Question

The number of integral values of $K$, for which the equation $7\, \cos \,x + 5\, \sin\, x = 2K + 1$ has a solution, is

• A. 4
• B. 8
• C. 10
• D. 12

Answer 1

Answer: (B)

$-\sqrt{7^{2}+5^{2}} \leq(7 \cos x+5 \sin x) \leq \sqrt{7^{2}+5^{2}}$
So, for solution, $-\sqrt{74} \leq(2 K+1) \leq \sqrt{74}$
$\Rightarrow -8.6 \leq(2 K+1) \leq 8.6$
$\Rightarrow -9.6 \leq 2 K \leq 7.6$
$\Rightarrow -4.8 \leq K \leq 3.8$
So, integral values of $K$ are
$-4,-3,-2,-1,0,1,2,3$ (eight values)