| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find second derivative |
| Difficulty | Easy -1.3 This is a straightforward C1 question testing basic differentiation and integration rules with no problem-solving required. Part (i) involves direct application of the power rule to find first and second derivatives of a simple polynomial. Part (ii) requires rewriting terms and applying standard integration formulas. All techniques are routine recall with no conceptual challenges. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (i)(a) \(15x^2 + 7\) | M1 A1 A1 | (3 marks) Allow any equivalent version of each term |
| (i)(b) \(30x\) | B1ft | (1 mark) |
| (ii) \(x + 2x^2 + x^{-1} + C\) | M1 A1 A1 A1 | (4 marks) A1: \(x + C\), A1: \(2x^2\), A1: \(x^{-1}\). Allow any equivalent version of each term |
**(i)(a)** $15x^2 + 7$ | M1 A1 A1 | (3 marks) Allow any equivalent version of each term
**(i)(b)** $30x$ | B1ft | (1 mark)
**(ii)** $x + 2x^2 + x^{-1} + C$ | M1 A1 A1 A1 | (4 marks) A1: $x + C$, A1: $2x^2$, A1: $x^{-1}$. Allow any equivalent version of each term
**Total: 8 marks**
---\begin{enumerate}[label=(\roman*)]
\item Given that $y = 5 x ^ { 3 } + 7 x + 3$, find\\
(a) $\frac { \mathrm { d } y } { \mathrm {~d} x }$, (b) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
\item Find $\int \left( 1 + 3 \sqrt { } x - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2005 Q2 [8]}}