| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Determine if function is increasing/decreasing |
| Difficulty | Standard +0.3 This is a straightforward multi-part differentiation question requiring polynomial differentiation, solving a quadratic inequality, finding a tangent equation, and solving a cubic. All techniques are standard C2 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(3x^2 - 6\) | 2 marks | 1 if one error |
| (ii) \(-\sqrt{2} < x < \sqrt{2}\) | 3 marks | M1 for using their \(y' = 0\); B1 f.t. for both roots found |
| (iii) subst \(x = -1\) in their \(y' [=-3]\) | B1 | f.t. |
| \(y = 7\) when \(x = -1\) | M1 | f.t. |
| \(y + 3x = 4\) | A1 | 3 terms |
| \(x^3 - 6x + 2 = -3x + 4\) | M1 | f.t. |
| \((2, -2)\) c.a.o. | A1, A1 | 6 marks total |
(i) $3x^2 - 6$ | 2 marks | 1 if one error
(ii) $-\sqrt{2} < x < \sqrt{2}$ | 3 marks | M1 for using their $y' = 0$; B1 f.t. for both roots found
(iii) subst $x = -1$ in their $y' [=-3]$ | B1 | f.t.
$y = 7$ when $x = -1$ | M1 | f.t.
$y + 3x = 4$ | A1 | 3 terms
$x^3 - 6x + 2 = -3x + 4$ | M1 | f.t.
$(2, -2)$ c.a.o. | A1, A1 | 6 marks total(i) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(ii) Find, in exact form, the range of values of $x$ for which $x ^ { 3 } - 6 x + 2$ is a decreasing function.\\
(iii) Find the equation of the tangent to the curve at the point $( - 1,7 )$.
Find also the coordinates of the point where this tangent crosses the curve again.
\hfill \mbox{\textit{OCR MEI C2 2006 Q11 [11]}}