| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Prove root count with given polynomial |
| Difficulty | Moderate -0.8 This is a structured, multi-part question that guides students through standard C1 techniques: sketching a cubic, applying the remainder theorem (substitute x=-1), using the factor theorem (show p(3)=0), and polynomial division. Each step is routine and clearly signposted with no novel problem-solving required. The 'hence' part is straightforward once the factorisation is complete. Easier than average for A-level. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| [Sketch of cubic curve with one max and one min (either way up)] | M1 | cubic curve with one max and one min (either way up) |
| [curve touching positive x-axis (either way up)] | A1 | curve touching positive x-axis (either way up) |
| [correct graph passing through O and touching x-axis at 2] | A1 | correct graph passing through O and touching x-axis at 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x(x^2 - 4x + 4) = 3\) | B1 | AG (must have = 0) |
| \(\Rightarrow x^3 - 4x^2 + 4x - 3 = 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(p(-1) = (-1)^3 - 4(-1)^2 + 4(-1) - 3\) | M1 | p(–1) attempted (condone one slip); or full long division to remainder |
| \((= -1 - 4 - 4 - 3)\) | ||
| \(= -12\) | A1 | must indicate remainder = –12 if long division used |
| Answer | Marks | Guidance |
|---|---|---|
| \(p(3) = 3^3 - 4 \times 3^2 + 4 \times 3 - 3\) | M1 | p(3) attempted (condone one slip); NOT long division |
| \(p(3) = 27 - 36 + 12 - 3\) | ||
| \(p(3) = 0 \Rightarrow x - 3\) is factor | A1 | shown = 0 plus statement |
| Answer | Marks | Guidance |
|---|---|---|
| Either \(b = -1\) (coefficient of x correct) or \(c = 1\) (constant term correct) | M1 | allow M1 for full attempt at long division or comparing coefficients if neither b nor c is correct |
| \(p(x) = (x - 3)(x^2 - x + 1)\) | A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Discriminant of their quadratic | M1 | numerical expression must be seen |
| \(= (-1)^2 - 4\) | ||
| Discriminant = –3 (or < 0) \(\Rightarrow\) no real roots | A1cso | must have correct quadratic and statement and all working correct |
| (Only real root is \(x = 3\)) | B1 | 3 |
| Total | 13 |
**5(a)(i)**
[Sketch of cubic curve with one max and one min (either way up)] | M1 | cubic curve with one max and one min (either way up)
[curve touching positive x-axis (either way up)] | A1 | curve touching positive x-axis (either way up)
[correct graph passing through O and touching x-axis at 2] | A1 | correct graph passing through O and touching x-axis at 2 | 3
**5(a)(ii)**
$x(x^2 - 4x + 4) = 3$ | B1 | AG (must have = 0)
$\Rightarrow x^3 - 4x^2 + 4x - 3 = 0$ | |
**5(b)(i)**
$p(-1) = (-1)^3 - 4(-1)^2 + 4(-1) - 3$ | M1 | p(–1) attempted (condone one slip); or full long division to remainder
$(= -1 - 4 - 4 - 3)$ | |
$= -12$ | A1 | must indicate remainder = –12 if long division used | 2
**5(b)(ii)**
$p(3) = 3^3 - 4 \times 3^2 + 4 \times 3 - 3$ | M1 | p(3) attempted (condone one slip); NOT long division
$p(3) = 27 - 36 + 12 - 3$ | |
$p(3) = 0 \Rightarrow x - 3$ is factor | A1 | shown = 0 plus statement | 2
**5(b)(iii)**
Either $b = -1$ (coefficient of x correct) or $c = 1$ (constant term correct) | M1 | allow M1 for full attempt at long division or comparing coefficients if neither b nor c is correct
$p(x) = (x - 3)(x^2 - x + 1)$ | A1 | 2
**5(c)**
Discriminant of their quadratic | M1 | numerical expression must be seen
$= (-1)^2 - 4$ | |
Discriminant = –3 (or < 0) $\Rightarrow$ no real roots | A1cso | must have correct quadratic and statement and all working correct
(Only real root is $x = 3$) | B1 | 3
| | Total | 13
---5
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Sketch the curve with equation $y = x ( x - 2 ) ^ { 2 }$.
\item Show that the equation $x ( x - 2 ) ^ { 2 } = 3$ can be expressed as
$$x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3 = 0$$
\end{enumerate}\item The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3$.
\begin{enumerate}[label=(\roman*)]
\item Find the remainder when $\mathrm { p } ( x )$ is divided by $x + 1$.
\item Use the Factor Theorem to show that $x - 3$ is a factor of $\mathrm { p } ( x )$.
\item Express $\mathrm { p } ( x )$ in the form $( x - 3 ) \left( x ^ { 2 } + b x + c \right)$, where $b$ and $c$ are integers.
\end{enumerate}\item Hence show that the equation $x ( x - 2 ) ^ { 2 } = 3$ has only one real root and state the value of this root.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2011 Q5 [13]}}