CAT 2020 Slot 1 QA Q6 Logarithms & Exponents

Question 6 of the Quantitative Ability (QA) section, CAT 2020 Slot 1, with CAT's official answer and a worked solution.

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

A
infinite
B
0
C
1
D
2

Answer: (B) 0 Logarithms & Exponents MCQ

Solution

The equation is \(2^x+2^{-x} = 2-(x-2)^2\).
By AM-GM, the left side is at least 2, with equality only at \(x=0\). The right side is at most 2, with equality only at \(x=2\).
Both sides can equal 2 only at the same x, which is impossible, so there are 0 solutions.