JEE Main 2022 (Online) 30th June Morning ShiftQuestion 1
Let $S_1=\{x\in[0,12\pi]:\sin^{5}x+\cos^{5}x=1\}$ and $S_2=\{x\in[0,8\pi]:\sin^{7}x+\cos^{7}x=1\}$. Then $n(S_1)-n(S_2)$ is equal to $\_\_\_\_\_\_$ .
Trigonometric Ratio & Identities
Answer$4$
Solution
Concept. For $n\ge3$, $|\sin x|^n\le\sin^2x$ and $|\cos x|^n\le\cos^2x$; the sum can be $1$ only when both are equalities. Formulas used. (1) For $0\le t\le1$ and $n\ge3$: $t^n\le t^2$, with equality iff $t\in\{0,1\}$Step 1 - Condition. $\sin^nx+\cos^nx\le|\sin x|^n+|\cos x|^n\le\sin^2x+\cos^2x=1$. Equality needs $(\sin x,\cos x)=(0,1)$ or $(1,0)$ (the values must also be positive). So $x=2k\pi$ or $x=\frac\pi2+2k\pi$. Step 2 - $S_1$ on $[0,12\pi]$. $x=2k\pi$: $k=0,\dots,6$ ($7$). $x=\frac\pi2+2k\pi$: $k=0,\dots,5$ ($6$). $n(S_1)=13$. Step 3 - $S_2$ on $[0,8\pi]$. $5+4=9$. Step 4 - Value. $13-9=4$. Answer: 4