| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Determine nature of stationary points |
| Difficulty | Standard +0.3 This is a structured multi-part question testing algebraic manipulation, differentiation of negative powers, and stationary point analysis. Part (a) is routine expansion and simplification; part (b) applies standard differentiation rules to the simplified form; parts (c) and (d) are straightforward applications of first and second derivative tests. While it requires multiple techniques, each step follows standard C2 procedures with no novel insight needed, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((x^4 - 6x^2 + 9)\) | M1 | |
| \((x^4 - 6x^2 + 9) \div x^3 = x - 6x^{-1} + 9x^{-3}\) | A1 | (2 marks) |
| (b) \(f'(x) = 1 + 6x^{-2} - 27x^{-4}\) | M1, A1, A1 | First A1: 2 terms correct (unsimplified); Second A1: all 3 correct (simplified) |
| (c) When \(x = \pm\sqrt{3}\), \(f'(x) = 1 + \frac{6}{(\sqrt{3})^2} - \frac{27}{(\sqrt{3})^4}\) | M1 | |
| \((= 1 + \frac{6}{3} - \frac{27}{9}) = 0\), ∴Stationary | A1 | (2 marks) |
| (d) \(f''(x) = -12x^{-3} + 108x^{-5}\) | M1 | M: Attempt to diff. \(f'(x)\), not \(g(x)f'(x)\) |
| \(f''(\sqrt{3}) = \frac{-12}{(\sqrt{3})^3} + \frac{108}{(\sqrt{3})^5}\) | A1 | |
| \((\approx -2.309 + 6.928 = 4.619)\) \((\approx -\frac{8}{-\sqrt{3}})\) | ||
| \(> 0\), ∴Minimum (not dependent on a numerical version of \(f''(x)\)) | A1ft | (3 marks) |
**(a)** $(x^4 - 6x^2 + 9)$ | M1 |
$(x^4 - 6x^2 + 9) \div x^3 = x - 6x^{-1} + 9x^{-3}$ | A1 | (2 marks)
**(b)** $f'(x) = 1 + 6x^{-2} - 27x^{-4}$ | M1, A1, A1 | First A1: 2 terms correct (unsimplified); Second A1: all 3 correct (simplified) | (3 marks)
**(c)** When $x = \pm\sqrt{3}$, $f'(x) = 1 + \frac{6}{(\sqrt{3})^2} - \frac{27}{(\sqrt{3})^4}$ | M1 |
$(= 1 + \frac{6}{3} - \frac{27}{9}) = 0$, ∴Stationary | A1 | (2 marks)
**(d)** $f''(x) = -12x^{-3} + 108x^{-5}$ | M1 | M: Attempt to diff. $f'(x)$, not $g(x)f'(x)$ |
$f''(\sqrt{3}) = \frac{-12}{(\sqrt{3})^3} + \frac{108}{(\sqrt{3})^5}$ | A1 |
$(\approx -2.309 + 6.928 = 4.619)$ $(\approx -\frac{8}{-\sqrt{3}})$ |
$> 0$, ∴Minimum (not dependent on a numerical version of $f''(x)$) | A1ft | (3 marks)$$f(x) = \frac{(x^2 - 3)^2}{x^3}, \quad x \neq 0.$$
\begin{enumerate}[label=(\alph*)]
\item Show that $f(x) = x - 6x^{-1} + 9x^{-3}$. [2]
\item Hence, or otherwise, differentiate $f(x)$ with respect to $x$. [3]
\item Verify that the graph of $y = f(x)$ has stationary points at $x = \pm\sqrt{3}$. [2]
\item Determine whether the stationary value at $x = \sqrt{3}$ is a maximum or a minimum. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q6 [10]}}