg( \frac{(3 + 4i)(2 + 3i)}{2 {}^{2} - (3i) {}^{2} } \bigg) \\ \\ \\ \longrightarrow \sf\frac{2 + i}{4 + 1} + \frac{2(1 + i)}{2} \bigg \{ \frac{6 - 12 + 17i}{4 + 9} \bigg \} \\ \\ \\ \longrightarrow \sf\frac{2 + i}{5} + \frac{(1 + i)\pm( - 6 + 17i)}{13} \\ \\ \\ \longrightarrow \sf \frac{2 + i}{5} + \frac{ - 6 + 17i - 6i - 17}{13} \: \bigg\{We\: \:Get \bigg\}\\ \\ \\ \longrightarrow \sf \frac{2 + i}{5} + \frac{ - 23 + 13i}{13} \: \:\bigg\{By\: \:Adding\bigg\}\\ \\ \\ \longrightarrow \sf \frac{13×(2 + i) + 5×( - 23 + 13i)}{65} \\ \\ \\ \longrightarrow \sf \frac{26 - 115 + 13i + 65i}{65} \\ \\ \\ \longrightarrow \sf \frac{ - 89 + 78i}{65} \: \:\: \:\bigg\{ \: \:By\: \:Adding\: \: \bigg\} \\ \\\\ \longrightarrow \sf \frac{ - 89}{65} + \frac{78}{65}i \: \:\: \:\bigg\{Dividing\: \: by\: \: 13\bigg\}\\ \\ \\ \longrightarrow\underline{ \sf \frac{ - 89}{65} + \frac{6}{5} i}\\ \\ \sf Thus,\: \: We \: Get\longrightarrow\underline{ \boxed{\sf \frac{ - 89}{65} + \frac{6}{5} i}}$