Question : In $\triangle \mathrm{ABC}$, $\angle \mathrm{ABC} = 90^{\circ}$, $\mathrm{BP}$ is drawn perpendicular to $\mathrm{AC}$. If $\angle \mathrm{BAP} = 50^{\circ},$ what is the value of $\angle \mathrm{PBC}?$

Option 1: $30^{\circ}$

Option 2: $45^{\circ}$

Option 3: $50^{\circ}$

Option 4: $60^{\circ}$

Question : $\triangle \mathrm{XYZ} \sim \triangle \mathrm{GST}$ and $\mathrm{XY}: \mathrm{GS}=2: 3, \mathrm{XV}$ is the median to the side $\mathrm{YZ}$, and $\mathrm{GD}$ is the median to the side ST. The value of $\left(\frac{\mathrm{YV}}{\mathrm{SD}}\right)^2$ is:

Option 1: $\frac{4}{9}$

Option 2: $\frac{3}{5}$

Option 3: $\frac{1}{4}$

Option 4: $\frac{2}{3}$

Question : In $\triangle X Y Z, \angle X=90^{\circ}, YZ=15 \mathrm{~cm}$, and $X Z=12 \mathrm{~cm}$. Then find $\cos Y$.

Option 1: $\frac{2}{5}$

Option 2: $\frac{4}{5}$

Option 3: $\frac{3}{5}$

Option 4: $\frac{3}{4}$

Question : If $\triangle \mathrm{BPQ} \cong \triangle \mathrm{ASR}$ and $\angle \mathrm{A}=\frac{1}{3} \angle \mathrm{R}=\angle \mathrm{S}$, then find $\angle \mathrm{Q}$. (All angles are in degrees).

Option 1: 108°

Option 2: 36°

Option 3: 72°

Option 4: 118°

Question : $\triangle \mathrm{PQR}$ is an equilateral triangle inscribed in a circle. $\mathrm{S}$ is any point on the arc $\mathrm{QR}$. Find the measure of $\angle \mathrm{PSQ}$.

Option 1: $30^{\circ}$

Option 2: $60^{\circ}$

Option 3: $90^{\circ}$

Option 4: $45^{\circ}$