| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Integrate after simplifying a quotient |
| Difficulty | Moderate -0.8 This is a straightforward C1 integration question requiring algebraic expansion of a binomial, term-by-term integration of powers, and using a boundary condition to find the constant. All steps are routine with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07i Differentiate x^n: for rational n and sums1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((3-x^2)^2 = 9 - 6x^2 + x^4\) | M1 | Attempt to expand numerator obtaining form \(9 \pm px^2 \pm qx^4\), \(p,q \neq 0\) |
| \(9x^{-2} + x^2\) | A1 | Must come from \(\frac{9+x^4}{x^2}\) |
| \(-6\) | A1 | Must come from \(\frac{-6x^2}{x^2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((f'(x) = 9x^{-2} - 6 + x^2)\) | ||
| \(-18x^{-3} + 2x\) | M1 A1ft | M1: \(x^n \to x^{n-1}\) on separate terms at least once. Do not award for \(A \to 0\) (integrating is M0). A1ft: \(-18x^{-3} + 2``B"x\) with numerical \(B\), no extra terms (\(A\) may have been incorrect or zero) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x) = -9x^{-1} - 6x + \frac{x^3}{3}(+c)\) | M1 A1ft | M1: \(x^n \to x^{n+1}\) on separate terms at least once (differentiating is M0). A1ft: \(-9x^{-1} + Ax + \frac{Bx^3}{3}(+c)\) with numerical \(A\) and \(B\), \(A,B \neq 0\) |
| \(10 = \frac{-9}{-3} - 6(-3) + \frac{(-3)^3}{3} + c\) so \(c = ...\) | M1 | Uses \(x=-3\) and \(y=10\) in what they think is \(f(x)\) — must be a changed function, i.e. not the original \(f'(x)\), to form a linear equation in \(c\). No \(+c\) gets M0 and A0 unless method implies correctly finding a constant |
| \(c = -2\) | A1 | cso |
| \(f(x) = -9x^{-1} - 6x + \frac{x^3}{3} + c\) (their \(c\)) | A1ft | Follow through their \(c\) in an otherwise correct expression. Allow \(-\frac{9}{x}\) for \(-9x^{-1}\) or \(\frac{9x^{-1}}{-1}\) |
## Question 9:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(3-x^2)^2 = 9 - 6x^2 + x^4$ | M1 | Attempt to expand numerator obtaining form $9 \pm px^2 \pm qx^4$, $p,q \neq 0$ |
| $9x^{-2} + x^2$ | A1 | Must come from $\frac{9+x^4}{x^2}$ |
| $-6$ | A1 | Must come from $\frac{-6x^2}{x^2}$ |
**Alternative 1:** Writes $\frac{(3-x^2)^2}{x^2}$ as $(3x^{-1}-x)^2$ and attempts to expand = M1, then A1A1.
**Alternative 2:** Sets $(3-x^2)^2 = 9 + Ax^2 + Bx^4$, expands $(3-x^2)^2$ and compares coefficients = M1 then A1A1. **(3 marks)**
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(f'(x) = 9x^{-2} - 6 + x^2)$ | | |
| $-18x^{-3} + 2x$ | M1 A1ft | M1: $x^n \to x^{n-1}$ on separate terms at least once. Do not award for $A \to 0$ (integrating is M0). A1ft: $-18x^{-3} + 2``B"x$ with numerical $B$, no extra terms ($A$ may have been incorrect or zero) |
**(2 marks)**
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x) = -9x^{-1} - 6x + \frac{x^3}{3}(+c)$ | M1 A1ft | M1: $x^n \to x^{n+1}$ on separate terms at least once (differentiating is M0). A1ft: $-9x^{-1} + Ax + \frac{Bx^3}{3}(+c)$ with numerical $A$ and $B$, $A,B \neq 0$ |
| $10 = \frac{-9}{-3} - 6(-3) + \frac{(-3)^3}{3} + c$ so $c = ...$ | M1 | Uses $x=-3$ and $y=10$ in what they think is $f(x)$ — must be a changed function, i.e. not the original $f'(x)$, to form a linear equation in $c$. No $+c$ gets M0 and A0 unless method implies correctly finding a constant |
| $c = -2$ | A1 | cso |
| $f(x) = -9x^{-1} - 6x + \frac{x^3}{3} + c$ (their $c$) | A1ft | Follow through their $c$ in an otherwise correct expression. Allow $-\frac{9}{x}$ for $-9x^{-1}$ or $\frac{9x^{-1}}{-1}$ |
**Note:** If they integrate in (b), no marks there, but if they then use that integration in (c), the marks for integration are available. **(5 marks) [10 marks total]**
---9.
$$f ^ { \prime } ( x ) = \frac { \left( 3 - x ^ { 2 } \right) ^ { 2 } } { x ^ { 2 } } , \quad x \neq 0$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { f } ^ { \prime } ( x ) = 9 x ^ { - 2 } + A + B x ^ { 2 }$,\\
where $A$ and $B$ are constants to be found.
\item Find $\mathrm { f } ^ { \prime \prime } ( x )$.
Given that the point $( - 3,10 )$ lies on the curve with equation $y = \mathrm { f } ( x )$,
\item find $\mathrm { f } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2013 Q9 [10]}}