$A$ crystal is made up of metal ions $M_1$ and $M_2$ and oxide ions. Oxide ions form a $ccp$ lattice structure. The cation $M_1$ occupies $50 \%$ of octahedral voids and the cation $M_2$ occupies $12.5 \%$ of tetrahedral voids of the oxide lattice. The oxidation numbers of $M_1$ and $M_2$ are,respectively:

If $f(x) = \frac{\cos^2 x + \sin^4 x}{\sin^2 x + \cos^4 x}$, for $x \in R$, then $f(2002)$ is equal to

A prism of angle $75^\circ$ and refractive index $\sqrt{3}$ is coated with thin film of refractive index $1.5$ only at the back exit surface. To have total internal reflection at the back exit surface the incident angle must be . . . . . . .
$(\sin 15^\circ = 0.25$ and $\sin 25^\circ = 0.43)$

$A$ rectangular loop of length $l$ and breadth $b$ is placed at a distance of $x$ from an infinitely long wire carrying current $i$ such that the direction of the current is parallel to the breadth. If the loop moves away from the current-carrying wire in a direction perpendicular to it with a velocity $v$,the magnitude of the induced emf in the loop is: ($\mu_0=$ permeability of free space)

The condition that $f(x)=a x^3+b x^2+c x+d$ has no extreme value, is

If $\frac{3x+2}{(x+1)(2x^2+3)} = \frac{A}{x+1} + \frac{Bx+C}{2x^2+3}$, then $A+C-B$ is equal to :