LIST-I | LIST-II | ||
---|---|---|---|
I | \(\text{H}_2\text{O}_2\) | P | \(\text{Mg(HCO}_3\text{)}_2 + \text{Ca(OH)}_2 \rightarrow \) |
II | \(\mathrm{Mg(OH)_2}\) | Q | \(\text{BaO}_2 + \text{H}_2\text{SO}_4 \rightarrow \) |
III | \(\text{BaCl}_2\) | R | \(\text{Ca(OH)}_2 + \text{MgCl}_2 \) |
IV | \(\text{CaCO}_3\) | S | \(\text{BaO}_2 + 2\text{HCl} \rightarrow\) |
T | \(\text{Ca(HCO}_3\text{)}_2 + 2\text{Ca(OH)}_2 \rightarrow \) |
I → Q; II → P; III → S; IV → R
I → T; II → P; III → Q; IV → R
I → T; II → R; III → Q; IV → P
I → Q; II → R; III → S; IV → P
The formation of hydrogen peroxide (\( \text{H}_2\text{O}_2 \)) happens by the reaction between \( \text{BaO}_2 \) and \( \text{H}_2\text{SO}_4 \).
This corresponds to reaction Q.
The formation of magnesium hydroxide (\( \text{Mg(OH)}_2 \)) occurs when \( \text{Ca(OH)}_2 \) reacts with \( \text{MgCl}_2 \).
This corresponds to reaction R.
Barium chloride (\( \text{BaCl}_2 \)) reacts with hydrochloric acid to form barium peroxide (\( \text{BaO}_2 \)), which corresponds to reaction S.
Calcium carbonate (\( \text{CaCO}_3 \)) is formed when \( \text{Mg(HCO}_3\text{)}_2 \) reacts with \( \text{Ca(OH)}_2 \), corresponding to reaction P.
The correct matches are:
The correct option is A: I → Q; II → R; III → S; IV → P.
Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
Then the value of $ pq $ is
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.