OCR MEI C2 — Question 2 11 marks

Exam BoardOCR MEI
ModuleC2 (Core Mathematics 2)
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStationary points and optimisation
TypeFind range where function increasing/decreasing
DifficultyModerate -0.3 This is a straightforward C2 integration and differentiation question with standard techniques. Part (i) requires polynomial integration and interpretation (net area), while part (ii) involves finding dy/dx, solving a quadratic using the formula, and identifying decreasing intervals. All steps are routine for C2 level with no novel problem-solving required, making it slightly easier than average.
Spec1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits
Fig. 11 shows the curve \(y = x^3 - 3x^2 - x + 3\). \includegraphics{figure_11}
  1. Use calculus to find \(\int_{-1}^{3} (x^3 - 3x^2 - x + 3) dx\) and state what this represents. [6]
  2. Find the \(x\)-coordinates of the turning points of the curve \(y = x^3 - 3x^2 - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x^3 - 3x^2 - x + 3\) is a decreasing function. [5]
Question 2:
AnswerMarks
2x4 x2
(i) x3  3x
4 2
their integral at 3 − their integral at 1
[= 2.25 − 1.75]
= −4 isw
represents area between curve and x
axis between x = 1 and 3
AnswerMarks
negative since below x-axisM2
M1
A1
B1
AnswerMarks
B1M1 if at least two terms correct
dependent on integration attemptedignore + c
M0 for evaluation of x3 3x2 x3 or of
differentiated version
B0 for area under or above curve between x = 1 and 3
AnswerMarks
2(ii) y = 3x2 − 6x − 1
their y = 0 soi
x = with a = 3, b = -
6 and c = -1 isw
x = or better as final answer
6 48 6 48
 x or ft their
6 6
AnswerMarks
final answerM1
M1
M1
A1
AnswerMarks
B1dependent on differentiation attempted
or 3(x – 1)2 - 4 [= 0] or better
2
eg A1 for 1 3
3
AnswerMarks
allow ≤ instead of <no follow through; NB or better stated without
working implies use of correct method
A0 for incorrect simplification, eg 1 ± √48
allow B1 if both inequalities are stated separately and
it’s clear that both apply
allow B1 if the terms and the signs are in reverse order
Question 2:
2 | x4 x2
(i) x3  3x
4 2
their integral at 3 − their integral at 1
[= 2.25 − 1.75]
= −4 isw
represents area between curve and x
axis between x = 1 and 3
negative since below x-axis | M2
M1
A1
B1
B1 | M1 if at least two terms correct
dependent on integration attempted | ignore + c
M0 for evaluation of x3 3x2 x3 or of
differentiated version
B0 for area under or above curve between x = 1 and 3
2 | (ii) y = 3x2 − 6x − 1
their y = 0 soi
x = with a = 3, b = -
6 and c = -1 isw
x = or better as final answer
6 48 6 48
 x or ft their
6 6
final answer | M1
M1
M1
A1
B1 | dependent on differentiation attempted
or 3(x – 1)2 - 4 [= 0] or better
2
eg A1 for 1 3
3
allow ≤ instead of < | no follow through; NB or better stated without
working implies use of correct method
A0 for incorrect simplification, eg 1 ± √48
allow B1 if both inequalities are stated separately and
it’s clear that both apply
allow B1 if the terms and the signs are in reverse order
Fig. 11 shows the curve $y = x^3 - 3x^2 - x + 3$.

\includegraphics{figure_11}

\begin{enumerate}[label=(\roman*)]
\item Use calculus to find $\int_{-1}^{3} (x^3 - 3x^2 - x + 3) dx$ and state what this represents. [6]
\item Find the $x$-coordinates of the turning points of the curve $y = x^3 - 3x^2 - x + 3$, giving your answers in surd form. Hence state the set of values of $x$ for which $y = x^3 - 3x^2 - x + 3$ is a decreasing function. [5]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C2  Q2 [11]}}
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Q1 4 Q2 11 Q3 3 Q4 4 Q5 10 Q6 5 Q7 4 Q8 5