| Exam Board | OCR MEI |
|---|---|
| Module | Further Extra Pure (Further Extra Pure) |
| Year | 2020 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Second-Order Homogeneous Recurrence Relations |
| Difficulty | Challenging +1.8 This is a Further Maths question on second-order recurrence relations requiring characteristic equation solution in complex form, conversion to trigonometric form using a given angle, and asymptotic analysis of a modified sequence. While the techniques are standard for Further Maths students (complex roots, De Moivre's theorem), the multi-step nature, need to work with complex exponentials/trigonometry, and the asymptotic behavior analysis across multiple cases elevates this significantly above average A-level difficulty. |
| Spec | 8.01c Sequence behaviour: periodic, convergent, divergent, oscillating, monotonic8.01d Sequence limits: limit of nth term as n tends to infinity, steady-states8.01g Second-order recurrence: solve with distinct, repeated, or complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (a) | u = kpn ⇒ p2 – 4p + 5 [= 0] |
| Answer | Marks |
|---|---|
| n | M1 |
| Answer | Marks |
|---|---|
| A1 | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | Auxiliary equation |
| Answer | Marks |
|---|---|
| n 2 | One sign error |
| Answer | Marks |
|---|---|
| [7] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | Auxiliary equation |
| Answer | Marks |
|---|---|
| n | M1 |
| => p = 2 ± i | M1 |
| r = 5, tanθ=0.5 soi | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (b) | If a = 0.1 then v converges to 0 as as n→∞. |
| Answer | Marks |
|---|---|
| ...and is oscillatory. | B1 |
| Answer | Marks |
|---|---|
| [5] | 2.5 |
| Answer | Marks |
|---|---|
| 2.2b | No need to mention oscillatory |
| Answer | Marks |
|---|---|
| positive and negative”). | Diagrams only not |
Question 3:
3 | (a) | u = kpn ⇒ p2 – 4p + 5 [= 0]
n
=> p = 2 ± i
r = 5, tanθ=0.5 soi
u = A(2 + i)n + B(2 – i)n
n
u = rn(αcos(nθ) + βsin(nθ)) soi
n
eg n = 0 [ ⇒ α = 0]
n
n = 1 => β = 1 so u =52sinnθ oe
n | M1
M1
M1
M1
A1
M1
A1 | 1.1
1.1
1.1
1.1
1.1
1.1
1.1 | Auxiliary equation
BC. Solving their auxiliary
equation
Finding mod/arg of at least one of
their roots
General solution in any form (can
be implied by correct real form)
FT
General solution in real form with
r and θ either specified or in situ
Substituting either initial
condition into their GS
n ( −1(1))
allow eg u =5 2sin ntan
n 2 | One sign error
Could be seen later
FT their A and B. r could
be seen as eg 2.24 and θ as
eg 0.46 or 26.6 here
One sign error
Could be seen later
FT their A and B. r could
be seen as eg 2.24 and θ as
eg 0.46 or 26.6 here
[7] | 1.1
1.1
1.1
1.1
1.1
1.1
1.1 | Auxiliary equation
BC. Solving their auxiliary
equation
Finding mod/arg of at least one
root
General solution in any form (can
be implied by correct real form)
Using one initial condition to find
an arbitrary constant
Solution given in mod/arg form
(Could also see )
𝑖𝑖𝑖𝑖
𝑒𝑒
Alternative Method
u = kpn ⇒ p2 – 4p + 5 = 0
n | M1
=> p = 2 ± i | M1
r = 5, tanθ=0.5 soi | M1
M1
u = A(2 + i)n + B(2 – i)n
n
M1
0 = A + B ⇒ A = -B
A(2 + i)n + B(2 – i)n = 1 ⇒ A = -½i, B = ½i
𝑢𝑢𝑛𝑛=
𝑛𝑛 𝑛𝑛
A1
1 𝑛𝑛 1 𝑛𝑛
− 𝑖𝑖√5 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐+𝑖𝑖𝑐𝑐𝑖𝑖𝑛𝑛𝑐𝑐) + 𝑖𝑖√5 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 −𝑖𝑖𝑐𝑐𝑖𝑖𝑛𝑛𝑐𝑐)
2 n 2
u =52sinnθ oe
n
A1
[7]
1.1
1.1
1.1
1.1
1.1
1.1
1.1
Auxiliary equation
BC. Solving their auxiliary
equation
Finding mod/arg of at least one
root
General solution in any form (can
be implied by correct real form)
Using one initial condition to find
an arbitrary constant
Solution given in mod/arg form
(Could also see )
3 | (b) | If a = 0.1 then v converges to 0 as as n→∞.
n
If a = 0.2 then v [does not converge...]
n
...and is bounded and oscillatory.
If a = 1 then v diverges...
n
...and is oscillatory. | B1
B1
B1
B1
B1
[5] | 2.5
2.2b
2.2b
2.2b
2.2b | No need to mention oscillatory
but must give the limit
Not “diverges”
Allow descriptions (eg “the sign
changes regularly” or “it goes
positive and negative” and “it is
bounded or “always between –1
and 1”).
Ignore “periodic”
Allow eg “the terms get bigger (in
size)”.
Allow descriptions (eg “the sign
changes regularly” or “it goes
positive and negative”). | Diagrams only not
sufficient for all three cases
B1 bounded
B1 oscillatory3 A sequence is defined by the recurrence relation $u _ { n + 2 } = 4 u _ { n + 1 } - 5 u _ { n }$ for $n \geqslant 0$, with $u _ { 0 } = 0$ and $u _ { 1 } = 1$.
\begin{enumerate}[label=(\alph*)]
\item Find an exact real expression for $u _ { n }$ in terms of $n$ and $\theta$, where $\tan \theta = \frac { 1 } { 2 }$.
A sequence is defined by $v _ { n } = a ^ { \frac { 1 } { 2 } n } u _ { n }$ for $n \geqslant 0$, where $a$ is a positive constant.
\item In each of the following cases, describe the behaviour of $v _ { n }$ as $n \rightarrow \infty$.
\begin{itemize}
\item $a = 0.1$
\item $a = 0.2$
\item $a = 1$
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Extra Pure 2020 Q3 [12]}}