CAT 2025 Slot 1 QA Q22 Inequalities & Modulus

Question 22 of the Quantitative Ability (QA) section, CAT 2025 Slot 1, with CAT's official answer and a worked solution.

Let \(3 \leq x \leq 6\) and \(\left[x^2\right]=[x]^2\), where \([x]\) is the greatest integer not exceeding \(x\). If set S represents all feasible values of \(x\), then a possible subset of S is
  1. A
    \((4, \sqrt{18}) \cup[5, \sqrt{27}) \cup\{6\}\)
  2. B
    \([3, \sqrt{10}] \cup[4, \sqrt{17}] \cup\{6\}\)
  3. C
    \([3, \sqrt{10}] \cup[5, \sqrt{26}]\)
  4. D
    \((3, \sqrt{10}) \cup[5, \sqrt{26}) \cup\{6\}\)

Answer: (D) (3, √(10)) [5, √(26)) 6 Inequalities & Modulus MCQ

Solution

  1. If \([x] = n\), we need \(n^2 \le x^2 < n^2 + 1\), that is \(n \le x < \sqrt{n^2 + 1}\).
  2. So \(S = [3, \sqrt{10}) \cup [4, \sqrt{17}) \cup [5, \sqrt{26}) \cup \{6\}\).
  3. Option D, \((3, \sqrt{10}) \cup [5, \sqrt{26}) \cup \{6\}\), lies inside \(S\). B and C include the excluded endpoints \(\sqrt{10}\) and \(\sqrt{26}\), and A includes values up to \(\sqrt{18}\), beyond \(\sqrt{17}\).

Shortcut. Right endpoints \(\sqrt{n^2+1}\) must be open: this eliminates B and C immediately.

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