331.

If a polygon of n sides has 275 diagonals, then n is equal to

• 25

• 35

• 20

• 15

$$\begin{gathered} \mathrm{A} \mathrm{polygon} \mathrm{of} \mathrm{n} \mathrm{sides} \mathrm{has} \mathrm{number} \mathrm{of} \mathrm{diagonals} \\ = \frac{\mathrm{n}\left(\mathrm{n} - 1\right)}{2} = 275 \left[\because \mathrm{as} \mathrm{given}\right] \\ \Rightarrow {\mathrm{n}}^2 - 3\mathrm{n} - 550 = 0 \\ \Rightarrow {\mathrm{n}}^2 - 25\mathrm{n} + 22\mathrm{n} - 550 = 0 \\ \Rightarrow \mathrm{n}\left(\mathrm{n} - 25\right) + 22\left(\mathrm{n} - 25\right) = 0 \\ \Rightarrow \left(\mathrm{n} - 25\right)\left(\mathrm{n} + 22\right) = 0 \\ \Rightarrow \mathrm{n} = 25 \left[\because \mathrm{n} = - 22 \mathrm{is} \mathrm{not} \mathrm{possible}\right] \end{gathered}$$