CAT 2020 Slot 1 QA Q15 Equations

Question 15 of the Quantitative Ability (QA) section, CAT 2020 Slot 1, with CAT's official answer and a worked solution.

How many distinct positive integer-valued solutions exist to the equation (x² - 7x + 11)^{(x² - 13x + 42)} = 1?

A
6
B
2
C
4
D
8

Answer: (A) 6 Equations MCQ

Solution

The equation, as set in CAT 2020, is \((x^2-7x+11)^{x^2-13x+42}=1\) (the second bracket is the exponent).
Exponent 0: \(x^2-13x+42=0 \Rightarrow x=6, 7\).
Base 1: \(x^2-7x+10=0 \Rightarrow x=2, 5\).
Base \(-1\) with even exponent: \(x^2-7x+12=0 \Rightarrow x=3, 4\); exponents are 12 and 6, both even, so both work.
Total: 6 positive integer solutions.