Question : The ratio of the length of each equal side and the third side of an isosceles triangle is 3 : 5. If the area of the triangle is $30 \sqrt{11}$ cm², then the length of the third side (in cm) is:

Option 1: $10\sqrt{6}$

Option 2: $5\sqrt{6}$

Option 3: $13\sqrt{6}$

Option 4: $11\sqrt{6}$

Question : The length of each side of a triangle is 12 cm. What is the length of the circumradius of the triangle?

Option 1: $8 \sqrt{3} \mathrm{~cm}$

Option 2: $2 \sqrt{3} \mathrm{~cm}$

Option 3: $6 \sqrt{3} \mathrm{~cm}$

Option 4: $4 \sqrt{3} \mathrm{~cm}$

Question : ABC is an equilateral triangle with a side of 12 cm. What is the length of the radius of the circle inscribed in it?

Option 1: $2 \sqrt{3}$ cm

Option 2: $8 \sqrt{3}$ cm

Option 3: $4 \sqrt{3} $ cm

Option 4: $6 \sqrt{3}$ cm

Question : The sum of the length and breadth of a rectangle is 6 cm. A square is constructed such that one of its sides is equal to a diagonal of the rectangle. If the ratio of areas of the square and rectangle is 5 : 2, the area of the square (in cm²) is:

Option 1: $20$

Option 2: $10$

Option 3: $4\sqrt{5}$

Option 4: $10\sqrt{2}$

Question : $\triangle PQR$ is right-angled at $Q$. The length of $PQ$ is 5 cm and $\angle P R Q=30^{\circ}$. Determine the length of the side $QR$.

Option 1: $5 \sqrt{3}~cm$

Option 2: $3 \sqrt{3}~cm$

Option 3: $\frac{1}{\sqrt{3}}~cm$

Option 4: $\frac{5}{\sqrt{3}}~cm$