cat 2022 Question 17

Question

The number of integer solutions of the equation $\left(x^2-10\right)^{\left(x^2-3 x-10\right)}=1$ is

Accepted Answer

Case 1: When $x^2$ − 3x − 10 = 0 and $x^2$ − 10 ≠ 0

$x^2$ − 3x − 10 = 0 or, (x−5) (x+2) = 0

or, x= 5 or -2

Case 2: $x^2$ − 10 = 1

$x^2$ − 11 = 0

No integer solutions

Case 3: $x^2$ − 10 = −1 and $x^2$ − 3x − 10 is even

$x^2$ − 9 = 0

or, (x+3)(x-3) = 0

or, x = -3 and 3

for x = -3 and +3$x^2$ − 3x − 10 is even

In total 4 values of x satisfy the equations

Question 17.