AS June 2023 Paper 1 Q5
5 The function f is defined by
\[\mathrm{f}(x) = 3x^2 \qquad 1 \leqslant x \leqslant 5\](a) Find the mean value of f [2 marks]
(b) The function g is defined by\[\mathrm{g}(x) = \mathrm{f}(x) + c \qquad 1 \leqslant x \leqslant 5\]
The mean value of g is 40
Calculate the value of the constant \(c\) [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| Evaluates the definite integral of f between 1 and 5 | M1 | 1.1a |
| Obtains 31 | A1 | 1.1b |
| (2) |
Typical solution
\[\begin{aligned} &\frac{1}{5 - 1}\int_1^5 3x^2\,\mathrm{d}x \\ &= \frac{1}{4} \times 124 \\ &= 31\end{aligned}\]| Scheme | Marks | AO |
|---|---|---|
| Sets up a correct equation to find \(c\) Follow through their part (a) answer. | M1 | 3.1a |
| Obtains the correct result. Follow through their part (a) answer. | A1F | 1.1b |
| (2) | ||
| (4 marks) |