| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Find curve from gradient |
| Difficulty | Moderate -0.5 This is a straightforward integration question requiring algebraic manipulation to split the fraction, then applying standard power rule integration and using the given point to find the constant. It's slightly easier than average as it's a routine C1 technique with no conceptual challenges, though the fractional powers require care. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{5x^3+2}{x^{\frac{1}{2}}} = 5x^2 + 2x^{-\frac{1}{2}}\) | M1 A1 | M1: One term correct; A1: Both terms correct, and no extra terms |
| \(f(x) = 3x + \frac{5x^{\frac{5}{2}}}{\left(\frac{5}{2}\right)} + \frac{2x^{\frac{1}{2}}}{\left(\frac{1}{2}\right)} (+C)\) | M1 A1ft | (+ C not required here) |
| \(6 = 3 + 2 + 4 + C\) | M1 | Use of \(x = 1\) and \(y = 6\) to form eqn. in \(C\) |
| \(C = -3\) | A1 cso | |
| \(3x + 2x^{\frac{5}{2}} + 4x^{\frac{1}{2}} - 3\) | A1 (ft C) | (simplified version required) |
| Answer | Marks |
|---|---|
| Guidance: For the integration: M1 requires evidence from just one term (e.g. \(3 \to 3x\)), but not just "+C". A1ft requires correct integration of at least 3 terms, with at least one of these terms having a fractional power. For the final A1, follow through on C only. |
$\frac{5x^3+2}{x^{\frac{1}{2}}} = 5x^2 + 2x^{-\frac{1}{2}}$ | M1 A1 | M1: One term correct; A1: Both terms correct, and no extra terms
$f(x) = 3x + \frac{5x^{\frac{5}{2}}}{\left(\frac{5}{2}\right)} + \frac{2x^{\frac{1}{2}}}{\left(\frac{1}{2}\right)} (+C)$ | M1 A1ft | (+ C not required here)
$6 = 3 + 2 + 4 + C$ | M1 | Use of $x = 1$ and $y = 6$ to form eqn. in $C$
$C = -3$ | A1 cso |
$3x + 2x^{\frac{5}{2}} + 4x^{\frac{1}{2}} - 3$ | A1 (ft C) | (simplified version required)
**Total: 7 marks**
| Guidance: For the integration: M1 requires evidence from just one term (e.g. $3 \to 3x$), but not just "+C". A1ft requires correct integration of at least 3 terms, with at least one of these terms having a fractional power. For the final A1, follow through on C only.
---\begin{enumerate}
\item The curve with equation $y = \mathrm { f } ( x )$ passes through the point $( 1,6 )$. Given that
\end{enumerate}
$$f ^ { \prime } ( x ) = 3 + \frac { 5 x ^ { 2 } + 2 } { x ^ { \frac { 1 } { 2 } } } , x > 0$$
find $\mathrm { f } ( x )$ and simplify your answer.\\
\hfill \mbox{\textit{Edexcel C1 2006 Q8 [7]}}