| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find second derivative |
| Difficulty | Moderate -0.8 This is a straightforward differentiation question requiring only the power rule applied to terms rewritten as negative powers. It's routine C1 material with no problem-solving element—students simply apply standard differentiation rules twice. Easier than average but not trivial since it requires rewriting fractions and handling negative indices correctly. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y = 5x^{-2} - \frac{1}{4}x^{-1} + x\) | M1 | \(x^{-2}\) used for \(\frac{1}{x^2}\) OR \(x^{-1}\) used for \(\frac{1}{x}\) soi, OR \(x\) correctly differentiated; Look out for: \(y = 5x^{-2} - 4x^{-1} + x\) followed by \(\frac{dy}{dx} = -10x^{-3} + 4x^{-2} + 1\) and then the correct answer. This is M1 A1 A1 A0; \(4x^{-1}\) is NOT a misread |
| \(\frac{dy}{dx} = -10x^{-3} + \frac{1}{4}x^{-2} + 1\) | A1 | \(kx^{-3}\) or \(kx^{-2}\) from differentiating |
| A1 | Two fully correct terms | |
| A1 | 4 marks; Completely correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{d^2y}{dx^2} = 30x^{-4} - \frac{1}{2}x^{-3}\) | M1 | Attempt to differentiate their \(\frac{dy}{dx}\) (one term correctly differentiated); Allow a sign slip in coefficient for M mark |
| A1 | 2 marks (total 6); Completely correct; NB Only penalise "\(+ c\)" first time seen in the question |
# Question 6:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $y = 5x^{-2} - \frac{1}{4}x^{-1} + x$ | M1 | $x^{-2}$ used for $\frac{1}{x^2}$ **OR** $x^{-1}$ used for $\frac{1}{x}$ soi, **OR** $x$ correctly differentiated; Look out for: $y = 5x^{-2} - 4x^{-1} + x$ followed by $\frac{dy}{dx} = -10x^{-3} + 4x^{-2} + 1$ and then the correct answer. This is **M1 A1 A1 A0**; $4x^{-1}$ is **NOT** a misread |
| $\frac{dy}{dx} = -10x^{-3} + \frac{1}{4}x^{-2} + 1$ | A1 | $kx^{-3}$ or $kx^{-2}$ from differentiating |
| | A1 | Two fully correct terms |
| | A1 | **4 marks**; Completely correct |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{d^2y}{dx^2} = 30x^{-4} - \frac{1}{2}x^{-3}$ | M1 | Attempt to differentiate their $\frac{dy}{dx}$ (one term correctly differentiated); Allow a sign slip in coefficient for M mark |
| | A1 | **2 marks** (total **6**); Completely correct; **NB** Only penalise "$+ c$" first time seen in the question |
---6 Given that $y = \frac { 5 } { x ^ { 2 } } - \frac { 1 } { 4 x } + x$, find\\
(i) $\frac { \mathrm { d } y } { \mathrm {~d} x }$,\\
(ii) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
\hfill \mbox{\textit{OCR C1 2011 Q6 [6]}}