Question

The number of integral values of $k$ for which the equation $7\cos x+5\sin x=2k+1$ has $a-$

• A. $4$
• B. $8$
• C. $10$
• D. $12$

Answer 1

The correct option is B $8$

To find the number of integral values of $k$, for which the equation
$7\cos x+5\sin x=2k+1$

As we know that , maximum value of $7\cos x+5\sin x$ is
$=\sqrt{7^2+5^2}$
$=\sqrt{74}=8.6$

Minimum value of $7\cos x+5\sin x$ is
$=-\sqrt{7^2+5^2}$
$=-8.6$
So, the value will vary between $-8.6$ and $8.6$
Since $2k+1$ is an odd integer
Odd integer between $-8.6$ and $8.6$ is
$-7,,-5,-3,-1,1,3,5,7$
and corresponding value of $k$ will be
$-4,-3,-2,-1,0,1,2,3$

Hence, there are $8$ values of $k$