Sequence And Series Question 378
Question: The value of x in $ (0,\pi ) $ which satisfy the equation $ {8^{1+| \cos ,x |+xos^{2}x+| {{\cos }^{3}}x |+…….to\infty ,}}=4^{3} $ is
Options:
A) $ { \frac{\pi }{2},\frac{3\pi }{4} } $
B) $ { \frac{\pi }{4},\frac{3\pi }{4} } $
C) $ { \frac{\pi }{3},\frac{2\pi }{3} } $
D) $ { \frac{\pi }{6},\frac{5\pi }{6} } $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] We have $ {8^{1+|\cos x|+|\cos x{{|}^{2}}+|\cos x{{|}^{3}}+…………….to\infty }}=4^{3} $ $ [\because {{\cos }^{2}}x=|{{\cos }^{2}}x| $ also $ |{{\cos }^{n}}x|=|\cos x{{|}^{n}}] $
$ \Rightarrow {8^{\frac{1}{1-|\cos x|}}}=4^{3}\Rightarrow {2^{\frac{3}{1-|\cos x|}}}=2^{6} $
$ \therefore ,\frac{3}{1-|\cos x|}=6\Rightarrow 1-|\cos x|=\frac{1}{2} $
$ \therefore |\cos x|=\frac{1}{2}\Rightarrow \cos x=\pm \frac{1}{2} $
$ \therefore x=\frac{\pi }{3},\frac{2\pi }{3} $
BETA
