| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| 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 application, part (ii) requires rewriting the root as a fractional power, and part (iii) needs expansion before differentiation—all standard textbook exercises with no problem-solving or insight required, making it easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{dy}{dx} = -50x^{-6}\) | M1 | \(kx^{-6}\) |
| Fully correct answer | A1 (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y = x^{\frac{1}{4}}\) | B1 | \(\sqrt[4]{x} = x^{\frac{1}{4}}\) soi |
| \(\frac{dy}{dx} = \frac{1}{4}x^c\) | B1 | \(\frac{1}{4}x^c\) |
| \(\frac{dy}{dx} = \frac{1}{4}x^{-\frac{3}{4}}\) | B1 (3) | \(kx^{-\frac{3}{4}}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y = (x^2+3x)(1-5x)\) expanded | M1 | Attempt to multiply out fully |
| \(= 3x - 14x^2 - 5x^3\) | A1 | Correct expression (may have 4 terms) |
| \(\frac{dy}{dx} = 3 - 28x - 15x^2\) | M1 | Two terms correctly differentiated from their expanded expression |
| Completely correct (3 terms) | A1 (4) |
## Question 5:
**Part (i):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{dy}{dx} = -50x^{-6}$ | M1 | $kx^{-6}$ |
| Fully correct answer | A1 (2) | |
**Part (ii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $y = x^{\frac{1}{4}}$ | B1 | $\sqrt[4]{x} = x^{\frac{1}{4}}$ soi |
| $\frac{dy}{dx} = \frac{1}{4}x^c$ | B1 | $\frac{1}{4}x^c$ |
| $\frac{dy}{dx} = \frac{1}{4}x^{-\frac{3}{4}}$ | B1 (3) | $kx^{-\frac{3}{4}}$ |
**Part (iii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $y = (x^2+3x)(1-5x)$ expanded | M1 | Attempt to multiply out fully |
| $= 3x - 14x^2 - 5x^3$ | A1 | Correct expression (may have 4 terms) |
| $\frac{dy}{dx} = 3 - 28x - 15x^2$ | M1 | Two terms correctly differentiated from their expanded expression |
| Completely correct (3 terms) | A1 (4) | |
---5 Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in each of the following cases:\\
(i) $y = 10 x ^ { - 5 }$,\\
(ii) $y = \sqrt [ 4 ] { x }$,\\
(iii) $y = x ( x + 3 ) ( 1 - 5 x )$.
\hfill \mbox{\textit{OCR C1 2009 Q5 [9]}}