| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Differentiate after index conversion |
| Difficulty | Moderate -0.8 This is a straightforward C1/C2 question requiring routine algebraic expansion of a binomial, simplification with index laws, then standard differentiation using power rule. All steps are mechanical with no problem-solving insight needed, making it easier than average but not trivial due to the multi-step nature. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x) = \frac{16 + 24\sqrt{x} + 9x}{x}\) | M1 | Expands numerator into three/four term quadratic in \(\sqrt{x}\) |
| \(f(x) = 16x^{-1} + 24x^{-\frac{1}{2}} + 9\) | M1A1A1 | Second M1: divides at least one term by \(x\) correctly; A1: two correct terms; A1: all terms correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f'(x) = -16x^{-2} - 12x^{-\frac{3}{2}}\) | M1 A1 | M1: evidence of differentiation \(x^n \to x^{n-1}\); A1: cao and cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| When \(x=4\), \(y=25\) | B1 | 25 only |
| \(f'(4) = -1 - \frac{12}{8} = -2\frac{1}{2}\) | M1 | Substitute \(x=4\) into derived function |
| \(y - 25 = -\frac{5}{2}(x-4)\) | M1 A1 | M1: uses "25" and gradient from calculus with \(x=4\); A1: any correct form e.g. \(y = -\frac{5}{2}x + 35\), \(5x + 2y - 70 = 0\) |
# Question 12:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x) = \frac{16 + 24\sqrt{x} + 9x}{x}$ | M1 | Expands numerator into three/four term quadratic in $\sqrt{x}$ |
| $f(x) = 16x^{-1} + 24x^{-\frac{1}{2}} + 9$ | M1A1A1 | Second M1: divides at least one term by $x$ correctly; A1: two correct terms; A1: all terms correct |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f'(x) = -16x^{-2} - 12x^{-\frac{3}{2}}$ | M1 A1 | M1: evidence of differentiation $x^n \to x^{n-1}$; A1: cao and cso |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| When $x=4$, $y=25$ | B1 | 25 only |
| $f'(4) = -1 - \frac{12}{8} = -2\frac{1}{2}$ | M1 | Substitute $x=4$ into derived function |
| $y - 25 = -\frac{5}{2}(x-4)$ | M1 A1 | M1: uses "25" and gradient from calculus with $x=4$; A1: any correct form e.g. $y = -\frac{5}{2}x + 35$, $5x + 2y - 70 = 0$ |
---12.
$$f ( x ) = \frac { ( 4 + 3 \sqrt { } x ) ^ { 2 } } { x } , \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { f } ( x ) = A x ^ { - 1 } + B x ^ { k } + C$, where $A , B , C$ and $k$ are constants to be determined.
\item Hence find $\mathrm { f } ^ { \prime } ( x )$.
\item Find an equation of the tangent to the curve $y = \mathrm { f } ( x )$ at the point where $x = 4$\\
2. LIIIII
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2016 Q12 [10]}}