Q3
The number of distinct integers \(n\) for which \(\log _{\left(\frac{1}{4}\right)}\left(n^2-7 n+11\right)>0\), is
Answer: B
Key idea: For a base between 0 and 1, \(\log_b y>0\) exactly when \(0<y<1\).
- We need \(0<n^2-7n+11<1\).
- Upper bound: \(n^2-7n+10<0 \Rightarrow (n-2)(n-5)<0 \Rightarrow 2<n<5\), so \(n=3\) or \(4\).
- Check the lower bound: \(n=3\) gives \(9-21+11=-1\); \(n=4\) gives \(16-28+11=-1\). Both are negative, so the logarithm is not even defined.
- No integer satisfies both conditions.
Tip: The quadratic has its minimum at \(n=3.5\) with value \(-0.25\), and it is negative for all \(n\) in \((2.38, 4.62)\), which swallows every candidate.
Hence, the answer is (B) 0.