| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Prove root count with given polynomial |
| Difficulty | Moderate -0.3 This is a structured multi-part C1 question with clear signposting through each step. Part (a) involves routine application of Factor Theorem, polynomial division, and discriminant analysis—all standard techniques. Part (b) requires integration and area calculation between curve and line, which are textbook exercises. While it has multiple parts (7 marks total), each individual step is straightforward with no novel problem-solving required, making it slightly easier than the average A-level question. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(1) = 1 + 4 - 5\) | M1 | must find \(f(1)\), NOT long division |
| \(\Rightarrow f(1) = 0 \Rightarrow (x-1)\) is factor | A1 | shown \(= 0\) plus a statement |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt at \(x^2 + x + 5\) | M1 | long division leading to \(x^2 \pm x + ...\) or equating coefficients |
| \(f(x) = (x-1)(x^2 + x + 5)\) | A1 | \(p = 1\), \(q = 5\) by inspection scores B1, B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((x =)\ 1\) is real root | B1 | |
| Consider \(b^2 - 4ac\) for their \(x^2 + x + 5\) | M1 | not the cubic! |
| \(b^2 - 4ac = 1^2 - 4 \times 5 = -19 < 0\) | ||
| Hence no real roots (or only real root is 1) | A1 | CSO; all values correct plus a statement |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int ... \, dx = \dfrac{x^4}{4} + 2x^2 - 5x \ (+c)\) | M1, A1, A1 | one term correct unsimplified; second term correct unsimplified; all correct unsimplified |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left[4 + 8 - 10\right] - \left[\dfrac{1}{4} + 2 - 5\right]\) | M1 | correct use of limits 1 and 2; \(F(2) - F(1)\) attempted |
| \(= 4\dfrac{3}{4}\) | A1 | |
| Area of \(\Delta = \dfrac{1}{2} \times 11 = 5\dfrac{1}{2}\) | B1 | correct unsimplified |
| \(\Rightarrow\) shaded area \(= 5\dfrac{1}{2} - 4\dfrac{3}{4}\) | combined integral of \(7x - 6 - x^3\) scores M1 for limits correctly used then | |
| \(= \dfrac{3}{4}\) | A1 | A3 correct answer with all working correct |
# Question 6(a)(i):
$f(1) = 1 + 4 - 5$ | M1 | must find $f(1)$, NOT long division
$\Rightarrow f(1) = 0 \Rightarrow (x-1)$ is factor | A1 | shown $= 0$ plus a statement
**Total: 2 marks**
---
# Question 6(a)(ii):
Attempt at $x^2 + x + 5$ | M1 | long division leading to $x^2 \pm x + ...$ or equating coefficients
$f(x) = (x-1)(x^2 + x + 5)$ | A1 | $p = 1$, $q = 5$ by inspection scores B1, B1
**Total: 2 marks**
---
# Question 6(a)(iii):
$(x =)\ 1$ is real root | B1 |
Consider $b^2 - 4ac$ for their $x^2 + x + 5$ | M1 | not the cubic!
$b^2 - 4ac = 1^2 - 4 \times 5 = -19 < 0$ | |
Hence no real roots (or only real root is 1) | A1 | CSO; all values correct plus a statement
**Total: 3 marks**
---
# Question 6(b)(i):
$\int ... \, dx = \dfrac{x^4}{4} + 2x^2 - 5x \ (+c)$ | M1, A1, A1 | one term correct unsimplified; second term correct unsimplified; all correct unsimplified
**Total: 3 marks**
---
# Question 6(b)(ii):
$\left[4 + 8 - 10\right] - \left[\dfrac{1}{4} + 2 - 5\right]$ | M1 | correct use of limits 1 and 2; $F(2) - F(1)$ attempted
$= 4\dfrac{3}{4}$ | A1 |
Area of $\Delta = \dfrac{1}{2} \times 11 = 5\dfrac{1}{2}$ | B1 | correct unsimplified
$\Rightarrow$ shaded area $= 5\dfrac{1}{2} - 4\dfrac{3}{4}$ | | combined integral of $7x - 6 - x^3$ scores M1 for limits correctly used then
$= \dfrac{3}{4}$ | A1 | A3 correct answer with all working correct
**Total: 4 marks**
---6
\begin{enumerate}[label=(\alph*)]
\item The polynomial $\mathrm { f } ( x )$ is given by $\mathrm { f } ( x ) = x ^ { 3 } + 4 x - 5$.
\begin{enumerate}[label=(\roman*)]
\item Use the Factor Theorem to show that $x - 1$ is a factor of $\mathrm { f } ( x )$.
\item Express $\mathrm { f } ( x )$ in the form $( x - 1 ) \left( x ^ { 2 } + p x + q \right)$, where $p$ and $q$ are integers.
\item Hence show that the equation $\mathrm { f } ( x ) = 0$ has exactly one real root and state its value.
\end{enumerate}\item The curve with equation $y = x ^ { 3 } + 4 x - 5$ is sketched below.\\
\includegraphics[max width=\textwidth, alt={}, center]{23f34515-3373-4644-a8a1-82b45809d934-4_505_959_868_529}
The curve cuts the $x$-axis at the point $A ( 1,0 )$ and the point $B ( 2,11 )$ lies on the curve.
\begin{enumerate}[label=(\roman*)]
\item Find $\int \left( x ^ { 3 } + 4 x - 5 \right) \mathrm { d } x$.
\item Hence find the area of the shaded region bounded by the curve and the line $A B$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2007 Q6 [14]}}