| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find derivative after algebraic simplification (fractional/mixed powers) |
| Difficulty | Easy -1.3 This is a straightforward C1 differentiation question testing basic power rule application. Part (i) is direct differentiation, (ii) requires expanding brackets first (or product rule), and (iii) needs rewriting the root as a fractional power. All are routine exercises with no problem-solving required, making this easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | Marks | Guidance |
| \(\frac{dy}{dx} = 2x^3 - 3\) | B1 | 1 term correct |
| B1 2 | Completely correct (\(+c\) is an error, but only penalise once) | |
| (ii) | Marks | Guidance |
| \(y = 2x^3 + 2x^2 + 3x + 3\) | M1 | Attempt to expand brackets |
| \(\frac{dy}{dx} = 6x^2 + 4x + 3\) | A1 | \(2x^3 + 2x^2 + 3x + 3\) |
| A1 | 2 terms correct | |
| A1 4 | Completely correct | |
| SR Recognisable attempt at product rule: M1, one part correct A1, second part correct A1, final simplified answer A1 | ||
| (iii) | Marks | Guidance |
| \(y = x^{\frac{1}{3}}\) | B1 | \(x^{\frac{1}{3}}\) soi |
| \(\frac{dy}{dx} = \frac{1}{5}x^{-\frac{4}{5}}\) | B1 | \(\frac{1}{5}x^{-\frac{4}{5}}\) |
| B1 3 | \(kx^{-\frac{4}{5}}\) | |
| 9 |
**(i)** | **Marks** | **Guidance**
---|---|---
$\frac{dy}{dx} = 2x^3 - 3$ | B1 | 1 term correct
| B1 2 | Completely correct ($+c$ is an error, but only penalise once)
**(ii)** | **Marks** | **Guidance**
---|---|---
$y = 2x^3 + 2x^2 + 3x + 3$ | M1 | Attempt to expand brackets
$\frac{dy}{dx} = 6x^2 + 4x + 3$ | A1 | $2x^3 + 2x^2 + 3x + 3$
| A1 | 2 terms correct
| A1 4 | Completely correct
| | **SR** Recognisable attempt at product rule: M1, one part correct A1, second part correct A1, final simplified answer A1
**(iii)** | **Marks** | **Guidance**
---|---|---
$y = x^{\frac{1}{3}}$ | B1 | $x^{\frac{1}{3}}$ soi
$\frac{dy}{dx} = \frac{1}{5}x^{-\frac{4}{5}}$ | B1 | $\frac{1}{5}x^{-\frac{4}{5}}$
| B1 3 | $kx^{-\frac{4}{5}}$
| **9** |7 Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in each of the following cases:\\
(i) $y = \frac { 1 } { 2 } x ^ { 4 } - 3 x$,\\
(ii) $y = \left( 2 x ^ { 2 } + 3 \right) ( x + 1 )$,\\
(iii) $y = \sqrt [ 5 ] { x }$.
\hfill \mbox{\textit{OCR C1 2005 Q7 [9]}}