The number of integral value of  for which the equation $7\mathrm{cosx}+5\mathrm{sinx}=2\mathrm{k}+1$  has a solution is    

$\frac{\left|2\mathrm{k}+1\right|}{\sqrt{49+25}}⩽\Rightarrow \frac{\left|2\mathrm{k}+1\right|}{\sqrt{74}}⩽1$
$\begin{array}{l}-\sqrt{74}\le 2\mathrm{k}+1\le \sqrt{74}\\ -1-\sqrt{74}\le 2\mathrm{k}\le -1+\sqrt{74}\\ \mathrm{k}=-4,-3,-2,-1,0,1,2,3\text{ satisfy }\end{array}$
No of integral values =8