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CAT 2025Slot 1QAQues & Sol

AlgebraHard

Question

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

Options

Solution

1. Concept Used

  • Topic: Logarithmic Inequalities — Properties of logarithms with base between 0 and 1
  • Formula: $\log_{b}(x) > 0 \implies 0 < x < 1 \quad \text{when} \quad 0 < b < 1$

2. Calculation

We are asked to find the number of distinct integers $n$ for which $\log_{\frac{1}{4}}(n^2 - 7n + 11) > 0$.

Step 1: Understand the base. The base is $\frac{1}{4}$, which lies in the interval $(0, 1)$. For logarithms with a base in $(0,1)$, the logarithm is positive only when the argument is strictly between 0 and 1. This is because the function $\log_{1/4}(x)$ is a decreasing function — it outputs a positive value only when $x < 1$, and the argument must still be positive for the log to be defined.

So the condition $\log_{\frac{1}{4}}(n^2 - 7n + 11) > 0$ requires: $0 < n^2 - 7n + 11 < 1$

Step 2: Analyze the expression $x = n^2 - 7n + 11$ for integer $n$. Since $n$ is an integer, the expression $n^2 - 7n + 11$ is always an integer (as it is a polynomial with integer coefficients evaluated at an integer).

Step 3: Check if an integer can lie strictly between 0 and 1. For the inequality to hold, we need: $0 < n^2 - 7n + 11 < 1$ But there is no integer that lies strictly between 0 and 1. The only integers satisfying $0 < k < 1$ would require a non-integer value — which is impossible.

Therefore, no integer value of $n$ can satisfy the given inequality.

Verification with a few values:

  • $n = 3$: $9 - 21 + 11 = -1$ → negative, log undefined
  • $n = 4$: $16 - 28 + 11 = -1$ → negative, log undefined
  • $n = 1$: $1 - 7 + 11 = 5$ → $\log_{1/4}(5) < 0$, not $> 0$
  • $n = 6$: $36 - 42 + 11 = 5$ → $\log_{1/4}(5) < 0$, not $> 0$

No integer value of $n$ makes the expression fall in $(0, 1)$, confirming zero valid integers.

3. Solution

Answer = Option D ✅

The final calculated value is 0. There are no distinct integers $n$ for which $\log_{\frac{1}{4}}(n^2 - 7n + 11) > 0$, since the expression $n^2 - 7n + 11$ is always an integer for integer $n$, and no integer lies strictly between 0 and 1.