Question : $\triangle PQR$ is an isosceles triangle and $PQ=PR=2a$ unit, $QR=a$ unit. Draw $PX \perp QR$, and find the length of $PX$.

Option 1: $\sqrt{5} a$

Option 2: $\frac{\sqrt{5} a}{2}$

Option 3: $\frac{\sqrt{10} a}{2}$

Option 4: $\frac{\sqrt{15} a}{2}$

Question : If the side of a square is $\frac{1}{2}(x+1)$ units and its diagonal is $\frac{3-x}{\sqrt{2}}$ units, then the length of the side of the square would be:

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

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

Option 3: 1 unit

Option 4: 2 units

Question : Find the area of the shaded portion of an equilateral triangle with sides 6 units shown in the following figure. A circle of radius 1 unit is centred at the midpoint of a side of the triangle.

Option 1: $\frac{1}{2}\left(9 \sqrt{3}-\frac{11}{7}\right)$ unit$^2$

Option 2: $\frac{1}{4}\left(9 \sqrt{3}-\frac{11}{7}\right)$ unit $^2$

Option 3: $\frac{1}{2}\left(6 \sqrt{3}-\frac{11}{7}\right)$ unit$^2$

Option 4: $\frac{1}{2}\left(9 \sqrt{3}-\frac{22}{7}\right)$ unit$^2$

Question : If for an isosceles triangle, the length of each equal side is $a$ units and that of the third side is $b$ units, then its area will be:

Option 1: $\frac{1}{4}\sqrt{4b^{2}-a^{2}}$ sq. units

Option 2: $\frac{a}{2}\sqrt{2a^{2}-b^{2}}$ sq. units

Option 3: $\frac{b}{4}\sqrt{4a^{2}-b^{2}}$ sq. units

Option 4: $\frac{b}{2}\sqrt{a^{2}-2b^{2}}$ sq. units

Question : $\frac{1}{\sqrt a}-\frac{1}{\sqrt b}=0$, then the value of $\frac{1}{a}+\frac{1}{ b}$ is:

Option 1: $\frac{1}{\sqrt{ab}}$

Option 2: $\sqrt{ab}$

Option 3: $\frac{2}{\sqrt{ab}}$

Option 4: $\frac{1}{2\sqrt{ab}}$