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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