CAIE P1 2024 June — Question 5 9 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2024
SessionJune
Marks9
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Mark schemeDownload PDF ↗
TopicArithmetic Sequences and Series
TypeTrigonometric arithmetic progression
DifficultyStandard +0.3 This question tests standard arithmetic and geometric progression formulas with trigonometric terms. Part (a) requires finding the common difference and applying the sum formula with substituted values. Part (b)(i) needs the sum to infinity formula with common ratio sin θ/tan θ = cos θ. Part (b)(ii) applies the finite GP sum formula. All steps are routine applications of memorized formulas with straightforward trigonometric substitutions—no novel insight or complex problem-solving required, making it slightly easier than average.
Spec1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<11.05g Exact trigonometric values: for standard angles
The first and second terms of an arithmetic progression are \(\tan\theta\) and \(\sin\theta\) respectively, where \(0 < \theta < \frac{1}{2}\pi\). \begin{enumerate}[label=(\alph*)] \item Given that \(\theta = \frac{1}{4}\pi\), find the exact sum of the first 40 terms of the progression. [4] \end enumerate} The first and second terms of a geometric progression are \(\tan\theta\) and \(\sin\theta\) respectively, where \(0 < \theta < \frac{1}{2}\pi\).
    1. Find the sum to infinity of the progression in terms of \(\theta\). [2]
    2. Given that \(\theta = \frac{1}{3}\pi\), find the sum of the first 10 terms of the progression. Give your answer correct to 3 significant figures. [3]
Question 5:
AnswerMarks Guidance
5(a)d sintan M1
term.
Condone incorrect evaluation before subtraction.
2
d  1
AnswerMarks Guidance
2A1 OE
Sight of 0.29 AWRT can be awarded M1A1.
40
S  2tan39sintan
AnswerMarks Guidance
40 2M1 Use of a correct formula for S . Condone use of
40
their, clearly identified, incorrect values for a and d
for this mark.
780
390 2740 or 740
AnswerMarks Guidance
2A1 ISW
If A0 then sight of 188 AWRT, 188.5 or 189
should be awarded M1A1M1A0.
4
AnswerMarks
5(b)(i)sin
r  cos
AnswerMarks Guidance
tanB1 Condone omission of .
tan sin tan2
S  or or
AnswerMarks Guidance
 1cos coscos2 tansinB1 ISW
Do not allow fractions within fractions nor omission
of .
2
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
5(b)(ii)1
a 3 1.73.. and r 
AnswerMarks Guidance
2B1 OE, SOI.
 1 10 
1  
S  3  2 
10  1 
1
 
 2 
AnswerMarks Guidance
 M1 This mark can be awarded for a correct formula with
their values for a and r or
110
sin
atan and r  or cos. Condone .
tan 2
AnswerMarks Guidance
= 3.46A1 AWRT
1023 3
Condone .
512
AnswerMarks
3Note: S gives the same answer but scores B1M0A0.
9
AnswerMarks Guidance
QuestionAnswer Marks 
Question 5:
--- 5(a) ---
5(a) | d sintan | M1 | For subtraction of their first term from their second
term.
Condone incorrect evaluation before subtraction.
2
d  1
2 | A1 | OE
Sight of 0.29 AWRT can be awarded M1A1.
40
S  2tan39sintan
40 2 | M1 | Use of a correct formula for S . Condone use of
40
their, clearly identified, incorrect values for a and d
for this mark.
780
390 2740 or 740
2 | A1 | ISW
If A0 then sight of 188 AWRT, 188.5 or 189
should be awarded M1A1M1A0.
4
--- 5(b)(i) ---
5(b)(i) | sin
r  cos
tan | B1 | Condone omission of .
tan sin tan2
S  or or
 1cos coscos2 tansin | B1 | ISW
Do not allow fractions within fractions nor omission
of .
2
Question | Answer | Marks | Guidance
--- 5(b)(ii) ---
5(b)(ii) | 1
a 3 1.73.. and r 
2 | B1 | OE, SOI.
 1 10 
1  
S  3  2 
10  1 
1
 
 2 
  | M1 | This mark can be awarded for a correct formula with
their values for a and r or
110
sin
atan and r  or cos. Condone .
tan 2
= 3.46 | A1 | AWRT
1023 3
Condone .
512
3 | Note: S gives the same answer but scores B1M0A0.
9
Question | Answer | Marks | Guidance
The first and second terms of an arithmetic progression are $\tan\theta$ and $\sin\theta$ respectively, where $0 < \theta < \frac{1}{2}\pi$.

\begin{enumerate}[label=(\alph*)]
\item Given that $\theta = \frac{1}{4}\pi$, find the exact sum of the first 40 terms of the progression. [4]
\end enumerate}

The first and second terms of a geometric progression are $\tan\theta$ and $\sin\theta$ respectively, where $0 < \theta < \frac{1}{2}\pi$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item \begin{enumerate}[label=(\roman*)]
\item Find the sum to infinity of the progression in terms of $\theta$. [2]

\item Given that $\theta = \frac{1}{3}\pi$, find the sum of the first 10 terms of the progression. Give your answer correct to 3 significant figures. [3]
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2024 Q5 [9]}}
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