Additional Mathematics 4037/22 — May/June 2022

Walkthrough

The question asks for the exact value of $y$ when $x = 6$, so we substitute $\sqrt{x} = \sqrt{6}$ into the fraction, giving $y = \frac{6 + \sqrt{6}}{3 + \sqrt{6}}$. This is not yet in the required form $a + b\sqrt{c}$ because of the surd in the denominator.

To remove the surd from the denominator, we multiply top and bottom by the conjugate of the denominator, $3 - \sqrt{6}$. The product $(3 + \sqrt{6})(3 - \sqrt{6}) = 9 - 6 = 3$, which is rational — that is exactly why the conjugate trick works.

Expanding the numerator carefully term by term:

$$(6 + \sqrt{6})(3 - \sqrt{6}) = 18 - 6\sqrt{6} + 3\sqrt{6} - 6 = 12 - 3\sqrt{6}$$

Dividing by the denominator $3$ gives $y = \frac{12 - 3\sqrt{6}}{3} = 4 - \sqrt{6}$, which is of the form $a + b\sqrt{c}$ with $a = 4$, $b = -1$ and $c = 6$.

Key Takeaways

• To write a fraction involving surds in the form $a + b\sqrt{c}$, rationalise the denominator by multiplying by the conjugate.
• The product of a surd expression and its conjugate removes the surd entirely: $(p + \sqrt{q})(p - \sqrt{q}) = p^2 - q$.
• Expand products of surds term by term, keeping each surd term separate until the final simplification.

Common Mistakes

• Forgetting to multiply the numerator as well as the denominator — multiplying only the bottom changes the value of the fraction.
• Sign slips when expanding $(6 + \sqrt{6})(3 - \sqrt{6})$: the middle terms are $-6\sqrt{6}$ and $+3\sqrt{6}$, combining to $-3\sqrt{6}$; getting one sign wrong gives a wrong final answer.
• The mark scheme awards the final A1 "not from wrong working" (nfww), so an answer of $4 - \sqrt{6}$ reached through incorrect algebra scores nothing.
• Failing to simplify $\frac{12 - 3\sqrt{6}}{3}$ fully — both terms must be divided by $3$.

Things to Be Careful About

• The question says "DO NOT USE A CALCULATOR", so all working must be exact — no decimal approximations of $\sqrt{6}$ at any stage.
• The answer must be left in exact form $4 - \sqrt{6}$; a decimal such as $1.55$ would score no accuracy mark.
• Show every line of the expansion: the M marks are for the conjugate multiplication step and the correct expansion, so skipping intermediate lines risks losing method marks.