The unbalanced chemical reactions given in List I show missing reagent or condition (?) which are provided in List II. Match List I with List II and select the correct answer using the code given below the lists :
| List I | List II | ||
|---|---|---|---|
| P. | $$Pb{O_2} + {H_2}S{O_4}\buildrel ? \over \longrightarrow PbS{O_4} + {O_2} + Other\,products$$ |
1. | NO |
| Q. | $$N{a_2}{S_2}{O_3} + {H_2}O\buildrel ? \over \longrightarrow NaHS{O_4} + Other\,products$$ |
2. | $${I_2}$$ |
| R. | $${N_2}{H_4}\buildrel ? \over \longrightarrow {N_2} + Other\,products$$ |
3. | Warm |
| S. | $$Xe{F_2}\buildrel ? \over \longrightarrow Xe + Other\,products$$ |
4. | $$C{l_2}$$ |
Match the chemical conversions in List I with the appropriate reagents in List II and select the correct answer using the code given below the lists :
$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ List $$I$$
(P.)$$\,\,\,\,$$ Volume of parallelopiped determined by vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ is $$2.$$ Then the volume of the parallelepiped determined by vectors $$2\left( {\overrightarrow a \times \overrightarrow b } \right),3\left( {\overrightarrow b \times \overrightarrow c } \right)$$ and $$\left( {\overrightarrow c \times \overrightarrow a } \right)$$ is
(Q.)$$\,\,\,\,$$ Volume of parallelopiped determined by vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ is $$5.$$ Then the volume of the parallelepiped determined by vectors $$3\left( {\overrightarrow a + \overrightarrow b } \right),\left( {\overrightarrow b + \overrightarrow c } \right)$$ and $$2\left( {\overrightarrow c + \overrightarrow a } \right)$$ is
(R.)$$\,\,\,\,$$ Area of a triangle with adjacent sides determined by vectors $${\overrightarrow a }$$ and $${\overrightarrow b }$$ is $$20.$$ Then the area of the triangle with adjacent sides determined by vectors $$\left( {2\overrightarrow a + 3\overrightarrow b } \right)$$ and $$\left( {\overrightarrow a - \overrightarrow b } \right)$$ is
(S.)$$\,\,\,\,$$ Area of a parallelogram with adjacent sides determined by vectors $${\overrightarrow a }$$ and $${\overrightarrow b }$$ is $$30.$$ Then the area of the parallelogram with adjacent sides determined by vectors $$\left( {\overrightarrow a + \overrightarrow b } \right)$$ and $${\overrightarrow a }$$ is
$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ List $$II$$
(1.)$$\,\,\,\,$$ $$100$$
(2.)$$\,\,\,\,$$ $$30$$
(3.)$$\,\,\,\,$$ $$24$$
(4.)$$\,\,\,\,$$ $$60$$
Let $\omega=\frac{\sqrt{3}+i}{2}$ and $P=\left\{\omega^n: n=1,2,3, \ldots\right\}$. Further
$\mathrm{H}_1=\left\{z \in \mathrm{C}: \operatorname{Re} z<\frac{1}{2}\right\}$ and
$\mathrm{H}_2=\left\{z \in \mathrm{C}: \operatorname{Re} z<\frac{-1}{2}\right\}$, where C is the
set of all complex numbers. If $z_1 \in \mathrm{P} \cap \mathrm{H}_1, z_2 \in$ $\mathrm{P} \cap \mathrm{H}_2$ and O
represents the origin, then $\angle z_1 \mathrm{O} z_2=$