Step 1: Identify compound A
The first reaction is:
A ⟶HNO₃ Pb(NO₃)₂
This indicates that compound A reacts with nitric acid to form lead nitrate. Hence, A must be metallic lead (Pb).
So, A = Pb
Step 2: Identify compound B
Pb(NO₃)₂ reacts with H₂SO₄ to give compound B. The reaction is: \[ \text{Pb(NO}_3)_2 + \text{H}_2\text{SO}_4 \rightarrow \text{PbSO}_4 \downarrow + 2\text{HNO}_3 \] PbSO₄ is a white precipitate formed by double displacement.
So, B = PbSO₄
Step 3: Identify compound C (Yellow precipitate)
Compound B is then treated to give a yellow precipitate C. The only classic yellow precipitate with lead is lead chromate (PbCrO₄), which is formed when Pb²⁺ reacts with chromate ions: \[ \text{Pb}^{2+} + \text{CrO}_4^{2-} \rightarrow \text{PbCrO}_4 \downarrow \, (\text{yellow ppt}) \] This confirms C is lead chromate.
So, C = PbCrO₄
Final Identifications:
A = Pb, B = PbSO₄, C = PbCrO₄
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:
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is: