OCR FP1 2012 January — Question 3 6 marks

Exam BoardOCR
ModuleFP1 (Further Pure Mathematics 1)
Year2012
SessionJanuary
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex Numbers Arithmetic
TypePure square root finding
DifficultyStandard +0.3 This is a standard Further Pure 1 question requiring the algebraic method for finding square roots of complex numbers (equating real and imaginary parts of (a+bi)² = 3+6√2i). While it's a Further Maths topic and requires careful algebraic manipulation with surds, it's a routine textbook exercise with a well-established method, making it slightly easier than average overall but typical for FP1 students.
Spec4.02h Square roots: of complex numbers
3 Use an algebraic method to find the square roots of \(3 + ( 6 \sqrt { 2 } )\) i. Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.
Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
\(x^2 - y^2 = 3\) and \(xy = 3\sqrt{2}\)M1, A1 Attempt to equate real and imaginary parts; Obtain both results
\(x^4 - 3x^2 - 18 = 0\) or \(y^4 + 3y^2 - 18 = 0\)M1 Eliminate to obtain quadratic in \(x^2\) or \(y^2\)
\(x = \pm\sqrt{6}\) or \(y = \pm\sqrt{3}\)M1, A1 Solve to obtain \(x\) or \(y\) value; Both values correct
\(\pm(\sqrt{6} + i\sqrt{3})\)A1 Correct answers as complex numbers
[6] 
## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 - y^2 = 3$ and $xy = 3\sqrt{2}$ | M1, A1 | Attempt to equate real and imaginary parts; Obtain both results |
| $x^4 - 3x^2 - 18 = 0$ or $y^4 + 3y^2 - 18 = 0$ | M1 | Eliminate to obtain quadratic in $x^2$ or $y^2$ |
| $x = \pm\sqrt{6}$ or $y = \pm\sqrt{3}$ | M1, A1 | Solve to obtain $x$ or $y$ value; Both values correct |
| $\pm(\sqrt{6} + i\sqrt{3})$ | A1 | Correct answers as complex numbers |
| **[6]** | | |

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3 Use an algebraic method to find the square roots of $3 + ( 6 \sqrt { 2 } )$ i. Give your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are exact real numbers.

\hfill \mbox{\textit{OCR FP1 2012 Q3 [6]}}
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