| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Curve properties and tangent/normal |
| Difficulty | Moderate -0.3 This is a straightforward C1 integration question requiring standard techniques: integrating powers of x (including x^{-1/2}), finding the constant using a given point, then finding a tangent equation. All steps are routine with no problem-solving insight needed, making it slightly easier than average but not trivial due to the multiple parts and algebraic manipulation required. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((y=)\frac{3x^2}{2} - \frac{5x^{\frac{1}{2}}}{\frac{1}{2}} - 2x\) \((+c)\) | M1A1A1 | Attempt to integrate \(x^n \to x^{n+1}\); 1st A1 at least 2 correct terms; 2nd A1 all 3 terms correct (condone missing \(+c\)) |
| \(f(4) = 5 \Rightarrow 5 = \frac{3}{2} \times 16 - 10 \times 2 - 8 + c\) | M1 | Using point \((4,5)\) to form linear equation for \(c\); must use \(x=4\), \(y=5\) and function must have "changed" |
| \(c = 9\) | A1 | |
| \(\left[f(x) = \frac{3}{2}x^2 - 10x^{\frac{1}{2}} - 2x + 9\right]\) | Final expression not required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(m = 3 \times 4 - \frac{5}{2} - 2 = 7.5\) or \(\frac{15}{2}\) | M1 | Attempt to evaluate \(f'(4)\); must have at least \(3 \times 4\) or \(-\frac{5}{2}\) and clearly using \(f'(x)\) with \(x = 4\) |
| \(y - 5 = \frac{15}{2}(x-4)\) | M1A1 | Using value of \(m\) to form equation through \((4,5)\); 1st A1 for any correct expression for line |
| \(2y - 15x + 50 = 0\) (o.e.) | A1 | Any correct equation with integer coefficients; "=" required e.g. \(2y = 15x - 50\) acceptable |
## Question 11:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(y=)\frac{3x^2}{2} - \frac{5x^{\frac{1}{2}}}{\frac{1}{2}} - 2x$ $(+c)$ | M1A1A1 | Attempt to integrate $x^n \to x^{n+1}$; 1st A1 at least 2 correct terms; 2nd A1 all 3 terms correct (condone missing $+c$) |
| $f(4) = 5 \Rightarrow 5 = \frac{3}{2} \times 16 - 10 \times 2 - 8 + c$ | M1 | Using point $(4,5)$ to form linear equation for $c$; must use $x=4$, $y=5$ and function must have "changed" |
| $c = 9$ | A1 | |
| $\left[f(x) = \frac{3}{2}x^2 - 10x^{\frac{1}{2}} - 2x + 9\right]$ | | Final expression not required |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $m = 3 \times 4 - \frac{5}{2} - 2 = 7.5$ or $\frac{15}{2}$ | M1 | Attempt to evaluate $f'(4)$; must have at least $3 \times 4$ or $-\frac{5}{2}$ and clearly using $f'(x)$ with $x = 4$ |
| $y - 5 = \frac{15}{2}(x-4)$ | M1A1 | Using value of $m$ to form equation through $(4,5)$; 1st A1 for any correct expression for line |
| $2y - 15x + 50 = 0$ (o.e.) | A1 | Any correct equation with integer coefficients; "=" required e.g. $2y = 15x - 50$ acceptable |
The images you've shared appear to be essentially blank pages — one is completely empty, and the other contains only publisher/contact information for Edexcel Publications (address, phone, fax, email, order code UA023696 Summer 2010, and company registration details).
There is **no mark scheme content** visible in these images. No questions, answers, mark allocations, or guidance notes are present.
Could you share the actual mark scheme pages? They may have been missed in the upload.\begin{enumerate}
\item The curve $C$ has equation $y = \mathrm { f } ( x ) , \quad x > 0$, where
\end{enumerate}
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x - \frac { 5 } { \sqrt { } x } - 2$$
Given that the point $P ( 4,5 )$ lies on $C$, find\\
(a) $\mathrm { f } ( x )$,\\
(b) an equation of the tangent to $C$ at the point $P$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
\hfill \mbox{\textit{Edexcel C1 2010 Q11 [9]}}