QJEEM22Maff63: JEE Main 2022 Mathematics - Class 11 JEEM22Maff63 Solution: Let and Then is equal to ______________.

Given,

${S_1} = \left\{ {x \in \left[ {0,12\pi } \right],\,{{\sin }^5}x + {{\cos }^5}x = 1} \right\}$

${S_2} = \left\{ {x \in \left[ {0,8\pi } \right],\,{{\sin }^7}x + {{\cos }^7}x = 1} \right\}$

(1) ${\sin ^5}x + {\cos ^5}x = 1$

This satisfies when $\sin x = 1$ and $\cos x = 0$

$\therefore$ $x = {\pi \over 2},{{5\pi } \over 2},{{9\pi } \over 2},{{13\pi } \over 2},{{17\pi } \over 2},{{21\pi } \over 2}$

It also satisfies when $\sin x = 0$ and $\cos x = 1$

$\therefore$ $x = 0,\,2\pi ,4\pi ,6\pi ,8\pi ,10\pi ,12\pi$

$\therefore$ Accepted values of x in $\left[ {0,12\pi } \right]$ is = 13

$\therefore$ $n({S_1}) = 13$

(2) ${\sin ^7}x + {\cos ^7}x = 1$

This satisfies when $\sin x = 1$ and $\cos x = 0$

For $x \in \left[ {0,\,8\pi } \right]$ , possible values

$x = {\pi \over 2},{{5\pi } \over 2},{{9\pi } \over 2},{{13\pi } \over 2}$

It also satisfies when $\sin x = 0$ and $\cos x = 1$

$x \in \left[ {0,8\pi } \right]$ , possible values

$x = 0,2\pi ,4\pi ,6\pi ,8\pi$

$\therefore$ Total accepted values of x in $\left[ {0,8\pi } \right]$ is = 9

$\therefore$ $n({S_2}) = 9$

$\therefore$ $n({S_1}) - n({S_2}) = 13 - 9 = 4$