JEE Main 2021 (Online) 26th February Morning ShiftQuestion 1
The number of integral values of '$k$' for which the equation $3\sin x+4\cos x=k+1$ has a solution, $k\in\mathbf{R}$ is $\_\_\_\_\_\_$ .
Trigonometric Ratio & Identities
Answer$11$
Solution
Concept. $3\sin x+4\cos x$ takes every value in $[-5,5]$. Formulas used. (1) $-\sqrt{a^2+b^2}\le a\sin x+b\cos x\le\sqrt{a^2+b^2}$Step 1 - Condition. $-5\le k+1\le5$, i.e. $-6\le k\le4$. Step 2 - Count. $k=-6,\dots,4$: $11$ integers. Answer: 11