| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find curve equation from derivative (extended problem with normals, stationary points, or further geometry) |
| Difficulty | Moderate -0.8 This is a straightforward C1 integration question requiring integration of x^(-2) and using a boundary condition to find the constant. The sketch and asymptotes are routine. Simpler than average A-level questions as it involves direct application of a standard integral with minimal problem-solving. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07l Derivative of ln(x): and related functions1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f(x) = \int \left(-\frac{1}{x^2}\right) dx\) | ||
| \(f(x) = x^{-1} + c\) | M1 A1 | |
| \((-1, 3) \therefore 3 = -1 + c\) | M1 | |
| \(c = 4\) | ||
| \(f(x) = x^{-1} + 4\) | A1 | |
| (b) Sketch showing asymptotes: \(x = 0\) and \(y = 4\) | B2 | (7) |
| B1 |
(a) $f(x) = \int \left(-\frac{1}{x^2}\right) dx$ | |
$f(x) = x^{-1} + c$ | M1 A1 |
$(-1, 3) \therefore 3 = -1 + c$ | M1 |
$c = 4$ | |
$f(x) = x^{-1} + 4$ | A1 |
(b) Sketch showing asymptotes: $x = 0$ and $y = 4$ | B2 | (7)
| B1 |\begin{enumerate}
\item The curve $y = \mathrm { f } ( x )$ passes through the point $P ( - 1,3 )$ and is such that
\end{enumerate}
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { x ^ { 2 } } , \quad x \neq 0 .$$
(a) Using integration, find $\mathrm { f } ( x )$.\\
(b) Sketch the curve $y = \mathrm { f } ( x )$ and write down the equations of its asymptotes.\\
\hfill \mbox{\textit{Edexcel C1 Q5 [7]}}