Bayer's process is used for the purification of Bauxite containing which of the following as a major impurity?

The following diagram indicates the energy levels of a certain atom. When the system moves from the $4E$ level to the $E$ level, a photon of wavelength $\lambda_1$ is emitted. The wavelength of the photon produced during its transition from the $\frac{7}{3}E$ level to the $E$ level is $\lambda_2$. The ratio $\frac{\lambda_1}{\lambda_2}$ will be:

What is the cause of genetic erosion?

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

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

The period of $\sin ^4 x + \cos ^4 x$ is

If $R$ is a relation from a finite set $A$ having $m$ elements to a finite set $B$ having $n$ elements,then the number of relations from $A$ to $B$ is