The correct match of the group reagents in List-I for precipitating the metal ion given in List-II from solutions is:
| List-I | List-II |
|---|---|
| (P) Passing H2S in the presence of NH4OH | (1) Cu2+ |
| (Q) (NH4)2CO3 in the presence of NH4OH | (2) Al3+ |
| (R) NH4OH in the presence of NH4Cl | (3) Mn2+ |
| (S) Passing H2S in the presence of dilute HCl | (4) Ba2+ (5) Mg2+ |
P → 5; Q → 3; R → 2; S → 4
(P) H\(_2\)S + NH\(_4\)OH → Precipitates Group IV cations
Group IV cations include Mn\(^{2+}\), Zn\(^{2+}\), Co\(^{2+}\), etc., which precipitate as sulfides in basic medium. \[ \Rightarrow \text{P} \rightarrow \boxed{3} \quad \text{(Mn}^{2+}\text{)} \] (Q) (NH\(_4\))\(_2\)CO\(_3\) + NH\(_4\)OH → Precipitates Group V cations
This group includes alkaline earth metals like Ba\(^{2+}\), Ca\(^{2+}\), Sr\(^{2+}\), which form carbonates. \[ \Rightarrow \text{Q} \rightarrow \boxed{4} \quad \text{(Ba}^{2+}\text{)} \] (R) NH\(_4\)OH + NH\(_4\)Cl → Precipitates Group III cations
Group III cations like Al\(^{3+}\), Cr\(^{3+}\), Fe\(^{3+}\) form hydroxides in weakly basic conditions. \[ \Rightarrow \text{R} \rightarrow \boxed{2} \quad \text{(Al}^{3+}\text{)} \] (S) H\(_2\)S + dilute HCl → Precipitates Group II cations
Group II includes Cu\(^{2+}\), Pb\(^{2+}\), Hg\(^{2+}\) which form sulfides in acidic medium. \[ \Rightarrow \text{S} \rightarrow \boxed{1} \quad \text{(Cu}^{2+}\text{)} \]
Final Answer: \( \boxed{\text{A}} \)
Among the following cations, the number of cations which will give characteristic precipitate in their identification tests with
\(K_4\)[Fe(CN)\(_6\)] is : \[ {Cu}^{2+}, \, {Fe}^{3+}, \, {Ba}^{2+}, \, {Ca}^{2+}, \, {NH}_4^+, \, {Mg}^{2+}, \, {Zn}^{2+} \]
Match List I with List II:
Choose the correct answer from the options given below:
As shown in the figures, a uniform rod $ OO' $ of length $ l $ is hinged at the point $ O $ and held in place vertically between two walls using two massless springs of the same spring constant. The springs are connected at the midpoint and at the top-end $ (O') $ of the rod, as shown in Fig. 1, and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $ f_1 $. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2, and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $ f_2 $. Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\frac{f_1}{f_2}$ is:
The reaction sequence given below is carried out with 16 moles of X. The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of S produced is ____. 
Use: Atomic mass (in amu): H = 1, C = 12, O = 16, Br = 80
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is