CAIE P1 2018 November — Question 5 7 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2018
SessionNovember
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicArithmetic Sequences and Series
TypeMixed arithmetic and geometric
DifficultyStandard +0.3 This is a simultaneous equations problem requiring students to apply the definitions of AP (constant difference) and GP (constant ratio). While it involves solving a quadratic, the setup is straightforward and the algebraic manipulation is routine for A-level. The problem is slightly above average difficulty due to the need to connect two progressions, but remains a standard textbook-style question.
Spec1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum
The first three terms of an arithmetic progression are \(4\), \(x\) and \(y\) respectively. The first three terms of a geometric progression are \(x\), \(y\) and \(18\) respectively. It is given that both \(x\) and \(y\) are positive.
  1. Find the value of \(x\) and the value of \(y\). [4]
  2. Find the fourth term of each progression. [3]
Question 5:
AnswerMarks Guidance
5(i)From the AP: x−4= y−x B1
Or equivalent statement e.g. y = 2x – 4 or x= .
2
y 18
From the GP: =
AnswerMarks Guidance
x yB1 y2
Or equivalent statement e.g. y2 = 18x or x= .
18
AnswerMarks Guidance
Simultaneous equations: y²−9y−36=0 or 2x²−17x+8=0M1 Elimination of either x or y to give a three term quadratic
(= 0)
OR
4+2d
4+d =x, 4+2d=y → =r oe
AnswerMarks
4+dB1
4+2d 2
( 4+d )   =18 → 2d2 −d −28=0
 
4+d
AnswerMarks Guidance
 M1 Uses ar2 = 18 to give a three term quadratic (= 0)
d = 4B1 −7
Condone inclusion of d = oe
2
AnswerMarks Guidance
QuestionAnswer Marks
5(i)OR
y 18
From the GP =
AnswerMarks
x yB1
y2 y2 y2
→x= → 4+ d = →d = – 4
AnswerMarks
18 18 18B1
 y2 
4+2 −4 = y → y²−9y−36=0
18
AnswerMarks Guidance
 M1
x = 8, y = 12.A1 1 
Needs both x and y. Condone  ,−3 included in final
2 
answer.
Fully correct answer www 4/4.
4
AnswerMarks Guidance
5(ii)AP 4th term = 16 B1
Condone inclusion of oe
2
3
12
GP 4th term = 8 ×
 
AnswerMarks Guidance
 8 M1 A valid method using their x and y from (i).
= 27A1 Condone inclusion of –108
Note: Answers from fortuitous x = 8, y = 12 in (i) can only
score M1.
Unidentified correct answer(s) with no working seen after
valid x = 8, y = 12 to be credited with appropriate marks.
3
AnswerMarks Guidance
QuestionAnswer Marks 
Question 5:
--- 5(i) ---
5(i) | From the AP: x−4= y−x | B1 | y+4
Or equivalent statement e.g. y = 2x – 4 or x= .
2
y 18
From the GP: =
x y | B1 | y2
Or equivalent statement e.g. y2 = 18x or x= .
18
Simultaneous equations: y²−9y−36=0 or 2x²−17x+8=0 | M1 | Elimination of either x or y to give a three term quadratic
(= 0)
OR
4+2d
4+d =x, 4+2d=y → =r oe
4+d | B1
4+2d 2
( 4+d )   =18 → 2d2 −d −28=0
 
4+d
  | M1 | Uses ar2 = 18 to give a three term quadratic (= 0)
d = 4 | B1 | −7
Condone inclusion of d = oe
2
Question | Answer | Marks | Guidance
5(i) | OR
y 18
From the GP =
x y | B1
y2 y2 y2
→x= → 4+ d = →d = – 4
18 18 18 | B1
 y2 
4+2 −4 = y → y²−9y−36=0
18
  | M1
x = 8, y = 12. | A1 | 1 
Needs both x and y. Condone  ,−3 included in final
2 
answer.
Fully correct answer www 4/4.
4
--- 5(ii) ---
5(ii) | AP 4th term = 16 | B1 | −13
Condone inclusion of oe
2
3
12
GP 4th term = 8 ×
 
 8  | M1 | A valid method using their x and y from (i).
= 27 | A1 | Condone inclusion of –108
Note: Answers from fortuitous x = 8, y = 12 in (i) can only
score M1.
Unidentified correct answer(s) with no working seen after
valid x = 8, y = 12 to be credited with appropriate marks.
3
Question | Answer | Marks | Guidance
The first three terms of an arithmetic progression are $4$, $x$ and $y$ respectively. The first three terms of a geometric progression are $x$, $y$ and $18$ respectively. It is given that both $x$ and $y$ are positive.

\begin{enumerate}[label=(\roman*)]
\item Find the value of $x$ and the value of $y$. [4]
\item Find the fourth term of each progression. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2018 Q5 [7]}}
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