• # question_answer 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

[b]  We have, $7\cos x+5\sin x=2k+1$ The minimum and maximum value of $7\cos x+5\sin x$ is $-\sqrt{{{7}^{2}}+{{5}^{2}}},\sqrt{{{7}^{2}}+{{5}^{2}}}$,$-\sqrt{74},\sqrt{74}$ $\therefore$$-\sqrt{74}<2k+1<\sqrt{74}$ $\Rightarrow$$-\,8<2k+1<8$ $\Rightarrow$$-\,4\le k\le 3$ $k=-\,4,-\,3,-\,2,-1,0,1,2,3$ Total 8 integral value.