JAMB Mathematics 2023 Paper

Concept:For a regular polygon with $k$ sides, each exterior angle is $\frac{360}{k}$ degrees. Use this formula to set up an equation from the given difference of exterior angles.Explanation:The exterior angle of an $(n-1)$-sided regular polygon is $\frac{360}{n-1}$.The exterior angle of an $(n+2)$-sided regular polygon is $\frac{360}{n+2}$.Their difference is $6^{\circ}$, so we write $\frac{360}{n-1} - \frac{360}{n+2} = 6$.Combine the left-hand side over a common denominator: $\frac{360(n+2) - 360(n-1)}{(n-1)(n+2)} = 6$.Simplify the numerator: $360n + 720 - 360n + 360 = 1080$.Thus, $\frac{1080}{(n-1)(n+2)} = 6$.Multiply both sides by $(n-1)(n+2)$: $1080 = 6(n-1)(n+2)$.Divide both sides by 6: $180 = (n-1)(n+2)$.Expand the right-hand side: $180 = n^{2} + 2n - n - 2$, so $n^{2} + n - 2 = 180$.Rearrange into standard quadratic form: $n^{2} + n - 182 = 0$.Factorise: $(n - 13)(n + 14) = 0$.Hence, $n = 13$ or $n = -14$. Since a polygon cannot have a negative number of sides, the valid value is $n = 13$.Answer:$n = 13$, which is option B.