At low pressure,the van der Waals equation becomes:

The equation $\sqrt{x + 1} - \sqrt{x - 1} = \sqrt{4x - 1}$ for $x \in R$ has:

The value of $c$ in the Lagrange's mean value theorem for $f(x)=\sqrt{x-2}$ in the interval $[2,6]$ is

If $x^y=y^x$, then $x(x-y \log x) \frac{d y}{d x}$ is equal to :

The condition for the coaxial system $x^2+y^2+2\lambda x+c=0$,where $\lambda$ is a parameter and $c$ is a constant,to have distinct limiting points,is

$A$ uniform metal rod of $2\,mm^2$ cross-section fixed between two walls is heated from $0\,^{\circ}C$ to $20\,^{\circ}C$. The coefficient of linear expansion of the rod is $12 \times 10^{-6}\,/^{\circ}C$. Its Young's modulus of elasticity is $10^{11}\,N/m^2$. The energy stored per unit volume of the rod will be ....... $J/m^3$.