| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2018 |
| Session | March |
| Marks | 9 |
| Topic | Complex Numbers Arithmetic |
| Type | Square roots with follow-up application |
| Difficulty | Standard +0.3 Part (i) is a standard FP1 technique for finding complex square roots by equating real and imaginary parts, requiring solving simultaneous equations. Part (ii) uses the conjugate root theorem and Vieta's formulas, which are routine applications for Further Maths students. Both parts follow well-established procedures with no novel insight required, making this slightly easier than average for an FP1 question. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02h Square roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((a + bi)^2 = a^2 - b^2 + 2abi\) | B1 | Seen or implied in solution |
| \(a^2 - b^2 = 528\) and \(2ab = 46\) | M1 | Comparing real and imaginary parts (no i) from a 3 term expansion |
| Eliminate \(b\) or \(a\) to obtain 3 term quadratic in \(a^2\) or \(b^2\) | M1 | Unknowns must not be in denominator and non-zero terms on same side. = 0 seen or implied by solution. |
| \(a^2 = 529\) or \(b^2 = 1\) only | A1 | |
| \(23 + i\) and \(-23 - i\) | A1 | Both roots. Can use \(\pm\) but not \(\pm 23 \pm i\) |
| [5] | ||
| (ii) Other root is \(3 - 2i\) | B1 | |
| \((x - (3 + 2i))(x - (3 - 2i))\) is a factor of \(x^3 - ax + 78\) | B1 | Or \(x^2 - 6x + 13\) |
| So \(x + 6\) is a factor | B1 | |
| \(a = 23\) | B1 | |
| [4] |
**(i)** $(a + bi)^2 = a^2 - b^2 + 2abi$ | B1 | Seen or implied in solution
$a^2 - b^2 = 528$ and $2ab = 46$ | M1 | Comparing real and imaginary parts (no i) from a 3 term expansion
Eliminate $b$ or $a$ to obtain 3 term quadratic in $a^2$ or $b^2$ | M1 | Unknowns must not be in denominator and non-zero terms on same side. = 0 seen or implied by solution.
$a^2 = 529$ or $b^2 = 1$ only | A1 |
$23 + i$ and $-23 - i$ | A1 | Both roots. Can use $\pm$ but not $\pm 23 \pm i$
| [5] |
**(ii)** Other root is $3 - 2i$ | B1 |
$(x - (3 + 2i))(x - (3 - 2i))$ is a factor of $x^3 - ax + 78$ | B1 | Or $x^2 - 6x + 13$
So $x + 6$ is a factor | B1 |
$a = 23$ | B1 |
| [4] |
---7 In this question you must show detailed reasoning.\\
(i) Find the square roots of the number $528 + 46 \mathrm { i }$ giving your answers in the form $a + b \mathrm { i }$.\\
(ii) $\quad 3 + 2 \mathrm { i }$ is a root of the equation $x ^ { 3 } - a x + 78 = 0$, where $a$ is a real number.
Find the value of $a$.
\hfill \mbox{\textit{OCR FP1 AS 2018 Q7 [9]}}