IIT-JEE 2002 ScreeningQuestion 1
The number of integral values of $k$ for which the equation $7\cos x+5\sin x=2k+1$ has a solution is
Trigonometric Ratio & Identities
Solution
Concept. $7\cos x+5\sin x$ takes every value in $\left[-\sqrt{74},\sqrt{74}\right]$. Formulas used. (1) Range of $p\cos x+q\sin x$: $\left[-\sqrt{p^2+q^2},\sqrt{p^2+q^2}\right]$Step 1 - Range. By (1), $\sqrt{49+25}=\sqrt{74}\approx8.60$. Step 2 - Odd integers. $2k+1$ must be an odd integer in $[-8.60,8.60]$: $-7,-5,\dots,5,7$. Step 3 - Count. $8$ values ($k=-4,\dots,3$). Answer: (B)