Which of the following compounds is consumed during the preparation of $Na_2CO_3$ by Solvay's process?

If the point $P(1,3)$ undergoes the following transformations successively:
$(i)$ Reflection with respect to the line $y=x$.
(ii) Translation through $3$ units along the positive direction of the $X$-axis.
(iii) Rotation through an angle of $\frac{\pi}{6}$ about the origin in the clockwise direction.
Then,the final position of the point $P$ is

$A$ pair of perpendicular lines passes through the origin and also through the points of intersection of the curve $x^2+y^2=4$ with $x+y=a$,where $a>0$. Then $a$ is equal to

At $T \text{ K}$,the ratio of kinetic energies of $4 \text{ g}$ of $H_{2(g)}$ and $8 \text{ g}$ of $O_{2(g)}$ is

If $\cos ^{-1}\left(\frac{x^2-y^2}{x^2+y^2}\right)=k$ (a constant),then $\frac{d y}{d x}$ is equal to

The tension of a stretched string is increased by $69\%$. In order to keep its frequency of vibration constant,its length must be increased by ..... $\%$