Question

If $a{\cos }^3\alpha +3a\cos \alpha .{\sin }^2\alpha =m$ and $a{\sin }^3\alpha +3a{\cos }^2\alpha .\sin \alpha =n$ then, $(m+n{)}^{2/3}+(m-n{)}^{2/3}$ is equal to:

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

Answer 1

The correct option is C $2a^{2/3}$

$m=a{cos}^3\alpha +3acos\alpha {sin}^2\alpha$

$n=a{sin}^3\alpha +3a{cos}^2\alpha sin\alpha$

$\therefore m+n=a({cos}^3\alpha +{sin}^3\alpha +3cos\alpha {sin}^2\alpha +3{cos}^2\alpha sin\alpha )$

$m+n=a[{cos}^3\alpha +{sin}^3\alpha +3sin\alpha cos\alpha (sin\alpha +cos\alpha \left)\right]$

$m+n=a{(cos\alpha +sin\alpha )}^3$ ...... $[\because (a+b{)}^3=a^3+b^3+3ab(a+b\left)\right]$

Similarly, $m-n=a{(cos\alpha -sin\alpha )}^3$

Now, ${(m+n)}^{2/3}+{(m-n)}^{2/3}=a^{2/3}[{(cos\alpha +sin\alpha )}^{3\times 2/3}+{(cos\alpha -sin\alpha )}^{3\times 2/3}]$

$=a^{2/3}[{(cos\alpha +sin\alpha )}^2+{(cos\alpha -sin\alpha )}^2]$

$=a^{2/3}[{cos}^2\alpha +{sin}^2\alpha +2sin\alpha cos\alpha +{cos}^2\alpha +{sin}^2\alpha -2sin\alpha cos\alpha ]$

$=a^{2/3}(1+1)$ ....... $(\because {cos}^2\alpha +{sin}^2\alpha =1)$

$=2a^{2/3}$