Choose the correct relation: (i) \(\sqrt{10} < 3.2\) (ii) \(\sqrt{11} < \sqrt{12}\) (iii) \(\sqrt{13} < \sqrt{14}\)
(i), (ii) and (iii)
Check each relation independently.
(i) Compare \(\sqrt{10}\) with \(3.2\) by squaring both sides (both positive):
\((\sqrt{10})^2 = 10,\quad (3.2)^2 = 10.24\)
Since \(10 < 10.24\), we have \(\sqrt{10} < 3.2\). True.
(ii) & (iii) The square-root function is strictly increasing for non-negative numbers. So \(11 < 12\) gives \(\sqrt{11} < \sqrt{12}\), and \(13 < 14\) gives \(\sqrt{13} < \sqrt{14}\). Both true.
Hence all three relations are correct — option (4).
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