OCR MEI Paper 2 2023 June — Question 10 5 marks

Exam BoardOCR MEI
ModulePaper 2 (Paper 2)
Year2023
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Parts
TypeDouble integration by parts
DifficultyStandard +0.3 This is a straightforward application of integration by parts with clear choices (u=4x, dv=cos 2x dx), requiring one application to reach the answer. The limits are standard and the arithmetic is clean. Slightly above average difficulty only because it requires careful execution of the technique and evaluation at π/4, but it's a textbook example with no conceptual challenges.
Spec1.08i Integration by parts
10 Determine the exact value of \(\int _ { 0 } ^ { \frac { \pi } { 4 } } 4 x \cos 2 x d x\).
Question 10:
AnswerMarks Guidance
\(4x \times \frac{1}{2}\sin 2x - \int \frac{1}{2}\sin 2x \times 4\ dx\)M1* (3.1a) allow sign errors only, condone omission of \(dx\)
\(2x\sin 2x - \int 2\sin 2x\ dx\) oeA1 (1.1) allow omission of \(dx\); may be unsimplified
\(F[x] = 2x\sin 2x + \cos 2x\)A1 (1.1) ignore \(+ c\)
\(F\left[\frac{\pi}{4}\right] - F[0]\)M1dep* (1.1) must see substitution if \(F[x]\) incorrect, otherwise may be implied by correct answer
\(\frac{\pi}{2} - 1\) or \(\frac{\pi - 2}{2}\) iswA1 (3.2a) allow recovery from bracket error 
## Question 10:
$4x \times \frac{1}{2}\sin 2x - \int \frac{1}{2}\sin 2x \times 4\ dx$ | M1* (3.1a) | allow sign errors only, condone omission of $dx$
$2x\sin 2x - \int 2\sin 2x\ dx$ oe | A1 (1.1) | allow omission of $dx$; may be unsimplified
$F[x] = 2x\sin 2x + \cos 2x$ | A1 (1.1) | ignore $+ c$
$F\left[\frac{\pi}{4}\right] - F[0]$ | M1dep* (1.1) | must see substitution if $F[x]$ incorrect, otherwise may be implied by correct answer
$\frac{\pi}{2} - 1$ or $\frac{\pi - 2}{2}$ isw | A1 (3.2a) | allow recovery from bracket error
10 Determine the exact value of $\int _ { 0 } ^ { \frac { \pi } { 4 } } 4 x \cos 2 x d x$.

\hfill \mbox{\textit{OCR MEI Paper 2 2023 Q10 [5]}}
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Q1 3 Q2 3 Q3 3 Q4 5 Q5 3 Q6 4 Q7 4 Q8 6 Q9 5 Q10 5 Q11 6 Q12 4 Q13 9 Q14 8 Q15 7 Q16 8 Q17 6 Q18 11