UPSC CDS I 2024 Math Paper

Concept:For any regular polygon with $n$ sides, the interior angle is $\frac{(n-2) \times 180^\circ}{n}$ and the exterior angle is $\frac{360^\circ}{n}$.The given difference between interior and exterior angles is $144^\circ$.Explanation:Step 1: Write the equation for the difference.$\frac{(n-2) \times 180}{n} - \frac{360}{n} = 144$.Step 2: Combine the fractions over a common denominator $n$.$\frac{180(n-2) - 360}{n} = 144$.Step 3: Simplify the numerator.$180(n-2) - 360 = 180n - 360 - 360 = 180n - 720$.So $\frac{180n - 720}{n} = 144$.Step 4: Multiply both sides by $n$.$180n - 720 = 144n$.Step 5: Bring $144n$ to the left side.$180n - 144n = 720$.$36n = 720$.Step 6: Divide both sides by $36$.$n = \frac{720}{36} = 20$.Thus the regular polygon has $20$ sides.Answer:The number of sides is $20$, which corresponds to option D.