| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find stationary points and nature |
| Difficulty | Moderate -0.3 This is a straightforward multi-part differentiation question requiring algebraic manipulation to expand brackets, simplification of fractional powers, and application of standard differentiation rules. While it involves multiple steps and fractional indices, each technique is routine for C1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks |
|---|---|
| (i) \(f(x) = \frac{x^2 - 8x + 16}{2x^{\frac{1}{2}}}\) | M1 |
| \(f(x) = \frac{1}{2}x^{\frac{3}{2}} - 4x^{\frac{1}{2}} + 8x^{-\frac{1}{2}}\), \(A = \frac{1}{2}, B = -4, C = 8\) | A2 |
| (ii) \(f'(x) = \frac{3}{4}x^{\frac{1}{2}} - 2x^{-\frac{1}{2}} - 4x^{-\frac{3}{2}}\) | M1, A2 |
| \(f'(x) = \frac{1}{4}x^{-\frac{1}{2}}(3x^2 - 8x - 16) = \frac{3x^2 - 8x - 16}{4x^{\frac{1}{2}}}\) | M1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\therefore (4, 0)\) | M1, M1, A1 | (11 marks) |
**(i)** $f(x) = \frac{x^2 - 8x + 16}{2x^{\frac{1}{2}}}$ | M1 |
$f(x) = \frac{1}{2}x^{\frac{3}{2}} - 4x^{\frac{1}{2}} + 8x^{-\frac{1}{2}}$, $A = \frac{1}{2}, B = -4, C = 8$ | A2 |
**(ii)** $f'(x) = \frac{3}{4}x^{\frac{1}{2}} - 2x^{-\frac{1}{2}} - 4x^{-\frac{3}{2}}$ | M1, A2 |
$f'(x) = \frac{1}{4}x^{-\frac{1}{2}}(3x^2 - 8x - 16) = \frac{3x^2 - 8x - 16}{4x^{\frac{1}{2}}}$ | M1, A1 |
**(iii)** $f'(x) = 0 \Rightarrow 3x^2 - 8x - 16 = 0$
$(3x + 4)(x - 4) = 0$
$x > 0 \therefore x = 4$
$\therefore (4, 0)$ | M1, M1, A1 | (11 marks)
---$$\text{f}(x) \equiv \frac{(x-4)^2}{2x^{\frac{1}{2}}}, \quad x > 0.$$
\begin{enumerate}[label=(\roman*)]
\item Find the values of the constants $A$, $B$ and $C$ such that
$$\text{f}(x) = Ax^{\frac{3}{2}} + Bx^{\frac{1}{2}} + Cx^{-\frac{1}{2}}.$$ [3]
\item Show that
$$\text{f}'(x) = \frac{3x^2 - 8x - 16}{4x^{\frac{3}{2}}}.$$ [5]
\item Find the coordinates of the stationary point of the curve $y = \text{f}(x)$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C1 Q8 [11]}}