CAT 2025 Slot 1 QA Question 17: The number of distinct integers $n$ for which $\log_{\frac{1}{4}}(n^2 - 7n + 11) 0$, is

Question

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

Accepted Answer

Let $A = n^{2} - 7n + 11$. Since the base $\dfrac{1}{4}$ is between 0 and 1,
$\log_{\frac{1}{4}} A > 0$
$0

First use the upper bound.
$n^{2} - 7n + 11 
$n^{2} - 7n + 10 
$(n - 2)(n - 5) 
$2 
So the only possible integer values are $n = 3$ and $n = 4$.

$n = 3: A = 9 - 21 + 11 = 2$
$n = 4: A = 16 -

The number of distinct integers nnn for which log⁡14(n2−7n+11)>0\log_{\frac{1}{4}}(n^2 - 7n + 11) > 0log41​​(n2−7n+11)>0, is