| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| 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 rules: linear function, negative power (requiring rewriting), and product that can be expanded. All three parts are routine applications of the power rule with no problem-solving required, making it easier than average but not trivial since part (ii) requires recognizing the algebraic manipulation needed. |
| Spec | 1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dy}{dx} = 5\) | B1 [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = 2x^{-2}\) | B1 | \(x^{-2}\) soi |
| \(\frac{dy}{dx} = -4x^{-3}\) | B1 | \(-4x^c\) |
| B1 [3] | \(kx^{-3}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = 10x^2 - 14x + 5x - 7\) | M1 | Expand the brackets to give an expression of form \(ax^2+bx+c\) (\(a\neq0, b\neq0, c\neq0\)) |
| \(y = 10x^2 - 9x - 7\) | A1 | Completely correct (allow 2 \(x\)-terms) |
| \(\frac{dy}{dx} = 20x - 9\) | B1 ft, B1 ft [4+1+3=8] | 1 term correctly differentiated; Completely correct (2 terms) |
## Question 7:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 5$ | B1 [1] | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 2x^{-2}$ | B1 | $x^{-2}$ soi |
| $\frac{dy}{dx} = -4x^{-3}$ | B1 | $-4x^c$ |
| | B1 [3] | $kx^{-3}$ |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 10x^2 - 14x + 5x - 7$ | M1 | Expand the brackets to give an expression of form $ax^2+bx+c$ ($a\neq0, b\neq0, c\neq0$) |
| $y = 10x^2 - 9x - 7$ | A1 | Completely correct (allow 2 $x$-terms) |
| $\frac{dy}{dx} = 20x - 9$ | B1 ft, B1 ft [4+1+3=8] | 1 term correctly differentiated; Completely correct (2 terms) |
---7 Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in each of the following cases.\\
(i) $y = 5 x + 3$\\
(ii) $y = \frac { 2 } { x ^ { 2 } }$\\
(iii) $y = ( 2 x + 1 ) ( 5 x - 7 )$
\hfill \mbox{\textit{OCR C1 2007 Q7 [8]}}