Find the length of the intercept cut by the pair of lines $2x^2+4xy-4y^2-6x-8y+7=0$ on the $Y$-axis.

Match the following columns and select the correct option:

$(a)$ Clostridium butylicum	$(i)$ Cyclosporin-$A$
$(b)$ Trichoderma polysporum	$(ii)$ Butyric acid
$(c)$ Monascus purpureus	$(iii)$ Citric acid
$(d)$ Aspergillus niger	$(iv)$ Blood cholesterol lowering agent

$(a)\quad (b)\quad (c)\quad (d)$

$\begin{aligned} & A(\alpha, \beta)=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & e^\beta\end{array}\right] \\ & \Rightarrow[A(\alpha, \beta)]^{-1}=\end{aligned}$

The activation energy for a reaction at the temperature $T \ K$ was found to be $2.303 \ RT \ J \ mol^{-1}$. The ratio of the rate constant to Arrhenius factor is

The product of the lengths of perpendiculars drawn from any point on the hyperbola $x^2 - 2y^2 - 2 = 0$ to its asymptotes is:

$\frac{\sqrt{5/2} + \sqrt{7 - 3\sqrt{5}}}{\sqrt{7/2} + \sqrt{16 - 5\sqrt{7}}} = ?$