| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Integration with given constant |
| Difficulty | Moderate -0.8 This is a straightforward C2 integration question requiring basic power rule application (rewriting x^{-2}), then substituting limits and solving a simple linear equation. Both parts are routine procedures with no problem-solving insight needed, making it easier than average but not trivial since it requires correct handling of the constant of integration and definite integral evaluation. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(\frac{1}{x^2} = x^{-2}\) seen or used. PI by correct integration of \(\frac{36}{x^2}\). | B1 |
| (a) | \(\int \left(\frac{36}{x^2} + ax\right) dx = \left[-\frac{36}{x} + \frac{ax^2}{2}\right] (+k)\) | M1 |
| (a) | A1 | |
| (b) | \(\int_1^3 \left(\frac{36}{x^2} + ax\right) dx = \left(-\frac{36}{3} + \frac{9a}{2}\right) - \left(-36 + \frac{a}{2}\right)\) | M1 |
| (b) | \(24 + 4a = 16, \quad (a = -2)\) | A1 |
| (b) | ||
| Total | 5 |
(a) | $\frac{1}{x^2} = x^{-2}$ seen or used. PI by correct integration of $\frac{36}{x^2}$. | B1 | Only one y-intercept, marked as 1 or coordinates (0,1) stated or 'y = 1 when x = 0'
(a) | $\int \left(\frac{36}{x^2} + ax\right) dx = \left[-\frac{36}{x} + \frac{ax^2}{2}\right] (+k)$ | M1 | Correct integration of either $\frac{36}{x^2}$ or $ax$.
(a) | | A1 | Correct integration of both terms, ACF, accept unsimplified. Condone missing +k.
(b) | $\int_1^3 \left(\frac{36}{x^2} + ax\right) dx = \left(-\frac{36}{3} + \frac{9a}{2}\right) - \left(-36 + \frac{a}{2}\right)$ | M1 | Attempt to find F(3) − F(1), following attempt at integration.
(b) | $24 + 4a = 16, \quad (a = -2)$ | A1 | M0 if F(x) = $\frac{36}{x^2} + ax$ or if uses an F(x) which contains part of the given integrand
(b) | | | $-2$ NMS scores 0/2
| **Total** | **5** |
**Note for (b):** Using the correct answer to (a), the example $-\frac{36}{3} + \frac{9a}{2} + 36 + \frac{a}{2}$ would get the M1 in (b) if no further evidence suggested the cand was working with F(3)+F(1).
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\begin{enumerate}[label=(\alph*)]
\item Find $\int \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x$, where $a$ is a constant.
\item Hence, given that $\int _ { 1 } ^ { 3 } \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x = 16$, find the value of the constant $a$.\\[0pt]
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2016 Q1 [5]}}