| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Determine if function is increasing/decreasing |
| Difficulty | Moderate -0.3 This is a straightforward C2 question combining routine integration and differentiation. Part (i) requires basic polynomial integration with no tricks. Part (ii) involves differentiating a cubic, solving a quadratic (yielding surds), and interpreting the derivative's sign—all standard techniques with no novel insight required. Slightly easier than average due to being purely procedural. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{x^4}{4} - x^3 - \frac{x^2}{2} + 3x\) | M2 | M1 if at least two terms correct |
| their integral at \(3\) \(-\) their integral at \(1\) \([= -2.25 - 1.75]\) | M1 | dependent on integration attempted |
| \(= -4\) isw | A1 | |
| represents area between curve and \(x\)-axis between \(x=1\) and \(3\) | B1 | |
| negative since below \(x\)-axis | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y' = 3x^2 - 6x - 1\) | M1 | |
| their \(y'=0\) soi | M1 | dependent on differentiation attempted |
| \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) with \(a=3,\ b=-6\) and \(c=-1\) isw | M1 | or \(3(x-1)^2 - 4 [=0]\) or better |
| \(x = \frac{6\pm\sqrt{48}}{6}\) or better as final answer | A1 | e.g. A1 for \(1\pm\frac{2}{3}\sqrt{3}\) |
| \(\frac{6-\sqrt{48}}{6} < x < \frac{6+\sqrt{48}}{6}\) or ft their final answer | B1 | allow \(\leq\) instead of \(<\) |
**Question 11(i):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{x^4}{4} - x^3 - \frac{x^2}{2} + 3x$ | **M2** | **M1** if at least two terms correct | ignore $+c$ |
| their integral at $3$ $-$ their integral at $1$ $[= -2.25 - 1.75]$ | **M1** | dependent on integration attempted | M0 for evaluation of $x^3-3x^2-x+3$ or of differentiated version |
| $= -4$ isw | **A1** | | |
| represents area between curve and $x$-axis between $x=1$ and $3$ | **B1** | | **B0** for area *under* or above curve between $x=1$ and $3$ |
| negative since below $x$-axis | **B1** | | |
---
**Question 11(ii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $y' = 3x^2 - 6x - 1$ | **M1** | | |
| their $y'=0$ soi | **M1** | dependent on differentiation attempted | |
| $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ with $a=3,\ b=-6$ and $c=-1$ isw | **M1** | or $3(x-1)^2 - 4 [=0]$ or better | no follow through; $\frac{6\pm\sqrt{48}}{6}$ or better stated without working implies use of correct method |
| $x = \frac{6\pm\sqrt{48}}{6}$ or better as final answer | **A1** | e.g. **A1** for $1\pm\frac{2}{3}\sqrt{3}$ | **A0** for incorrect simplification, e.g. $1\pm\sqrt{48}$ |
| $\frac{6-\sqrt{48}}{6} < x < \frac{6+\sqrt{48}}{6}$ or ft their final answer | **B1** | allow $\leq$ instead of $<$ | allow **B1** if *both* inequalities are stated separately and it's clear that both apply; allow **B1** if the terms and the signs are in reverse order |
---(i) Use calculus to find $\int _ { 1 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } - x + 3 \right) \mathrm { d } x$ and state what this represents.\\
(ii) Find the $x$-coordinates of the turning points of the curve $y = x ^ { 3 } - 3 x ^ { 2 } - x + 3$, giving your answers in surd form. Hence state the set of values of $x$ for which $y = x ^ { 3 } - 3 x ^ { 2 } - x + 3$ is a decreasing function.
\hfill \mbox{\textit{OCR MEI C2 2011 Q11 [11]}}