Easy -1.2 This is a straightforward rationalizing the denominator question requiring multiplication by the conjugate (2-√3) and simplification. It's a standard AS-level technique with routine algebraic manipulation, making it easier than average but not trivial due to the arithmetic involved with the numerator 1+4√3.
4 In this question you must show detailed reasoning.
Express \(\frac { 1 + 4 \sqrt { 3 } } { 2 + \sqrt { 3 } }\) in the form \(\mathrm { a } + \mathrm { b } \sqrt { 3 }\), where \(a\) and \(b\) are integers to be determined.
Attempt to multiply \(\frac{1+4\sqrt{3}}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}} = \ldots\)
M1
Multiplying by the correct conjugate, and making a valid attempt at the resulting multiplication on either numerator or denominator - condoning sign/bracket errors only
\(\frac{2-\sqrt{3}+8\sqrt{3}-12}{2^2-\sqrt{3}^2}\) or better with attempt at numerator
A1
A valid attempt at multiplication on the numerator must be made to score this mark (condoning one sign or coeff. error only). If they write \((1+4\sqrt{3})(2-\sqrt{3}) = -10+7\sqrt{3}\) and \((2+\sqrt{3})(2-\sqrt{3})=1\) then M0A0A0 as detailed reasoning required.
\(-10+7\sqrt{3}\)
A1
Correct result from correct work. At least one correct intermediate line of working required before the final answer.
## Question 4:
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to multiply $\frac{1+4\sqrt{3}}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}} = \ldots$ | M1 | Multiplying by the correct conjugate, **and** making a valid attempt at the resulting multiplication on **either** numerator or denominator - condoning sign/bracket errors only |
| $\frac{2-\sqrt{3}+8\sqrt{3}-12}{2^2-\sqrt{3}^2}$ or better with attempt at numerator | A1 | A valid attempt at multiplication on the numerator must be made to score this mark (condoning one sign or coeff. error only). If they write $(1+4\sqrt{3})(2-\sqrt{3}) = -10+7\sqrt{3}$ and $(2+\sqrt{3})(2-\sqrt{3})=1$ then M0A0A0 as detailed reasoning required. |
| $-10+7\sqrt{3}$ | A1 | Correct result from correct work. At least one correct intermediate line of working required before the final answer. |
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4 In this question you must show detailed reasoning.\\
Express $\frac { 1 + 4 \sqrt { 3 } } { 2 + \sqrt { 3 } }$ in the form $\mathrm { a } + \mathrm { b } \sqrt { 3 }$, where $a$ and $b$ are integers to be determined.
\hfill \mbox{\textit{OCR MEI AS Paper 2 2024 Q4 [3]}}