The limiting radius ratio for a tetrahedral void is:

Atom $X$ occupies the $fcc$ lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in $\%$) of the resultant solid is closest to

The anions $(A)$ of a $C_{p}A_{q}$ molecule form an $fcc$ lattice. Cations $(C)$ are positioned at the body center and half of the edge centers. The formula of the molecule is:

The packing efficiency of the face-centered cubic $(fcc)$,body-centered cubic $(bcc)$,and simple/primitive cubic $(pc)$ lattices follows the order:

In hexagonal systems of crystals,a frequently encountered arrangement of atoms is described as a hexagonal prism. Here,the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. $A$ space-filling model of this structure,called hexagonal close-packed $(HCP)$,is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of these three spheres touches three spheres of the bottom layer. Finally,the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be $r$.
$1.$ The number of atoms on this $HCP$ unit cell is
$(A)$ $4$ $(B)$ $6$ $(C)$ $12$ $(D)$ $17$
$2.$ The volume of this $HCP$ unit cell is
$(A)$ $24 \sqrt{2} r^3$ $(B)$ $16 \sqrt{2} r^3$
$(C)$ $12 \sqrt{2} r^3$ $(D)$ $\frac{64 r^3}{3 \sqrt{3}}$
$3.$ The empty space in this $HCP$ unit cell is
$(A)$ $74 \%$ $(B)$ $47.6 \%$ $(C)$ $32 \%$ $(D)$ $26 \%$
Give the answer for questions $1, 2$ and $3.$

In a $ccp$ structure of a mixed oxide,$1/8$ of the tetrahedral voids are occupied by divalent $X^{+2}$ ions and $50\%$ of the octahedral voids are occupied by trivalent $Y^{+3}$ ions. The formula of the oxide is: