| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Mean value of function |
| Difficulty | Moderate -0.8 This is a straightforward application of the mean value formula for functions. Part (a) requires computing (1/(b-a))∫f(x)dx with a simple polynomial integrand. Part (b) uses the linearity property that adding a constant c to a function adds c to its mean value. Both parts are routine calculations with no problem-solving insight required, making this easier than average. |
| Spec | 4.08e Mean value of function: using integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{5-1}\int_1^5 3x^2\,dx\) | M1 | Evaluates the definite integral of f between 1 and 5 |
| \(= \frac{1}{4} \times 124 = 31\) | A1 | Obtains 31 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(31 + c = 40\) | M1 | Sets up a correct equation to find \(c\); follow through their part (a) answer |
| \(c = 9\) | A1F | Obtains the correct result; follow through their part (a) answer |
## Question 5:
### Part 5(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{5-1}\int_1^5 3x^2\,dx$ | M1 | Evaluates the definite integral of f between 1 and 5 |
| $= \frac{1}{4} \times 124 = 31$ | A1 | Obtains 31 |
### Part 5(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $31 + c = 40$ | M1 | Sets up a correct equation to find $c$; follow through their part (a) answer |
| $c = 9$ | A1F | Obtains the correct result; follow through their part (a) answer |
---5 The function f is defined by
$$f ( x ) = 3 x ^ { 2 } \quad 1 \leq x \leq 5$$
5
\begin{enumerate}[label=(\alph*)]
\item Find the mean value of f\\
5
\item The function g is defined by
$$\mathrm { g } ( x ) = \mathrm { f } ( x ) + c \quad 1 \leq x \leq 5$$
The mean value of $g$ is 40\\
Calculate the value of the constant $c$
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2023 Q5 [4]}}