UPSC NDA 2 2016 Mathematics Solved Paper

Concept:The sum of interior angles of an $n$-sided polygon is $(n-2)\times 180^\circ$.The angles are in arithmetic progression with first term $120^\circ$ and common difference $5^\circ$.We equate the sum from the AP formula to the polygon sum and solve for $n$.Explanation:Let the number of sides be $n$.Sum of interior angles of polygon: $(n-2)\times 180^\circ$.Sum of $n$ terms of AP: $\frac{n}{2}[2\times 120 + (n-1)\times 5]$.Equating: $\frac{n}{2}[240 + 5n - 5] = (n-2)\times 180$.Simplify: $\frac{n}{2}[5n + 235] = 180n - 360$.Multiply both sides by 2: $5n^2 + 235n = 360n - 720$.Rearrange: $5n^2 - 125n + 720 = 0$.Divide by 5: $n^2 - 25n + 144 = 0$.Factorize: $(n-16)(n-9)=0$.So $n=16$ or $n=9$.Check $n=16$: 16th angle = $120 + 15\times 5 = 195^\circ > 180^\circ$, which is impossible for an interior angle.Hence $n=9$ is the only valid value.Answer:One