Test Index

CAT 2019 Shift 1 Paper

Question : 84 of 100

Marks: +1, -0

2\cos (x(x+1)) = 2^{x} + 2^{-x}

Solution:

For any real value of

x

, the expression

2\cos (x(x+1))=2^{x}+2^{-x}

would always be positive.

Lets find the maximum value of

2\cos (x(x+1))=2^{x}+2^{-x}

Applying AM-GM inequality we have

\frac{2^{x}+2^{-x}}{2} \geq \sqrt{2^{x} \times 2^{-x}}

\Rightarrow 2^{x}+2^{-x} \geq 2 \sqrt{2^{0}}

\Rightarrow 2^{x}+2^{-x} \geq 2

Therefore,

2\cos (x(x+1)) \geq 2

It is known that

-1 \leq \cos \theta \leq 1

\Rightarrow 2\cos (x(x+1))=2

Hence, the expression is valid only if

2^{x}+2^{-x}=2,

which is true for only one value of

x

i.e.

0 .

Therefore, the expression has only one real solution.

Go to Question:

Prev Question

Next Question