| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Verify factor then sketch or analyse curve |
| Difficulty | Moderate -0.3 This is a straightforward C1 question combining factorization, differentiation, and solving equations. Part (a) requires factoring a cubic (which factors easily as x(x²+3x-4)), part (b) needs basic differentiation to find the tangent at the origin, and part (c) involves solving a cubic equation by substitution. All techniques are standard C1 procedures with no novel insight required, making it slightly easier than average but still requiring multiple steps across 11 marks. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02q Use intersection points: of graphs to solve equations1.07m Tangents and normals: gradient and equations |
| Answer | Marks |
|---|---|
| (a) \(x(x^2 + 3x - 4) = 0\) | M1 |
| \(x(x + 4)(x - 1) = 0\) | M1 |
| \(x = 0\) (at O), \(-4, 1\) | |
| \(\therefore (-4, 0), (1, 0)\) | A1 |
| (b) \(\frac{dy}{dx} = 3x^2 + 6x - 4\) | M1 A1 |
| grad \(= -4\) | M1 |
| \(\therefore y = -4x\) | A1 |
| (c) \(x^3 + 3x^2 - 4x = -4x\) | M1 |
| \(x^3 + 3x^2 = 0\) | |
| \(x^2(x + 3) = 0\) | M1 |
| \(x = 0\) (at O), \(-3\) | A1 |
| \(\therefore (-3, 12)\) | A1 |
**(a)** $x(x^2 + 3x - 4) = 0$ | M1 |
$x(x + 4)(x - 1) = 0$ | M1 |
$x = 0$ (at O), $-4, 1$ |
$\therefore (-4, 0), (1, 0)$ | A1 |
**(b)** $\frac{dy}{dx} = 3x^2 + 6x - 4$ | M1 A1 |
grad $= -4$ | M1 |
$\therefore y = -4x$ | A1 |
**(c)** $x^3 + 3x^2 - 4x = -4x$ | M1 |
$x^3 + 3x^2 = 0$ |
$x^2(x + 3) = 0$ | M1 |
$x = 0$ (at O), $-3$ | A1 |
$\therefore (-3, 12)$ | A1 |
**Total: 11 marks**
---\includegraphics{figure_1}
Figure 1 shows the curve $C$ with the equation $y = x^3 + 3x^2 - 4x$ and the straight line $l$.
The curve $C$ crosses the $x$-axis at the origin, $O$, and at the points $A$ and $B$.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of $A$ and $B$. [3]
\end{enumerate}
The line $l$ is the tangent to $C$ at $O$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find an equation for $l$. [4]
\item Find the coordinates of the point where $l$ intersects $C$ again. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q9 [11]}}