Identify $B$ in the following reaction:
$H_4SiO_4$ $\xrightarrow[1000^{\circ}C]{-2H_2O} A$ $\xrightarrow[\Delta]{\text{Carbon}} B + CO$

The ratio in which $\hat{i}+2 \hat{j}+3 \hat{k}$ divides the join of $-2 \hat{i}+3 \hat{j}+5 \hat{k}$ and $7 \hat{i}-\hat{k}$ is

If the lines $2x - 3y = 5$ and $3x - 4y = 7$ are two diameters of a circle of radius $7$,then the equation of the circle is

Match the following molecules in Column-$I$ with the number of lone pairs on their central atom in Column-$II$.

Column-$I$ (Molecules)	Column-$II$ (Number of lone pairs on central atom)
$A$. $NH_3$	$1$. Two
$B$. $H_2O$	$2$. Three
$C$. $XeF_2$	$3$. Zero
$D$. $CH_4$	$4$. Four
	$5$. One

The percentage composition of an organic compound $A$ is: carbon $85.71 \%$ and hydrogen $14.29 \%$. Its vapour density is $14$. Consider the following reaction sequence:
$A$ $\xrightarrow{Cl_2 / H_2O} B$ $\xrightarrow[(ii) H_3O^+]{(i) KCN / EtOH} C$
Identify $\underline{C}$.

If $\int \frac{\sin x}{\cos x(1+\cos x)} d x=f(x)+c$,then $f(x)$ is equal to