Easy -1.2 This is a straightforward application of rationalizing the denominator by multiplying by the conjugate (√3 + 1)/(√3 + 1), followed by routine algebraic simplification. It requires only standard technique with no problem-solving insight, making it easier than average but not trivial since students must execute multiple algebraic steps correctly.
Allow \(-0.5\) and \(1.5\). Apply isw if correct answer seen. Answer only (very unlikely) is full marks if correct
(4)
Alternative method:
Answer
Marks
Guidance
Answer/Working
Marks
Guidance
\((p+q\sqrt{3})(\sqrt{3}-1)=5-2\sqrt{3}\), form simultaneous equations in \(p\) and \(q\)
M1
\(-p+3q=5\) and \(p-q=-2\)
A1
Solve to give \(p=-\frac{1}{2}\) and \(q=\frac{3}{2}\)
M1 A1
## Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{5-2\sqrt{3}}{\sqrt{3}-1} \times \frac{(\sqrt{3}+1)}{(\sqrt{3}+1)}$ | M1 | For multiplying numerator and denominator by same correct expression |
| $= \frac{\cdots}{2}$ denominator of 2 | A1 | For a correct denominator as a single number (NB depends on M mark) |
| Numerator $= 5\sqrt{3}+5-2\sqrt{3}\cdot\sqrt{3}-2\sqrt{3}$ | M1 | For an attempt to multiply the numerator by $(\sqrt{3}\pm1)$ and get 4 terms with at least 2 correct |
| $\frac{5-2\sqrt{3}}{\sqrt{3}-1} = -\frac{1}{2}+\frac{3}{2}\sqrt{3}$ | A1 | Allow $-0.5$ and $1.5$. Apply isw if correct answer seen. Answer only (very unlikely) is full marks if correct |
| | **(4)** | |
**Alternative method:**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(p+q\sqrt{3})(\sqrt{3}-1)=5-2\sqrt{3}$, form simultaneous equations in $p$ and $q$ | M1 | |
| $-p+3q=5$ and $p-q=-2$ | A1 | |
| Solve to give $p=-\frac{1}{2}$ and $q=\frac{3}{2}$ | M1 A1 | |