Moderate -0.3 This question requires simplifying √45 to 3√5, collecting like terms involving √5 and rational terms separately, then solving a linear equation. While it tests understanding of surd manipulation and algebraic rearrangement across multiple steps, it follows a standard procedure without requiring novel insight or complex problem-solving, making it slightly easier than average.
2 In this question you must show detailed reasoning.
Solve the equation \(x \sqrt { 5 } + 32 = x \sqrt { 45 } + 2 x\). Give your answer in the form \(a \sqrt { 5 } + b\), where \(a\) and \(b\) are integers to be determined.
(solve by completing the square or quadratic formula)
M1dep*
Solve by completing the square or using quadratic formula
\(x = 4\sqrt{5} - 4\) only
A1
## Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| $x\sqrt{5} + 32 = x\sqrt{45} + 2x$ then $x(\sqrt{45} + 2 - \sqrt{5}) = 32$ | M1* | Re-arranging and factorising out $x$ |
| $\sqrt{45} = 3\sqrt{5}$ | B1 | Replacing $\sqrt{45} = 3\sqrt{5}$ (or $\sqrt{45} \times \sqrt{5} = 15$ if multiplying through by $\sqrt{5}$). Could appear at any point. |
| $x = \left(\frac{32}{2\sqrt{5}+2}\right)\left(\frac{2\sqrt{5}-2}{2\sqrt{5}-2}\right)$ | M1dep* | Correct method for rationalising the surd of the denominator with $x$ taking the form $\frac{k_1}{k_2\sqrt{5}+k_3}$ o.e. |
| $x = \frac{32(2\sqrt{5}-2)}{20-4} = 4\sqrt{5} - 4$ | A1 | cao where $a=4, b=-4$. Need to see some correct working. |
**Alternative Scheme:**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\sqrt{45} = 3\sqrt{5}$ | B1 | Could appear at any point |
| $(32-2x)^2 = (2\sqrt{5}x)^2 \Rightarrow 16x^2 + 128x - 1024 (= 0)$ | M1* | Rearranging and squaring leading to a 3TQ |
| (solve by completing the square or quadratic formula) | M1dep* | Solve by completing the square or using quadratic formula |
| $x = 4\sqrt{5} - 4$ only | A1 | |
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2 In this question you must show detailed reasoning.
Solve the equation $x \sqrt { 5 } + 32 = x \sqrt { 45 } + 2 x$. Give your answer in the form $a \sqrt { 5 } + b$, where $a$ and $b$ are integers to be determined.
\hfill \mbox{\textit{OCR PURE Q2 [4]}}