Question

JEE Mains Maths question is in the image

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

$\displaystyle \: { \bigg( \: \frac{1 + i}{1 - i} \bigg)}^{ \frac{m}{2} } = { \bigg( \: \frac{1 + i}{1 - i} \bigg)}^{ \frac{n}{3} } = 1$

FORMULA TO BE IMPLEMENTED

$\displaystyle \: { \: \frac{1 + i}{1 - i} }$

$= \displaystyle \: { \: \frac{i(1 + i)}{i(1 - i)} }$

$= \displaystyle \: { \: \frac{i(1 + i)}{(i - {i}^{2} )} }$

$= \displaystyle \: { \: \frac{i(1 + i)}{(i + 1 )} }$

$= \displaystyle \: { \: \frac{i(1 + i)}{(1 + i )} }$

$= i$

$\displaystyle \: { \bigg( \: \frac{1 + i}{1 - i} \bigg)}^{ p } =1 \: \: \: gives$

$\implies\displaystyle \: {i}^{p} = 1$

$\implies\displaystyle \: {i}^{p} = {(i)}^{4k}$

So p is a multiple of 4

$\implies\displaystyle \: p = 4k$

CALCULATION

$\displaystyle \: { \bigg( \: \frac{1 + i}{1 - i} \bigg)}^{ \frac{m}{2} } = { \bigg( \: \frac{1 + i}{1 - i} \bigg)}^{ \frac{n}{3} } = 1$

Hence

$\displaystyle \: \frac{m}{2} \: \: is \: a \: multiple \: \: of \: 4$

And

$\displaystyle \: \frac{n}{3} \: \: is \: a \: multiple \: \: of \: 4$

Therefore

$\displaystyle \: m \: \: is \: a \: multiple \: \: of \: \: 8$

And

$\displaystyle \: n \: \: is \: a \: multiple \: \: of \: \: 12$

RESULT

$\red{ \boxed{Hence \: HCF \: ( m, n ) \: = 4 \: }}$

Answer 2

Answer:

4

Step-by-step explanation:

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