Question

The number of integral values of k for which the equation $7cosx+5sinx=2k+1$ has a solution is

• A. 4
• B. 8
• C. 10
• D. 12

Answer 1

The correct option is B 8

The given equation can be written as $rcos(x-\alpha )=2k+1$ where $rcos\alpha =7,rsin\alpha =5$
$\Rightarrow cos(x-\alpha )=\frac{2k+1}{\sqrt{74}}as{r}^2=7^2+5^2=74$
$\Rightarrow -1\le \frac{2k+1}{\sqrt{74}}\le 1$
$\Rightarrow -\sqrt{74}\le 2k+1\le \sqrt{74}$
$\Rightarrow -8\le 2k+1\le 8$ (For integral values of k)
$\Rightarrow -4\le k\le 3$
$\Rightarrow k=-4,-3,-2,-1,0,1,2,3$
Which gives 8 integral values of k.