Question

$Ifaco{s}^3\alpha +3acos\alpha si{n}^2\alpha =mandasi{n}^3\alpha +3aco{s}^2\alpha sin\alpha =n,then(m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}isequalto$

• A. $2a^2$
• B. $2a^{\frac{1}{3}}$
• C. $2a^{\frac{2}{3}}$
• D. $2a^3$

Answer 1

The correct option is C

$2a^{\frac{2}{3}}$

From the given relations, we get $m+n=aco{s}^3\alpha +3acos\alpha si{n}^2\alpha +3aco{s}^2\alpha sin\alpha +asi{n}^3\alpha$ $=a(cos\alpha +sin\alpha {)}^3$
$Similarlym-n=a(cos\alpha -sin\alpha {)}^3\therefore (m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}=a^{\frac{2}{3}}\left[\right(cos\alpha +sin\alpha {)}^2+(cos\alpha -sin\alpha {)}^2]$