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3 g of activated charcoal was added to 50 mL of 0.06 N acetic acid. After filtration, the filtrate had a normality of 0.042 N. How much acetic acid was adsorbed per gram of charcoal?
Initial moles of acetic acid: $0.06 \times 0.050 = 3 \times 10^{-3}\ \text{mol}$
Remaining moles: $0.042 \times 0.050 = 2.1 \times 10^{-3}\ \text{mol}$
Adsorbed moles $= (3 - 2.1) \times 10^{-3} = 9 \times 10^{-4}\ \text{mol}$
Mass adsorbed (M of AcOH $= 60$) $= 9 \times 10^{-4} \times 60 = 0.054\ \text{g} = 54\ \text{mg}$
Per gram of charcoal $= \dfrac{54}{3} = \mathbf{18\ \text{mg/g}}$
If $g(x) = \begin{cases}k\sqrt{x+1}, & 0\le x\le3\\ mx+2, & 3
For continuity at $x=3$: $k\sqrt{4}=3m+2\Rightarrow 2k=3m+2$ ...(i)
For differentiability at $x=3$: Left derivative $=\frac{k}{2\sqrt{x+1}}\big|_{x=3}=\frac{k}{4}$; Right derivative $=m$.
So $\frac{k}{4}=m$ ...(ii)
From (ii): $k=4m$. Substitute in (i): $8m=3m+2\Rightarrow5m=2\Rightarrow m=\frac{2}{5}$, $k=\frac{8}{5}$.
$k+m=\frac{8}{5}+\frac{2}{5}=\frac{10}{5}=\mathbf{2}$
So correct answer is $k+m=2$, which is option A (index 0). But official JEE answer is $\frac{10}{3}$... Re-checking: $5m=2 \Rightarrow m=2/5$. $k=4(2/5)=8/5$. $k+m = 8/5+2/5=2$. Answer is $\mathbf{2}$.
Identify the longest chain containing the double bond: 5 carbons (pent), with double bond at C2 (pent-2-ene).
Number from the end that gives the double bond the lower locant. A methyl group appears at C3 and Br at C4.
IUPAC name: 4-Bromo-3-methylpent-2-ene
This is a 5-carbon chain ($n = 5$: pent) with double bond at position 2 (C2=C3), methyl at C3, and Br at C4.
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