The modulus of the expression {4+{2} i}{3-{2} i} {4+{2} i}{3-{2} i} can be written as -

Correct Answer is : 3221132211{3{22}}{11} {3{22}}{11}1132211322

Correct Answer is : 3221132211{3{22}}{11} {3{22}}{11}1132211322

Test Index

UPSC NDA Math Model Paper 3

Question : 1 of 120

Marks: +1, -0

\frac{4+\sqrt{2} i}{3-\sqrt{2} i} \frac{4+\sqrt{2} i}{3-\sqrt{2} i}

can be written as -

Solution:

Given expression is

\frac{4+\sqrt{2} i}{3-\sqrt{2} i} \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}-(\sqrt{2} i)^{2}}

\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}-(\sqrt{2} i)^{2}}

\Rightarrow \frac{10+7\sqrt{2} i}{11} = \frac{10}{11} + \frac{7\sqrt{2}}{11} i

\Rightarrow \frac{10+7\sqrt{2} i}{11} = \frac{10}{11} + \frac{7\sqrt{2}}{11} i

\therefore |z| = \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}

\therefore |z| = \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}

Go to Question:

Next Question