There is a rectangular sheet of dimension $(2m-1) \times (2n-1)$,where $m > 0, n > 0$. It has been divided into squares of unit area by drawing lines perpendicular to the sides. Find the number of rectangles having sides of odd unit length.

$A$ monoatomic ideal gas initially at temperature $T_1$ is enclosed in a cylinder fitted with a massless,frictionless piston. By releasing the piston suddenly,the gas is allowed to expand adiabatically to a temperature $T_2$. If $L_1$ and $L_2$ are the lengths of the gas columns before and after expansion respectively,then $\frac{T_2}{T_1}$ is

$A$ mass $M$ attached to a horizontal spring executes $S.H.M.$ of amplitude $A_1$. When the mass $M$ passes through its mean position,a smaller mass $m$ is placed over it and both of them move together with amplitude $A_2$. The ratio $\left(\frac{A_1}{A_2}\right)$ is

The solution of the differential equation $(3xy + y^2)dx + (x^2 + xy)dy = 0$ is

The area (in square units) of the region bounded by $x^2=8y$,$x=4$ and the $X$-axis is:

Match the following components of a cell with their respective percentage of the total cellular mass:

Components	$\%$ of the total cellular mass
$A$. Water	$1$. $10-15$
$B$. Proteins	$2$. $70-90$
$C$. Carbohydrates	$3$. $5-7$
$D$. Lipids	$4$. $3$
$E$. Nucleic acids	$5$. $2$
$F$. Ions	$6$. $1$