If the pair of lines $2x^2 + 3xy + y^2 = 0$ makes angles $\theta_1$ and $\theta_2$ with the positive direction of the $X$-axis,then $|\tan(\theta_1 - \theta_2)| = $

In a reaction,$A + B \rightleftharpoons C + D$,$40 \%$ of $B$ has reacted at equilibrium,when $1 \ mole$ of $A$ was heated with $1 \ mole$ of $B$ in a $10 \ L$ closed vessel. The value of $K_C$ is

The partial fraction of $\frac{6x^4 + 5x^3 + x^2 + 5x + 2}{1 + 5x + 6x^2}$ is:

Steam at $100^{\circ} C$ is passed into $1 \ kg$ of water contained in a calorimeter of water equivalent $0.2 \ kg$ at $9^{\circ} C$ until the temperature of the calorimeter and water in it increases to $90^{\circ} C$. The mass of steam condensed in $kg$ is nearly (specific heat of water $= 1 \ cal/g^{\circ} C$,latent heat of vaporisation $= 540 \ cal/g$)

If $f(x) = \frac{x}{1+|x|}$, for $x \in \mathbb{R}$, then $f^{\prime}(0)$ is equal to :

For the hyperbola $\frac{x^2}{\cos^2 \alpha} - \frac{y^2}{\sin^2 \alpha} = 1$,which of the following remains constant if $\alpha$ varies?