JAMB Mathematics 2023 Paper

Concept:Rationalising the denominator removes the surd from the bottom, allowing the expression to be written in the required form $a + b\sqrt{3}$.Explanation:Start with $\frac{3 - \sqrt{3}}{2 + \sqrt{3}}$.Multiply the numerator and denominator by the conjugate of the denominator, which is $2 - \sqrt{3}$:$$\frac{3 - \sqrt{3}}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}}$$Expand the numerator carefully:$$(3 - \sqrt{3})(2 - \sqrt{3}) = 6 - 3\sqrt{3} - 2\sqrt{3} + (\sqrt{3})^2$$This simplifies to:$$6 - 5\sqrt{3} + 3 = 9 - 5\sqrt{3}$$Now expand the denominator:$$(2 + \sqrt{3})(2 - \sqrt{3}) = 4 - (\sqrt{3})^2 = 4 - 3 = 1$$Therefore, the given expression becomes:$$\frac{9 - 5\sqrt{3}}{1} = 9 - 5\sqrt{3}$$Write this as $9 + (-5)\sqrt{3}$, so we can compare it with $a + b\sqrt{3}$.Thus, $a = 9$ and $b = -5$.Answer:$a = 9$, $b = -5$.