Correct Answer - Option 2 :
\(\frac{{3\sqrt {22} }}{11}\)
CONCEPT:
Let z = a + ib be a complex number. Then, the modulus of z, denoted by |z|, is defined to be the non-negative real number \(\sqrt {{a^2} + {b^2}} .\)
CALCULATION:
Given expression is \(\frac{{4 + \sqrt 2 i}}{{3 - \sqrt 2 i}}\)
\( \Rightarrow \frac{{4 + \sqrt 2 i}}{{3 - \sqrt 2 i}} = \frac{{4 + \sqrt 2 i}}{{3 - \sqrt 2 i}} \times \frac{{3 + \sqrt 2 i}}{{3 + \sqrt 2 i}} = \frac{{12 + 4\sqrt 2 i + 3\sqrt 2 i - 2}}{{{3^2} - {{\left( {\sqrt 2 i} \right)}^2}}}\)
\( \Rightarrow \frac{{10 + 7\sqrt 2 i}}{11} = \frac{{10}}{11} + \frac{{7\sqrt 2 }}{11}i\)
\(\therefore \left| z \right| = \sqrt {{a^2} + {b^2}} = \sqrt {{{\left( {\frac{{10}}{11}} \right)}^2} + {{\left( {\frac{{7\sqrt 2 }}{11}} \right)}^2}} = \sqrt {\frac{{100 + 49 \times 2}}{{{11^2}}}} = \frac{{\sqrt {198} }}{11} = \frac{{3\sqrt {22} }}{11}\)
