CAT 2025 Slot 1 · Quant · Question 3
CAT 2025 Slot 1 previous year question on Logarithms, with the answer and a step-by-step solution.
The number of distinct integers n for which {\log}_{\frac{1}{4}}\left({n}^{2}-7n+11 \right)>0, is
- Ainfinite
- B0
- C2
- D1
Pick an option, then check.
Answer & solution
Answer: Option B
A logarithm carries two conditions, and this question turns on remembering both.
The base is \frac{1}{4}, which is less than 1, so the logarithm is decreasing and the inequality flips when we remove it: {n}^{2}-7n+11<{\left( \frac{1}{4}\right)}^{0}=1.
That gives {n}^{2}-7n+10<0, so \left( n-2 \right)\left( n-5 \right)<0 and 2<n<5.
The integers are n=3 and n=4.
But the argument of a logarithm must also be positive.
At n=3: 9-21+11=-1.
At n=4: 16-28+11=-1.
Both are negative, so neither is admissible.
No integer works, and the answer is 0.
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