An ideal gas is taken around the cycle $ABCA$ as shown in the $P-V$ diagram. The work done during the cycle is

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=x-[x]$ and $g(x)=[x]$ for $x \in R$,where $[x]$ is the greatest integer not exceeding $x$,then for every $x \in R, f(g(x))$ is equal to

If the line $lx + my = 1$ is a normal to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,then $\frac{a^2}{l^2} - \frac{b^2}{m^2}$ is equal to

In a metre bridge experiment,the ratio of the left gap resistance to the right gap resistance is $2:3$. The balance point from the left is: (in $~cm$)

The horizontal and vertical displacements of a projectile at time $t$ are $x=36 t$ and $y=48 t-4.9 t^2$,respectively. Initial velocity of the projectile in $m/s$ is

What is $Z$ in the following reaction? $CH_3-CH_2-CO_2^{\ominus} Na^{\oplus} \stackrel{NaOH / CaO}{\longrightarrow} Z$