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CAT 2020 Quant Question [Slot 1] with Solution 09

Question

How many distinct positive integer-valued solutions exist to the equation ${\left( {{x^2} - 7x + 11} \right)^{\left( {{x^2} - 13x + 42} \right)}} = 1?$

1. 6
2. 8
3. 2
4. 4
Option: 1
Solution:

${\left( {{x^2} - 7x + 11} \right)^{\left( {{x^2} - 13x + 42} \right)}} = 1$

We know if ${a^b} = 1$

$\Rightarrow a = 1$ and b is any number

or $a = - 1$ and b is even

$a > 0$ and  b is 0

case $1:{x^2} - 13x + 42 = 0 \Rightarrow x = 6,7$

case $2:{x^2} - 7x + 11 = 1 \Rightarrow {x^2} - 7x + 10 = 0 \Rightarrow$ x=2 or 5

case $3:{x^2} - 7x + 11 = - 1 \Rightarrow {x^2} - 7x + 12 = 0$

$\Rightarrow$ x=4 or 3

Hence number of solutions are 6

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