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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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