| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Curve properties and tangent/normal |
| Difficulty | Standard +0.3 This is a straightforward integration question requiring standard power rule techniques (rewriting surds as fractional powers) followed by using a point to find the constant. Part (b) requires understanding that parallel lines have equal gradients and that normal gradient = -1/tangent gradient, then solving a simple equation. Slightly above average due to the two-part structure and surd manipulation, but all techniques are routine C1 content. |
| Spec | 1.07m Tangents and normals: gradient and equations1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x) = x^{\frac{3}{2}} - \frac{9}{2}x^{\frac{1}{2}} + 2x\) | M1 | \(x^n \rightarrow x^{n+1}\); two terms in \(x\) correct, simplification not required in coefficients or powers |
| All terms in \(x\) correct | A1 | Simplification not required in coefficients or powers |
| \((+c)\) | A1 | \(+c\) is not required |
| Sub \(x=4\), \(y=9\) into \(f(x) \Rightarrow c = \ldots\) | M1 | Sub \(x=4\), \(y=9\) into \(f(x)\) to obtain value for \(c\). If no \(+c\) then M0. Use of \(x=9\), \(y=4\) is M0 |
| \((f(x)=)x^{\frac{3}{2}} - \frac{9}{2}x^{\frac{1}{2}} + 2x + 2\) | A1 | Must be all on one line and simplified. Accept equivalents e.g. \(f(x) = x^2 - 4.5\sqrt{x} + 2x + 2\). Allow \(x\sqrt{x}\) for \(x^{\frac{3}{2}}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Gradient of normal is \(-\frac{1}{2} \Rightarrow\) gradient of tangent \(= +2\) | M1A1 | \(2y + x = 0\) is \(\pm\frac{1}{2}(m) \Rightarrow \frac{dy}{dx} = -\frac{1}{\pm\frac{1}{2}}\); A1: gradient of tangent \(= +2\) (may be implied) |
| \(\frac{3\sqrt{x}}{2} - \frac{9}{4\sqrt{x}} + 2 = 2 \Rightarrow \frac{3\sqrt{x}}{2} - \frac{9}{4\sqrt{x}} = 0\) | M1 | Sets the given \(f'(x)\) or their \(f'(x)\) = their changed \(m\) and not their \(m\) where \(m\) has come from \(2y + x = 0\) |
| \(\times 4\sqrt{x} \Rightarrow 6x - 9 = 0 \Rightarrow x = \ldots\) | M1 | \(\times 4\sqrt{x}\) or equivalent correct algebraic processing (allow sign/arithmetic errors only) and attempt to solve to obtain value for \(x\). If \(f'(x) \neq 2\) they need to be solving a three term quadratic in \(\sqrt{x}\) correctly and square to obtain value for \(x\). Must be using the given \(f'(x)\) for this mark. |
| \(x = 1.5\) | A1 | \(x = \frac{3}{2}\) (1.5). Accept equivalents e.g. \(x = \frac{9}{6}\). If any 'extra' values are not rejected, score A0. |
## Question 10:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) = x^{\frac{3}{2}} - \frac{9}{2}x^{\frac{1}{2}} + 2x$ | M1 | $x^n \rightarrow x^{n+1}$; two terms in $x$ correct, simplification not required in coefficients or powers |
| All terms in $x$ correct | A1 | Simplification not required in coefficients or powers |
| $(+c)$ | A1 | $+c$ is not required |
| Sub $x=4$, $y=9$ into $f(x) \Rightarrow c = \ldots$ | M1 | Sub $x=4$, $y=9$ into $f(x)$ to obtain value for $c$. If no $+c$ then M0. Use of $x=9$, $y=4$ is M0 |
| $(f(x)=)x^{\frac{3}{2}} - \frac{9}{2}x^{\frac{1}{2}} + 2x + 2$ | A1 | Must be all on one line **and simplified**. Accept equivalents e.g. $f(x) = x^2 - 4.5\sqrt{x} + 2x + 2$. Allow $x\sqrt{x}$ for $x^{\frac{3}{2}}$ |
**Total: (5 marks)**
---
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Gradient of normal is $-\frac{1}{2} \Rightarrow$ gradient of tangent $= +2$ | M1A1 | $2y + x = 0$ is $\pm\frac{1}{2}(m) \Rightarrow \frac{dy}{dx} = -\frac{1}{\pm\frac{1}{2}}$; A1: gradient of tangent $= +2$ (may be implied) |
| $\frac{3\sqrt{x}}{2} - \frac{9}{4\sqrt{x}} + 2 = 2 \Rightarrow \frac{3\sqrt{x}}{2} - \frac{9}{4\sqrt{x}} = 0$ | M1 | Sets the given $f'(x)$ **or their** $f'(x)$ = their **changed** $m$ and **not** their $m$ where $m$ has come from $2y + x = 0$ |
| $\times 4\sqrt{x} \Rightarrow 6x - 9 = 0 \Rightarrow x = \ldots$ | M1 | $\times 4\sqrt{x}$ or equivalent correct algebraic processing (allow sign/arithmetic errors only) and attempt to solve to obtain value for $x$. If $f'(x) \neq 2$ they need to be solving a three term quadratic in $\sqrt{x}$ correctly and square to obtain value for $x$. **Must be using the given $f'(x)$ for this mark.** |
| $x = 1.5$ | A1 | $x = \frac{3}{2}$ (1.5). Accept equivalents e.g. $x = \frac{9}{6}$. **If any 'extra' values are not rejected, score A0.** |
**Total: (5 marks)**
---A curve with equation $y = \mathrm { f } ( x )$ passes through the point $( 4,9 )$.
Given that
$$f ^ { \prime } ( x ) = \frac { 3 \sqrt { } x } { 2 } - \frac { 9 } { 4 \sqrt { } x } + 2 , \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item find $\mathrm { f } ( x )$, giving each term in its simplest form.
Point $P$ lies on the curve.
The normal to the curve at $P$ is parallel to the line $2 y + x = 0$
\item Find the $x$ coordinate of $P$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2015 Q10 [10]}}