Question : ABCD is a square. Draw an equilateral $\triangle $PBC on side BC considering BC is a base and an equilateral $\triangle $QAC on diagonal AC considering AC is a base. Find the value of $\frac{\text{area of $\triangle PBC$}}{\text{area of $\triangle QAC$}}$.

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

Option 2: $1$

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

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

Question : The altitude of an equilateral triangle of side $\frac{2}{\sqrt3}$ cm is:

Option 1: $\frac{4}{3}$ cm

Option 2: $\frac{4}{\sqrt3}$ cm

Option 3: $\frac{2}{3}$ cm 

Option 4: $1$ cm

Question : If $\frac{\tan\theta +\cot\theta }{\tan\theta -\cot\theta }=2, (0\leq \theta \leq 90^{0})$, then the value of $\sin\theta$ is:

Option 1: $\frac{2}{\sqrt3}$

Option 2: $\frac{\sqrt3}{2}$

Option 3: $\frac{1}{2}$

Option 4: $1$

Question : If the numerical values of the height and the area of an equilateral triangle are the same, then the length of each side of the triangle is:

Option 1: 2 units

Option 2: 4 units

Option 3: 5 units

Option 4: 8 units

Question : If $N=\frac{(\sqrt7-\sqrt3)}{(\sqrt7+\sqrt3)}$, then what is the value of $(N+\frac{1}{N})$?

Option 1: $2\sqrt2$

Option 2: $5$

Option 3: $10$

Option 4: $13$