| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area under polynomial curve |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing routine C2 skills: factorising to find intercepts, differentiation (including second derivative), identifying stationary points, and direct integration with substitution of limits. All techniques are standard with no problem-solving insight required, making it easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x^2(9 - x^2) = 0\) | B1 | \(9 \times 0^2 - 0^4 = 0\); B0 if correct answer appears from clearly incorrect working |
| \(x = 0\) and \(\pm 3\), [so \(a = 3\) or \(a = -3\)] | B1 | \(9 \times 3^2 - 3^4 = 0\) and \(9 \times (-3)^2 - (-3)^4 = 0\); \(a = \pm 3\) without working does not score |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y' = 18x - 4x^3\) | B1 | |
| \(y'' = 18 - 12x^2\) or ft | B1 | |
| their \(y' = 0\) | M1 | |
| \(2x(9 - 2x^2) = 0\) so \(x = 0\) | A1 | or \(18 \times 0 - 4 \times 0^3 = 0\) |
| \(x = 0\), \(y'' = 18\) cao so minimum | B1 | or evaluation of \(y'\) at \(\pm h\) where \(h < \sqrt{4.5}\) |
| \(x = \pm\sqrt{4.5}\) oe eg \(\pm\dfrac{3\sqrt{2}}{2}\) | A1 | accept 2.12 or better for \(\sqrt{4.5}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_0^3 (9x^2 - x^4)\,dx\) soi or ft | M1 | condone omission of, or wrong limits |
| \(3x^3 - 0.2x^5\) | A1 | correct answer implies M1; ignore \(+ c\) |
| F[their positive \(a\)] \(-\) [F[0]] or (not and) F[0] \(-\) F[their negative \(a\)] | M1 | dependent on at least one term correct; M0 if neither limit is 0; M0 for F[0] \(-\) F[their positive \(a\)]; M0 for use of Trapezium Rule |
| \(32.4\) oe cao | A1 |
## Question 4(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x^2(9 - x^2) = 0$ | B1 | $9 \times 0^2 - 0^4 = 0$; B0 if correct answer appears from clearly incorrect working |
| $x = 0$ and $\pm 3$, [so $a = 3$ or $a = -3$] | B1 | $9 \times 3^2 - 3^4 = 0$ and $9 \times (-3)^2 - (-3)^4 = 0$; $a = \pm 3$ without working does not score |
**[2 marks]**
---
## Question 4(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $y' = 18x - 4x^3$ | B1 | |
| $y'' = 18 - 12x^2$ or ft | B1 | |
| their $y' = 0$ | M1 | |
| $2x(9 - 2x^2) = 0$ so $x = 0$ | A1 | or $18 \times 0 - 4 \times 0^3 = 0$ |
| $x = 0$, $y'' = 18$ cao so minimum | B1 | or evaluation of $y'$ at $\pm h$ where $h < \sqrt{4.5}$ |
| $x = \pm\sqrt{4.5}$ oe eg $\pm\dfrac{3\sqrt{2}}{2}$ | A1 | accept 2.12 or better for $\sqrt{4.5}$ |
**[6 marks]**
---
## Question 4(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_0^3 (9x^2 - x^4)\,dx$ soi or ft | M1 | condone omission of, or wrong limits |
| $3x^3 - 0.2x^5$ | A1 | correct answer implies M1; ignore $+ c$ |
| F[their positive $a$] $-$ [F[0]] **or** (not **and**) F[0] $-$ F[their negative $a$] | M1 | dependent on at least one term correct; M0 if neither limit is 0; M0 for F[0] $-$ F[their positive $a$]; M0 for use of Trapezium Rule |
| $32.4$ oe cao | A1 | |
**[4 marks]**
---4 The equation of a curve is $y = 9 x ^ { 2 } - x ^ { 4 }$.\\
(i) Show that the curve meets the $x$-axis at the origin and at $x = \pm a$, stating the value of $a$.\\
(ii) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
Hence show that the origin is a minimum point on the curve. Find the $x$-coordinates of the maximum points.\\
(iii) Use calculus to find the area of the region bounded by the curve and the $x$-axis between $x = 0$ and $x = a$, using the value you found for $a$ in part (i).
\hfill \mbox{\textit{OCR MEI C2 Q4 [12]}}