| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch then find derivative/gradient/tangent |
| Difficulty | Moderate -0.3 This is a straightforward C1 curve sketching question requiring factorization of a cubic, finding a tangent using differentiation, and solving a cubic equation. All techniques are standard for this level, though the multi-part structure and final intersection calculation require more work than the most basic questions, placing it slightly below average difficulty. |
| 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 |
|---|---|
| \(x(x^2 + 3x - 4) = 0\) | M1 |
| \(x(x+4)(x-1) = 0\) | M1 |
| \(x = 0\) (at \(O\)), \(-4\), \(1\) | A1 |
| \(\therefore (-4, 0), (1, 0)\) | A1 |
| Answer | Marks |
|---|---|
| \(\frac{dy}{dx} = 3x^2 + 6x - 4\) | M1 A1 |
| \(\text{grad} = -4\) | M1 |
| \(\therefore y = -4x\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x^3 + 3x^2 - 4x = -4x\) | M1 | |
| \(x^3 + 3x^2 = 0\) | M1 | |
| \(x^2(x+3) = 0\) | M1 | |
| \(x = 0\) (at \(O\)), \(-3\) | A1 | |
| \(\therefore (-3, 12)\) | A1 | (11) |
## (i)
$x(x^2 + 3x - 4) = 0$ | M1 |
$x(x+4)(x-1) = 0$ | M1 |
$x = 0$ (at $O$), $-4$, $1$ | A1 |
$\therefore (-4, 0), (1, 0)$ | A1 |
## (ii)
$\frac{dy}{dx} = 3x^2 + 6x - 4$ | M1 A1 |
$\text{grad} = -4$ | M1 |
$\therefore y = -4x$ | A1 |
## (iii)
$x^3 + 3x^2 - 4x = -4x$ | M1 |
$x^3 + 3x^2 = 0$ | M1 |
$x^2(x+3) = 0$ | M1 |
$x = 0$ (at $O$), $-3$ | A1 |
$\therefore (-3, 12)$ | A1 | (11) |
---\includegraphics{figure_8}
The diagram 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=(\roman*)]
\item Find the coordinates of $A$ and $B$. [3]
\end{enumerate}
The line $l$ is the tangent to $C$ at $O$.
\begin{enumerate}[label=(\roman*)]
\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{OCR C1 Q8 [11]}}