| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reduction Formulae |
| Type | Trigonometric power reduction |
| Difficulty | Challenging +1.3 This is a standard reduction formula question from Further Maths requiring integration by parts to derive the recurrence relation, then applying it systematically. While it requires careful algebraic manipulation and multiple steps, the technique is well-practiced in Further Maths courses and follows a predictable pattern. The twist of converting cos²θ sin⁵θ using the identity adds modest complexity but remains within standard Further Maths territory. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)8.06a Reduction formulae: establish, use, and evaluate recursively |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (i) | DR |
| Answer | Marks |
|---|---|
| n | M1 |
| Answer | Marks |
|---|---|
| [5] | 2.1 |
| Answer | Marks |
|---|---|
| 1.1 | Correct splitting and attempt at (cid:179)n by |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (ii) | (a) |
| Answer | Marks |
|---|---|
| = 2 | M1 |
| Answer | Marks |
|---|---|
| [2] | i |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (ii) | (b) |
| Answer | Marks |
|---|---|
| 7 5 3 1 105 | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.1a |
| 2.1 | Repeated use of RF soi |
Question 5:
5 | (i) | DR
I (cid:32)(cid:179) (cid:83) sin(cid:84).sinn(cid:16)1(cid:84)d(cid:84)
n
0
(cid:83)
(cid:32)(cid:170)– cos(cid:84). sinn–1(cid:84)(cid:186) (cid:16)
(cid:172) (cid:188)
0
(cid:83)
...(cid:179) (cid:16)cos(cid:84).(n(cid:16)1)sinn(cid:16)2(cid:84).cos(cid:84)d(cid:84)
0
(cid:83)(cid:11) (cid:12)
(cid:32)0(cid:14)(cid:11)n(cid:16)1(cid:12)(cid:180) (cid:181) 1(cid:16)sin2(cid:84) sinn(cid:16)2(cid:84)d(cid:84)
(cid:182)
0
(cid:32) (cid:11) n(cid:16)1(cid:94)I (cid:16)I (cid:96)(cid:12)
n(cid:16)2 n
n(cid:16)1
(cid:159)I (cid:32) I
n n(cid:16)2
n | M1
A1
M1
A1
E1
[5] | 2.1
1.1
2.1
1.1
1.1 | Correct splitting and attempt at (cid:179)n by
parts
n
Use of trigonometric identity in order
e
to obtain …
... substitution for I
m
AG
5 | (ii) | (a) | DR
(cid:83)
I (cid:32)(cid:179) sin(cid:84)d(cid:84)
1
0
= 2 | M1
e
A1
[2] | i
c
3.1a
1.1
5 | (ii) | (b) | p
DR
S
Using c2 (cid:32)1(cid:16)s2, I (cid:32)I (cid:16)I
5 7
I (cid:16)I (cid:32) (cid:11) 1(cid:16)6(cid:12) I (cid:32)1(cid:117)4I
5 7 7 5 7 5 3
(cid:32) 1(cid:117)4(cid:117)2I (cid:32) 16
7 5 3 1 105 | M1
E1
[2] | 3.1a
2.1 | Repeated use of RF soi
Correct answer with sufficient detail to
show the reduction process5 In this question you must show detailed reasoning.\\
It is given that $I _ { n } = \int _ { 0 } ^ { \pi } \sin ^ { n } \theta \mathrm {~d} \theta$ for $n \geq 0$.
\begin{enumerate}[label=(\roman*)]
\item Prove that $I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$ for $n \geq 2$.
\item (a) Evaluate $I _ { 1 }$.\\
(b) Use the reduction formula to determine the exact value of $\int _ { 0 } ^ { \pi } \cos ^ { 2 } \theta \sin ^ { 5 } \theta \mathrm {~d} \theta$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure Q5 [9]}}