| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Differentiate after index conversion |
| Difficulty | Moderate -0.3 This is a straightforward C1 differentiation and integration question requiring standard application of power rule twice for part (a), algebraic substitution for part (b), and expansion followed by term-by-term integration for part (c). While it has multiple parts and requires careful algebra, all techniques are routine for C1 level with no problem-solving insight needed, making it slightly easier than average. |
| 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 |
|---|---|---|
| (a) \(\frac{dy}{dx} = 3x^{\frac{1}{2}}\) | M1 A1 | |
| \(\frac{d^2y}{dx^2} = \frac{3}{2}x^{-\frac{1}{2}}\) | A1 | |
| (b) LHS \(= 4x^2\left(\frac{3}{2}x^{-\frac{1}{2}}\right) - 3(2x^{\frac{1}{2}} - 1) = 6x^{\frac{3}{2}} - 6x^{\frac{1}{2}} + 3 = 3\) | M1, A1 | [\(k = 3\)] |
| (c) \(= \int (2x^{\frac{1}{2}} - 1)^2 \, dx = \int (4x - 4x^{\frac{1}{2}} + 1) \, dx\) | M1 A1 | |
| \(= x^4 - \frac{8}{5}x^{\frac{3}{2}} + x + c\) | M1 A3 | (11 marks) |
**(a)** $\frac{dy}{dx} = 3x^{\frac{1}{2}}$ | M1 A1 |
$\frac{d^2y}{dx^2} = \frac{3}{2}x^{-\frac{1}{2}}$ | A1 |
**(b)** LHS $= 4x^2\left(\frac{3}{2}x^{-\frac{1}{2}}\right) - 3(2x^{\frac{1}{2}} - 1) = 6x^{\frac{3}{2}} - 6x^{\frac{1}{2}} + 3 = 3$ | M1, A1 | [$k = 3$]
**(c)** $= \int (2x^{\frac{1}{2}} - 1)^2 \, dx = \int (4x - 4x^{\frac{1}{2}} + 1) \, dx$ | M1 A1 |
$= x^4 - \frac{8}{5}x^{\frac{3}{2}} + x + c$ | M1 A3 | (11 marks)\begin{enumerate}
\item Given that
\end{enumerate}
$$y = 2 x ^ { \frac { 3 } { 2 } } - 1$$
(a) find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$,\\
(b) show that
$$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 y = k$$
where $k$ is an integer to be found,\\
(c) find
$$\int y ^ { 2 } \mathrm {~d} x$$
\hfill \mbox{\textit{Edexcel C1 Q8 [11]}}