Question
For $$ - \pi < x < \pi ,$$ the values of $$x$$ which satisfy the relation $${11^{1 + \left| {\cos x} \right| + {{\cos }^2}x + \left| {{{\cos }^3}x} \right| + .....{\text{ upto }}\infty }} = 121$$ are given by
A.
$$ \pm \frac{\pi }{3}, \pm \frac{{2\pi }}{3}$$
B.
$$\frac{\pi }{3},\frac{{2\pi }}{4}$$
C.
$$\frac{\pi }{4},\frac{{3\pi }}{4}$$
D.
None of these
Answer :
$$ \pm \frac{\pi }{3}, \pm \frac{{2\pi }}{3}$$
Solution :
Since, $$0 < x < \pi , - 1 < \cos x < 1$$
$$ \Rightarrow 0 \leqslant \left| {\cos x} \right| < 1.$$
We can write the given expression as $${11^{\frac{1}{{\left( {1 - \left| {\cos x} \right|} \right)}}}} = 121$$
$$\eqalign{
& \Rightarrow \frac{1}{{1 - \left| {\cos x} \right|}} = 2 \cr
& \Rightarrow 1 - \left| {\cos x} \right| = \frac{1}{2} \cr
& \Rightarrow \left| {\cos x} \right| = \frac{1}{2} \cr
& \Rightarrow \cos x = \pm \frac{1}{2} \cr
& \Rightarrow x = \pm \frac{\pi }{3}, \pm \frac{{2\pi }}{3} \cr} $$