| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Integration with algebraic manipulation |
| Difficulty | Moderate -0.8 This C1 question requires expanding a binomial with a square root, solving a simple equation, substituting a value, and integrating a polynomial after expansion. All steps are routine algebraic manipulations with no problem-solving insight needed, making it easier than average but not trivial due to the surd manipulation required. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((2 - \sqrt{x})^2 = 0\) | ||
| \(\sqrt{x} = 2\) | M1 | |
| \(x = 4\) | A1 | |
| (b) \(= (2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3}\) | M1 A1 | |
| (c) \(= \int (2 - \sqrt{x})^2 \, dx\) | ||
| \(= \int (4 - 4\sqrt{x} + x) \, dx\) | B1 | |
| \(= 4x - \frac{8}{3}x^{3/2} + \frac{1}{2}x^2 + c\) | M1 A2 | (8) |
**(a)** $(2 - \sqrt{x})^2 = 0$ | |
$\sqrt{x} = 2$ | M1 |
$x = 4$ | A1 |
**(b)** $= (2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3}$ | M1 A1 |
**(c)** $= \int (2 - \sqrt{x})^2 \, dx$ | |
$= \int (4 - 4\sqrt{x} + x) \, dx$ | B1 |
$= 4x - \frac{8}{3}x^{3/2} + \frac{1}{2}x^2 + c$ | M1 A2 | (8)$\text{f}(x) = (2 - \sqrt{x})^2, \quad x > 0$.
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $\text{f}(x) = 0$. [2]
\item Find $\text{f}(3)$, giving your answer in the form $a + b\sqrt{3}$, where $a$ and $b$ are integers. [2]
\item Find
$$\int \text{f}(x) \, dx.$$ [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q5 [8]}}