Let’s analyze each statement individually:
Statement A: \( \text{Mn}_2\text{O}_7 \) is an oil at room temperature. - \( \text{Mn}_2\text{O}_7 \) (manganese heptoxide) is indeed a dark green oil that is highly unstable and explosive at room temperature. It is a covalent oxide and behaves as an acidic oxide due to its high oxidation state. Therefore, this statement is correct.
Statement B: \( \text{V}_2\text{O}_4 \) reacts with acid to give \( \text{VO}_2^{2+} \). - \( \text{V}_2\text{O}_4 \) (vanadium(IV) oxide) is an amphoteric oxide. It can react with acids, undergoing oxidation, to form the \( \text{VO}_2^{2+} \) ion (vanadyl ion). This reaction occurs as:
\[ \text{V}_2\text{O}_4 + 2\text{H}^+ \to 2\text{VO}_2^{2+} + \text{H}_2\text{O} \]
Therefore, this statement is also correct.
Statement C: \( \text{CrO} \) is a basic oxide. - \( \text{CrO} \) (chromium(II) oxide) is a basic oxide. It reacts readily with acids to form salts and water. For example:
\[ \text{CrO} + 2\text{HCl} \to \text{CrCl}_2 + \text{H}_2\text{O} \]
This confirms that \( \text{CrO} \) behaves as a basic oxide, so this statement is correct.
Statement D: \( \text{V}_2\text{O}_5 \) does not react with acid. - \( \text{V}_2\text{O}_5 \) (vanadium(V) oxide) is an amphoteric oxide, meaning it can react with both acids and bases. In fact, \( \text{V}_2\text{O}_5 \) can react with acidic solutions, such as hydrochloric acid, and undergo reduction. Therefore, this statement is incorrect.
Since statements A and C are correct, the correct answer is Option (2).
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: