If $f(x) = \left|\begin{array}{ccc} 1 & x & x+1 \\ 2x & x(x-1) & x(x+1) \\ 3x(x-1) & x(x-1)(x-2) & (x-1)x(x+1) \end{array}\right|$,then $f(2012)$ is equal to:

$\int \left( \frac{2 - \sin 2x}{1 - \cos 2x} \right) e^x \, dx$ is equal to

Identify the plant shown in the given figure.

The range of the function $f(x)=-\sqrt{-x^2-6x-5}$ is

Molar volume $( V_m )$ of a van der Waals gas can be calculated by expressing the van der Waals equation as a cubic equation with $V_m$ as the variable. The ratio $($in $mol dm ^{-3} )$ of the coefficient of $V_m^2$ to the coefficient of $V_m$ for a gas having van der Waals constants $a=6.0 dm ^6 atm \ mol ^{-2}$ and $b=0.060 dm ^3 mol^{-1}$ at $300 K$ and $300 atm$ is $....$ .
Use: Universal gas constant $(R)=0.082 dm ^3 atm \ mol { }^{-1} K^{-1}$

For the parabola $y^2+6y-2x+5=0$:
$(I)$ The vertex is $(-2,-3)$
$(II)$ The directrix is $y+3=0$
Which of the following is correct?