Question 7 - A-Level Maths

OCR MEI AS Paper 1 2023 June — Question 7 6 marks

Exam Board	OCR MEI
Module	AS Paper 1 (AS Paper 1)
Year	2023
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Areas by integration
Type	Area under polynomial curve
Difficulty	Moderate -0.8 This is a straightforward area under a curve question requiring students to find roots of a quadratic, set up a definite integral, and evaluate it. While it requires showing detailed reasoning and finding an exact answer, the steps are all standard AS-level techniques with no conceptual challenges or novel problem-solving required.
Spec	1.08e Area between curve and x-axis: using definite integrals

7 In this question you must show detailed reasoning. \includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-5_643_716_303_242} Find the exact area of the shaded region shown in the diagram, enclosed by the \(x\)-axis and the curve \(y = - 3 x ^ { 2 } + 7 x - 2\).

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Question 7:

Answer	Marks	Guidance
Answer	Marks	Guidance
Points of intersection with \(x\)-axis when \(-3x^2 + 7x - 2 = 0\)	M1	Attempt to find intersection with \(x\)-axis
\(x = \frac{1}{3},\ 2\)	A1	Both exact roots seen
\(\text{Area} = \int_{\frac{1}{3}}^{2}(-3x^2 + 7x - 2)\,dx\)	M1*	Allow for indefinite integral also
\(= \left[-x^3 + \frac{7}{2}x^2 - 2x\right]_{\frac{1}{3}}^{2}\)	A1	Correct indefinite integral
\(\left(-8 + \frac{7\times4}{2} - 2\times2\right) - \left(-\left(\frac{1}{3}\right)^3 + \frac{7}{2\times9} - \frac{2}{3}\right)\)	M1(dep)	Substitution of their limits into their cubic expression must be seen
\(= \frac{125}{54}\)	A1	Must be exact. Allow mixed number \(2\frac{17}{54}\) or recurring decimal \(2.3\dot{1}4\dot{8}\) www

## Question 7:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Points of intersection with $x$-axis when $-3x^2 + 7x - 2 = 0$ | M1 | Attempt to find intersection with $x$-axis |
| $x = \frac{1}{3},\ 2$ | A1 | Both exact roots seen |
| $\text{Area} = \int_{\frac{1}{3}}^{2}(-3x^2 + 7x - 2)\,dx$ | M1* | Allow for indefinite integral also |
| $= \left[-x^3 + \frac{7}{2}x^2 - 2x\right]_{\frac{1}{3}}^{2}$ | A1 | Correct indefinite integral |
| $\left(-8 + \frac{7\times4}{2} - 2\times2\right) - \left(-\left(\frac{1}{3}\right)^3 + \frac{7}{2\times9} - \frac{2}{3}\right)$ | M1(dep) | Substitution of their limits into their cubic expression must be seen |
| $= \frac{125}{54}$ | A1 | Must be exact. Allow mixed number $2\frac{17}{54}$ or recurring decimal $2.3\dot{1}4\dot{8}$ www |

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7 In this question you must show detailed reasoning.\\
\includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-5_643_716_303_242}

Find the exact area of the shaded region shown in the diagram, enclosed by the $x$-axis and the curve $y = - 3 x ^ { 2 } + 7 x - 2$.

\hfill \mbox{\textit{OCR MEI AS Paper 1 2023 Q7 [6]}}

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