Question

The number of integral value 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

The correct option is B $8$

$-\sqrt{a^2+b^2}\le acosx+bsinx\le \sqrt{a^2+b^2}$

$\Rightarrow -\sqrt{7^2+5^2}\le 7cosx+5sinx\le \sqrt{7^2+5^2}=\sqrt{74}$

$\Rightarrow -8\le 7cosx+5sinx\le 8$

$\Rightarrow -8\le 2k+1\le 8\Rightarrow -9\le 2k\le 7\Rightarrow \frac{-9}{2}\le k\le \frac{7}{2}$

So, Integral Values Lying in between $-4.5\le k\le 3.5$ are $=-4,-3-2,-1,0,1,2,3$

Therefore, $8$ integral values of $k$ satisfies the Equation