Step 1: Definition of Aldehydes and Ketones - Aldehydes (\(-CHO\)) have a hydrogen attached to the carbonyl group.
- Ketones (\(R-CO-R'\)) have two alkyl groups attached to the carbonyl.
Step 2: Distinguishing Tests (1) Tollens' Test: Silver Mirror Test - Principle: Aldehydes reduce Tollens' reagent (\( AgNO_3 + NH_3 + H_2O \)) to form a silver mirror. Ketones do not react. \[ R-CHO + 2[Ag(NH_3)_2]^+ + H_2O \rightarrow R-COOH + 2Ag + 4NH_3 \] - Observation: A silver mirror appears inside the test tube for aldehydes. No reaction occurs with ketones. \vspace{0.3cm}
(2) Fehling's Test - Principle: Aldehydes reduce Fehling's solution (Cu\(^{2+}\) in alkaline medium) to red Cu\(_2\)O precipitate. Ketones do not react. \[ R-CHO + 2Cu^{2+} + 5OH^- \rightarrow R-COO^- + Cu_2O + 3H_2O \] - Observation: A brick-red precipitate forms for aldehydes. No reaction for ketones.
Find the values of \( x, y, z \) if the matrix \( A \) satisfies the equation \( A^T A = I \), where
\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]