Higher November 2023 Paper 4 Q22
22 You are given this identity.
\[\frac{2 - 3\sqrt{18}}{\sqrt{18} + 4} = a\sqrt{2} + b\]Find the value of \(a\) and the value of \(b\).
You must show each step in your working. [6]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| [\(a =\)] 21 [\(b =\)] −31 with correct working | 6 | “Correct working” requires at least M1M1M1M1 | |
| M1 for \(\frac{2 - 3\sqrt{18}}{\sqrt{18} + 4} \times \frac{\sqrt{18} - 4}{\sqrt{18} - 4}\) M1 for multiplying their numerator e.g. \(2\sqrt{18} - 8 - 3 \times 18 + 12\sqrt{18}\) oe or better M1 for simplifying their numerator e.g. \(14\sqrt{18} - 62\) M1 for multiplying their denominator e.g. \(18 + 4\sqrt{18} - 4\sqrt{18} - 16\) oe or better e.g. 18 – 16 or 2 M1 for \(\sqrt{18} = 3\sqrt{2}\) at any stage and may be implied in working A1dep for [a =] 21 or [b =] −31 dep. on only M4 awarded | alternative : M1 for \(\sqrt{18} = 3\sqrt{2}\) M1 for \(\frac{2 - 9\sqrt{2}}{3\sqrt{2} + 4} \times \frac{3\sqrt{2} - 4}{3\sqrt{2} - 4}\) M1 for multiplying their numerator \(6\sqrt{2} - 8 - 3 \times 18 + 36\sqrt{2}\) oe or better M1 for simplifying their numerator e.g. \(42\sqrt{2} - 62\) M1 for \(18 + 12\sqrt{2} - 12\sqrt{2} - 16\) oe or better e.g. 18 – 16 or 2 A1dep for [a =] 21 or [b =] −31 dep. on only M4 awarded | ||
| Alternative: M1 for \([2 - 3\sqrt{18} =] (a\sqrt{2} + b)(\sqrt{18} + 4)\) M1 for \(a\sqrt{2}\sqrt{18} + 4a\sqrt{2} + b\sqrt{18} + 4b\) oe or better M1 for \(\sqrt{2}\sqrt{18} = 6\) or \(\sqrt{18} = 3\sqrt{2}\) M1 for \(6a + 4a\sqrt{2} + 3b\sqrt{2} + 4b\) M1 for \(2 = 6a + 4b\) oe and \(-9 = 4a + 3b\) oe A1dep for [a =] 21 or [b =] −31 dep. on only M4 awarded If 0 scored SC1 for [a =] 21 and [b =] −31 | Note : working may be implied by use rather than explicitly seen and follow through from any errors if subsequent working is correct | ||