| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area under polynomial curve |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question covering standard C2 techniques: factorising to find intercepts, differentiation for stationary points, and direct integration for area. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
The equation of a curve is $y = 9x^2 - x^4$.
(i) Show that the curve meets the $x$-axis at the origin and at $x = \pm a$, stating the value of $a$. [2]
(ii) Find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$.
Hence show that the origin is a minimum point on the curve. Find the $x$-coordinates of the maximum points. [6]
(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). [4]12 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 2012 Q12 [12]}}