QJEEM20Ma2478: JEE Main 2020 Mathematics - Class 12 JEEM20Ma2478 Solution: If y = , then at x = 0 is _______.

Question

Verified Solution • 2025–26 Session moderate

If y =

\sum\limits_{k = 1}^6 {k{{\cos }^{ - 1}}\left\{ {{3 \over 5}\cos kx - {4 \over 5}\sin kx} \right\}}

,

then

{{dy} \over {dx}}

at x = 0 is _______.

Accepted Answer

Put,

\cos \alpha = {3 \over 5},\sin \alpha = {4 \over 5}

\therefore {3 \over 5}\cos kx - {4 \over 5}\sin \,kx

= \cos \alpha .\cos kx - \sin \alpha .\sin kx

= \cos \left( {\alpha + kx} \right)

So,

y = \sum\limits_{k = 1}^6 {k{{\cos }^{ - 1}}\left( {\cos \left( {\alpha + kx} \right)} \right)}

= \sum\limits_{k = 1}^6 {\left( {{k^2}x + kx} \right)}

\Rightarrow {{dy} \over {dx}} = \sum\limits_{k = 1}^6 {{k^2}}

= {{6 \times 7 \times 13} \over 6} = 91

Step-by-Step Verified Solution