| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Parametric polynomials with root conditions |
| Difficulty | Standard +0.3 Part (a) is routine quadratic formula with complex numbers. Part (b)(i) requires understanding that complex conjugate roots occur when coefficients are real—a standard FP1 concept. Part (b)(ii) involves substituting w = p + 2i and using sum/product of roots, requiring algebraic manipulation but following predictable steps. This is slightly easier than average A-level difficulty due to being a standard FP1 complex numbers question with clear structure. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((x+2)^2 - 4 + 20 = 0\) | M1 | OE eg \((x+2)^2 = -16\) |
| \(x + 2 = \pm 4i\) | B1 | \(\sqrt{-16} = 4i\) |
| \((x =) -2 \pm 4i\) | A1 | NMS \(-2 \pm 4i\) scores 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(\begin{Bmatrix}(x =) \frac{-4 \pm\sqrt{16-4(20)}}{2} & = & \frac{-4 \pm\sqrt{-64}}{2}\end{Bmatrix}\) | (M1) | Correct substitution into quadratic formula |
| \(= \frac{-4 \pm 8i}{2}\) | (B1) | \(\sqrt{-64} = 8i\) or \(\frac{2}{2} = 4i\) |
| \((x =) -2 \pm 4i\) | (A1) | \((c = -2, d = \pm 4)\) |
| (b)(i) Roots are complex conjugates (and coeff. of \(z^2\) and constant term are both real) so coefficients of quadratic are all real | E1 | |
| \((4+i+qi)\) is real ie for real \(q\) \((1+q)i = 0 \Rightarrow q\) must be \(-1\) | E1 | Indep of previous E1 but must refer to \((4+i+qi)\) or coefficient of \(z\) being 'real' and \(q = -1\) |
| Answer | Marks | Guidance |
|---|---|---|
| (b)(ii) Roots \(p+2i\) and \(p-2i\) | B1 | PI by subst of both \(p+2i\) and \(p-2i\) for \(z\) or \((p+2i)(p-2i)\) seen or \((p+2i)+(p-2i)\) seen |
| \((p+2i)(p-2i) = 20 \Rightarrow p^2 = 16\) | M1 | Either or equivalent |
| \((p+2i) + (p-2i) = -4-i-qi\) \(\Rightarrow \pm 8 = -4-i-qi\) | M1 | OE eg \(q\) must be in the form \(-1+ki\), where \(k\) is real. \(\pm 8 = -4+k\) |
| \(q = -1 + 12i\) \(q = -1 - 4i\) | A1 | A1 |
**(a)** $(x+2)^2 - 4 + 20 = 0$ | M1 | OE eg $(x+2)^2 = -16$
$x + 2 = \pm 4i$ | B1 | $\sqrt{-16} = 4i$
$(x =) -2 \pm 4i$ | A1 | NMS $-2 \pm 4i$ scores 3 marks
**Total for (a): 3 marks**
**Alternative (a):**
$\begin{Bmatrix}(x =) \frac{-4 \pm\sqrt{16-4(20)}}{2} & = & \frac{-4 \pm\sqrt{-64}}{2}\end{Bmatrix}$ | (M1) | Correct substitution into quadratic formula
$= \frac{-4 \pm 8i}{2}$ | (B1) | $\sqrt{-64} = 8i$ or $\frac{2}{2} = 4i$
$(x =) -2 \pm 4i$ | (A1) | $(c = -2, d = \pm 4)$
**(b)(i)** Roots are complex conjugates (and coeff. of $z^2$ and constant term are both real) so coefficients of quadratic are all real | E1 |
$(4+i+qi)$ is real ie for real $q$ $(1+q)i = 0 \Rightarrow q$ must be $-1$ | E1 | Indep of previous E1 but must refer to $(4+i+qi)$ or coefficient of $z$ being 'real' and $q = -1$
**Total for (b)(i): 2 marks**
**(b)(ii)** Roots $p+2i$ and $p-2i$ | B1 | PI by subst of both $p+2i$ and $p-2i$ for $z$ or $(p+2i)(p-2i)$ seen or $(p+2i)+(p-2i)$ seen
$(p+2i)(p-2i) = 20 \Rightarrow p^2 = 16$ | M1 | Either or equivalent
$(p+2i) + (p-2i) = -4-i-qi$ $\Rightarrow \pm 8 = -4-i-qi$ | M1 | OE eg $q$ must be in the form $-1+ki$, where $k$ is real. $\pm 8 = -4+k$
$q = -1 + 12i$ $q = -1 - 4i$ | A1 | A1
**Total for (b)(ii): 5 marks**
**Overall Total: 10 marks**
**Alternative (a):**
$2c = -4$, $c^2 + d^2 = 20$ (M1 need both); $c = -2$ (B1) $d = \pm 4$; $-2 \pm 4i$ (A1)
---\begin{enumerate}[label=(\alph*)]
\item Solve the equation $x^2 + 4x + 20 = 0$, giving your answers in the form $c + di$, where $c$ and $d$ are integers.
[3 marks]
\item The roots of the quadratic equation
$$z^2 + (4 + i + qi)z + 20 = 0$$
are $w$ and $w^*$.
\begin{enumerate}[label=(\roman*)]
\item In the case where $q$ is real, explain why $q$ must be $-1$.
[2 marks]
\item In the case where $w = p + 2i$, where $p$ is real, find the possible values of $q$.
[5 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2016 Q7 [10]}}