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Q.
The number of integral values of $ k $ for which the equation $ 7\cos x+5\sin x=2k+1 $ has a solution is
Jharkhand CECEJharkhand CECE 2013
Solution:
We know that,
$ a\cos \theta +b\sin \theta =c $
has one solution only. $ |c|\le \sqrt{{{a}^{2}}+{{b}^{2}}} $
Then, for the given, equation, we must have
$ |2k+1|\,\,\le \sqrt{74} $
$ \Rightarrow $ $ -\sqrt{74}<2k+1<\sqrt{74} $
$ \Rightarrow $ $ -8<2k+1<8 $
$ \therefore $ $ k=-4,\,\,-3,\,\,-2,\,\,-1,\,\,0,\,\,1,\,\,2,\,\,3 $
Thus, there are $ 8 $ values which will satisfy the above inequality