Edexcel P2 2022 October — Question 3 7 marks

Exam BoardEdexcel
ModuleP2 (Pure Mathematics 2)
Year2022
SessionOctober
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicArithmetic Sequences and Series
TypePeriodic or repeating sequence
DifficultyStandard +0.3 This question requires recognizing that cos²(nπ/3) is periodic with period 6, evaluating a few trigonometric values (standard angles), and summing a repeating pattern. While it involves multiple steps, the techniques are straightforward: basic trig evaluation and arithmetic of a repeating sequence. The 'hence' structure guides students through the solution, making it slightly easier than average.
Spec1.04e Sequences: nth term and recurrence relations1.04g Sigma notation: for sums of series1.05g Exact trigonometric values: for standard angles
  1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$a _ { n } = \cos ^ { 2 } \left( \frac { \mathrm { n } \pi } { 3 } \right)$$ Find the exact values of
    1. \(a _ { 1 }\)
    2. \(a _ { 2 }\)
    3. \(a _ { 3 }\)
  2. Hence find the exact value of 50 $$n + \cos ^ { 2 } \frac { n \pi } { 3 }$$ You must make your method clear.
Question 3(a)(i):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(a_1=\dfrac{1}{4}\)B1 Accept 0.25; note degrees mode gives \(\approx0.9997\) and scores no marks
Question 3(a)(ii):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(a_2=\dfrac{1}{4}\)B1 Accept 0.25
Question 3(a)(iii):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(a_3=1\)B1 Must clearly not come from rounded degrees calculation
Question 3(b):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\dfrac{50}{2}[2+49]\ (=1275)\)M1 Correct attempt to find sum \(1+2+3+\ldots+50\)
\(1275\)A1 Any correct numerical expression or 1275; correct answer of 1275 implies both M1A1
\(\displaystyle\sum_{n=1}^{50}\cos^2\!\left(\dfrac{n\pi}{3}\right)=34\times\dfrac{1}{4}+16\times1\)M1 Correct attempt to find \(\sum\cos^2\!\left(\tfrac{n\pi}{3}\right)\); must be correct method for correct sequence; writing just \(\tfrac{49}{2}\) scores M0
\(1275+\dfrac{49}{2}=\dfrac{2599}{2}\)A1 Accept 1299.5 or exact equivalent; isw once correct answer seen; correct answer only scores no marks 
# Question 3(a)(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $a_1=\dfrac{1}{4}$ | B1 | Accept 0.25; note degrees mode gives $\approx0.9997$ and scores no marks |

# Question 3(a)(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $a_2=\dfrac{1}{4}$ | B1 | Accept 0.25 |

# Question 3(a)(iii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $a_3=1$ | B1 | Must clearly not come from rounded degrees calculation |

# Question 3(b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\dfrac{50}{2}[2+49]\ (=1275)$ | M1 | Correct attempt to find sum $1+2+3+\ldots+50$ |
| $1275$ | A1 | Any correct numerical expression or 1275; correct answer of 1275 implies both M1A1 |
| $\displaystyle\sum_{n=1}^{50}\cos^2\!\left(\dfrac{n\pi}{3}\right)=34\times\dfrac{1}{4}+16\times1$ | M1 | Correct attempt to find $\sum\cos^2\!\left(\tfrac{n\pi}{3}\right)$; must be correct method for correct sequence; writing just $\tfrac{49}{2}$ scores M0 |
| $1275+\dfrac{49}{2}=\dfrac{2599}{2}$ | A1 | Accept 1299.5 or exact equivalent; isw once correct answer seen; correct answer only scores no marks |

---
\begin{enumerate}
  \item A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by
\end{enumerate}

$$a _ { n } = \cos ^ { 2 } \left( \frac { \mathrm { n } \pi } { 3 } \right)$$

Find the exact values of\\
(a) (i) $a _ { 1 }$\\
(ii) $a _ { 2 }$\\
(iii) $a _ { 3 }$\\
(b) Hence find the exact value of

50

$$n + \cos ^ { 2 } \frac { n \pi } { 3 }$$

You must make your method clear.

\hfill \mbox{\textit{Edexcel P2 2022 Q3 [7]}}
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