CAT 2022 Question Paper | Quant Slot 2

CAT 2022 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 17 : The number of integer solutions of the equation \(\left(x^2-10\right)^{\left(x^2-3 x-10\right)}=1\) is

Video Explanation

Explanatory Answer

We have \( \left( x ^ { 2 } - 10 \right) ^ { \left( x ^ { 2 } - 3 x - 10 \right) } = 1 \).
Whenever \( a ^ { b } = 1 \).
It could be because of one of the three cases…
Case 1)
b = 0
\( x ^ { 2 } - 3 x - 10 = 0 \)
x = 5 or x = -2
Case 2)
a = 1
\( x ^ { 2 } - 10 = 1 \)
\( x ^ { 2 } = 11 \)
x is not an integer.
Case 3)
a = -1 & b is even.
\( x ^ { 2 } - 10 = - 1 \)
\( x = \pm 3 \)
For both values of x, \( x = \pm 3 \)
b = \( x ^ { 2 } - 3 x - 10 \) is even.
Therefore x can be 5, -2, +3 or -3.

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

Hence, the answer is '4'