CAT 2025 Slot 1 QA Q3 Logarithms & Exponents

Question 3 of the Quantitative Ability (QA) section, CAT 2025 Slot 1, with CAT's official answer and a worked solution.

The number of distinct integers \(n\) for which \(\log _{\left(\frac{1}{4}\right)}\left(n^2-7 n+11\right)>0\), is

A
infinite
B
0
C
2
D
1

Answer: (B) 0 Logarithms & Exponents MCQ

Solution

With base \(\frac14 < 1\), \(\log_{1/4} y > 0\) exactly when \(0 < y < 1\).
\(y < 1\): \(n^2 - 7n + 10 < 0 \Rightarrow (n-2)(n-5) < 0 \Rightarrow n = 3\) or \(4\).
But \(y(3) = 9 - 21 + 11 = -1\) and \(y(4) = 16 - 28 + 11 = -1\), both negative, so the log is undefined.
No integer works: the answer is 0.

Shortcut. Check that the argument is positive for the candidates before counting them.