Question : The sides of similar triangle $\triangle ABC$ and $\triangle DEF$ are in the ratio of $\frac{\sqrt{3}}{\sqrt{5}}$. If the area of $\triangle ABC$ is $90 \text{ cm}^2$, then the area of $\triangle DFF\left(\right.$ in $\left.\text{cm}^2\right)$ is:

Option 1: 150

Option 2: 152

Option 3: 154

Option 4: 156

Question : ABC is an isosceles right-angled triangle with $\angle$B = 90°. On the sides AC and AB, two equilateral triangles ACD and ABE have been constructed. The ratio of the area of $\triangle$ABE and $\triangle$ACD is:

Option 1: $1 : 3$

Option 2: $2 : 3$

Option 3: $1 : 2$

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

Question : $\triangle\mathrm{ABC}$ is a right angled triangle. $\angle \mathrm{A}=90°$, $AB = 4$ cm, and $BC = 5$ cm. What is the value of $\cos B + \cot C$?

Option 1: $\frac{17}{20}$

Option 2: $\frac{29}{20}$

Option 3: $\frac{23}{20}$

Option 4: $\frac{31}{20}$

Question : If $\sin(A+B)=\sin A\cos B+\cos A \sin B$, then the value of $\sin75°$ is:

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

Option 2:

$\frac{\sqrt{2}+1}{2\sqrt{2}}$

Option 3:

$\frac{\sqrt{3}+1}{2\sqrt{2}}$

Option 4:

$\frac{\sqrt{3}+1}{2}$

Question : Let $ABC$ and $PQR$ be two congruent right-angled triangles such that $\angle A=\angle P=90^{\circ}$. If $BC=13\ \text{cm}$ and $PR=12\ \text{cm}$, then find the length of $AB$.

Option 1: 25 cm

Option 2: 20 cm

Option 3: 10 cm

Option 4: 5 cm