| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find derivative of simple polynomial (integer powers) |
| Difficulty | Easy -1.3 This is a straightforward C1 differentiation question requiring only the power rule to find f'(x) = 3 + 3x², followed by solving a simple quadratic equation. Both parts are routine procedural tasks with no problem-solving or conceptual challenge, making it easier than the average A-level question. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (a) \(f'(x) = 3 + 3x^2\) | M1A1 (2) | M1: attempt to differentiate \(x^n \to x^{n-1}\); just one term will do; A1: fully correct; must be \(3\) not \(3x^0\); if \(+c\) present scores A0 |
| (b) \(3 + 3x^2 = 15\) and attempt to simplify | M1 | 1st M1: forming correct equation and trying to rearrange \(f'(x)=15\) |
| \(x^2 = k \to x = \sqrt{k}\) (ignore \(\pm\)) | M1 | 2nd M1: dependent on \(f'(x)\) being of form \(a+bx^2\); attempting to solve \(a+bx^2=15\); correct processing leading to \(x=\ldots\) |
| \(x = 2\) (ignore \(x=-2\)) | A1 (3) | A1: \(x=2\) |
# Question 4:
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(a)** $f'(x) = 3 + 3x^2$ | M1A1 (2) | M1: attempt to differentiate $x^n \to x^{n-1}$; just one term will do; A1: fully correct; must be $3$ not $3x^0$; if $+c$ present scores A0 |
| **(b)** $3 + 3x^2 = 15$ and attempt to simplify | M1 | 1st M1: forming correct equation and trying to rearrange $f'(x)=15$ |
| $x^2 = k \to x = \sqrt{k}$ (ignore $\pm$) | M1 | 2nd M1: dependent on $f'(x)$ being of form $a+bx^2$; attempting to solve $a+bx^2=15$; correct processing leading to $x=\ldots$ |
| $x = 2$ (ignore $x=-2$) | A1 (3) | A1: $x=2$ |
---4.
$$\mathrm { f } ( x ) = 3 x + x ^ { 3 } , \quad x > 0 .$$
\begin{enumerate}[label=(\alph*)]
\item Differentiate to find $\mathrm { f } ^ { \prime } ( x )$.
Given that $\mathrm { f } ^ { \prime } ( x ) = 15$,
\item find the value of $x$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2008 Q4 [5]}}