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CAT 2020 Slot 1 QA Question & Solution

AlgebraMedium

Question

The number of real-valued solutions of the equation $2^{x}+2^{-x}=2-(x-2)^{2}$ is:

Options

1
2
infinite
0

Solution

Question Figure

The graphs of $2^{x}+2^{-x} \ \ and \ \ 2-(x-2)^{2}$ never intersect. So, number of solutions = $0$.

Alternate method:

We notice that the minimum value of the term in the LHS will be greater than or equal to 2 {at x=0; LHS = 2}.

However, the term in the RHS is less than or equal to 2 {at x=2; RHS = 2}. The values of x at which both the sides become 2 are distinct; hence, there are zero real-valued solutions to the above equation.