Find $\alpha$ and $\beta$, if $x+1$ and $x+2$ are factors of $x^3+3x^2-2\alpha x+\beta$.

In the given figure, $\text{PT}$ is tangent to the circle at $\text{P}$ and $\text{QR}$ is a diameter of the circle. If $\text{∠RPT=50°}$. Find $\text{∠QRP}$(in $°$).

A tower subtends an angle $\mathrm{\alpha }$ at a point $\text{A}$ In the plane of its base and the angle of depression of the foot of the tower at a point $\mathrm{b} \text{metres}$ just above $\text{A}$ is $\mathrm{\beta }$. Prove that the height of the tower is $\mathrm{b} \mathrm{tan\alpha } \mathrm{cot\beta }$.

Find the modal value from the graph drawn for the monthly profits in rupees of $100$ shops are distributed as follows: