To determine the number of ions that can liberate hydrogen from a dilute acid, we need to analyze their reducing abilities. Strong reducing agents can donate electrons to \(\text{H}^+\), reducing it to \(\text{H}_2\).
Step 1: Evaluate \(\text{Ti}^{2+}\)
Titanium(II) (\(\text{Ti}^{2+}\)) has a strong tendency to get oxidized to \(\text{Ti}^{3+}\), making it a strong reducing agent.
\(\text{Ti}^{2+}\) can react with \(\text{H}^+\) from dilute acids to liberate \(\text{H}_2\).
Step 2: Evaluate \(\text{Cr}^{2+}\)
Chromium(II) (\(\text{Cr}^{2+}\)) is a strong reducing agent and can be oxidized to \(\text{Cr}^{3+}\).
\(\text{Cr}^{2+}\) reacts with \(\text{H}^+\) from dilute acids to liberate \(\text{H}_2\):
\[2\text{Cr}^{2+} (\text{aq}) + 2\text{H}^+ (\text{aq}) \rightarrow 2\text{Cr}^{3+} (\text{aq}) + \text{H}_2 (\text{g}).\]
Step 3: Evaluate \(\text{V}^{2+}\)
Vanadium(II) (\(\text{V}^{2+}\)) is also a strong reducing agent and can be oxidized to \(\text{V}^{3+}\).
\(\text{V}^{2+}\) reacts with \(\text{H}^+\) from dilute acids to liberate \(\text{H}_2\).
Conclusion:
All three ions (\(\text{Ti}^{2+}\), \(\text{Cr}^{2+}\), and \(\text{V}^{2+}\)) can liberate \(\text{H}_2\) from dilute acids.
Final Answer: (3).
Let \( S = \left\{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \right\} \), where
\[ A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \]Then \( n(S) \) is equal to ______.
Let \( T_r \) be the \( r^{\text{th}} \) term of an A.P. If for some \( m \), \( T_m = \dfrac{1}{25} \), \( T_{25} = \dfrac{1}{20} \), and \( \displaystyle\sum_{r=1}^{25} T_r = 13 \), then \( 5m \displaystyle\sum_{r=m}^{2m} T_r \) is equal to: