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CAT 2020 Quant Question [Slot 1] with Solution 07

Question

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

  1. infinite
  2. 1
  3. 0
  4. 2
Option: 3
Solution:

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

The minimum value of \({2^x} + {2^{ - x}}\) is 2 when \(x = 0\)

But \(x = 0;2 - {(x - 2)^2} =  - 2\)

The maximum value of \(2 - {(x - 2)^2}\) is 2 when \(x = 2\)

But \(x = 2\quad {2^x} + {2^{ - x}} = \frac{{17}}{4}\)

Hence there is no value of \(x,{2^x} + {2^{ - x}} = 2 - {(x - 2)^2}\)

The number of solutions is 0

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