| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find second derivative |
| Difficulty | Easy -1.8 This is a straightforward C2 question requiring only direct application of basic differentiation and integration rules with no problem-solving element. Part (a) involves two routine differentiations of a simple polynomial, and part (b) is standard definite integration—both are mechanical procedures well below average A-level difficulty. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f'(x) = 3x^2 + 6x\) | B1 | Acceptable alternatives: \(3x^2 + 6x^1\); \(3x^2 + 3\times2x\); \(3x^2 + 6x + 0\). Adding constant (e.g. \(+c\)) is B0 |
| \(f''(x) = 6x + 6\) | M1 | Attempt to differentiate \(f'(x)\); \(x^n \to x^{n-1}\) seen in at least one term. Coefficient of \(x\) ignored for method mark. \(x^2 \to x^1\) and \(x \to x^0\) acceptable |
| A1 cao | Acceptable alternatives: \(6x^1 + 6x^0\); \(3\times2x + 3\times2\). Adding constant is A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int(x^3 + 3x^2 + 5)\,dx = \frac{x^4}{4} + \frac{3x^3}{3} + 5x\) | M1, A1 | Attempt to integrate \(f(x)\); \(x^n \to x^{n+1}\). Ignore incorrect notation. Acceptable alternatives include \(\frac{x^4}{4} + x^3 + 5x\); \(\frac{x^4}{4} + \frac{3x^3}{3} + 5x^1\); with or without \(+c\) |
| \(\left[\frac{x^4}{4} + x^3 + 5x\right]_1^2 = 4 + 8 + 10 - \left(\frac{1}{4} + 1 + 5\right)\) | M1 | Substituting 2 and 1 into any function other than \(x^3+3x^2+5\) and subtracting either way round. Must substitute for all \(x\) but could make a slip. \(4+8+10-\frac{1}{4}+1+5\) acceptable as evidence of subtraction |
| \(= 15\frac{3}{4}\) o.e. | A1 | e.g. \(15\frac{3}{4}\), \(15.75\), \(\frac{63}{4}\). Must be a single number. Answer only is M0A0M0A0 |
## Question 1:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f'(x) = 3x^2 + 6x$ | B1 | Acceptable alternatives: $3x^2 + 6x^1$; $3x^2 + 3\times2x$; $3x^2 + 6x + 0$. Adding constant (e.g. $+c$) is B0 |
| $f''(x) = 6x + 6$ | M1 | Attempt to differentiate $f'(x)$; $x^n \to x^{n-1}$ seen in at least one term. Coefficient of $x$ ignored for method mark. $x^2 \to x^1$ and $x \to x^0$ acceptable |
| | A1 cao | Acceptable alternatives: $6x^1 + 6x^0$; $3\times2x + 3\times2$. Adding constant is A0 |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int(x^3 + 3x^2 + 5)\,dx = \frac{x^4}{4} + \frac{3x^3}{3} + 5x$ | M1, A1 | Attempt to integrate $f(x)$; $x^n \to x^{n+1}$. Ignore incorrect notation. Acceptable alternatives include $\frac{x^4}{4} + x^3 + 5x$; $\frac{x^4}{4} + \frac{3x^3}{3} + 5x^1$; with or without $+c$ |
| $\left[\frac{x^4}{4} + x^3 + 5x\right]_1^2 = 4 + 8 + 10 - \left(\frac{1}{4} + 1 + 5\right)$ | M1 | Substituting 2 and 1 into any function other than $x^3+3x^2+5$ and subtracting either way round. Must substitute for all $x$ but could make a slip. $4+8+10-\frac{1}{4}+1+5$ acceptable as evidence of subtraction |
| $= 15\frac{3}{4}$ o.e. | A1 | e.g. $15\frac{3}{4}$, $15.75$, $\frac{63}{4}$. Must be a single number. Answer only is M0A0M0A0 |
---1.
$$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 5$$
Find
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { f } ^ { \prime \prime } ( x )$,
\item $\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2007 Q1 [7]}}