CAIE P3 2018 November — Question 8 10 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2018
SessionNovember
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGeneralised Binomial Theorem and Partial Fractions
TypePartial fractions with repeated linear factor
DifficultyStandard +0.3 This is a standard two-part question combining partial fractions with binomial expansion. Part (i) requires decomposing into the form A/(1-2x) + B/(2-x) + C/(2-x)^2, which is routine but involves a repeated linear factor. Part (ii) applies standard binomial expansions to each term. While requiring careful algebra across multiple steps (10 marks total), both techniques are core P3/C4 content with no novel problem-solving required.
Spec1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions
Let \(f(x) = \frac{7x^2 - 15x + 8}{(1 - 2x)(2 - x)^2}\).
  1. Express \(f(x)\) in partial fractions. [5]
  2. Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
Question 8:
AnswerMarks
8(i)A B C
State or imply the form + +
( )2
AnswerMarks
1−2x 2−x 2−xB1
Use a correct method for finding a constant
M1 is available following a single slip in working from their form
AnswerMarks Guidance
but no A marks (even if a constant is “correct”)M1 7= A+2B
−15=−4A−5B−2C
8=4A+2B+C
AnswerMarks
Obtain one of A = 1, B = 3, C = – 2A1
Obtain a second valueA1
Obtain the third valueA1
A Dx+E
[Mark the form + , where A = 1, D = – 3 and
( )2
1−2x 2−x
E = 4, B1M1A1A1A1 as above.]
5
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
8(ii)Use a correct method to find the first two terms of the expansion of
−1 −2
( )−1 ( )−1  1  ( )−2  1 
1−2x , 2−x , 1− x , 2−x or 1− x
AnswerMarks Guidance
 2   2 M1 Symbolic coefficients are not sufficient for the M1
Obtain correct unsimplified expansions up to the term in x2of each
AnswerMarks Guidance
partial fractionA3ft 1+2x+4x2
The ft is on A, B, C. 3 + 3x+ 3x2
2 4 8
−1 − 1x−3x2
2 2 8
9
Obtain final answer 2+ x+4x2
AnswerMarks
4A1
[For the A, D, E form of fractions give M1A2ft for the expanded
partial fractions, then, if D ≠ 0, M1 for multiplying out fully, and
AnswerMarks
A1 for the final answer.][The ft is on A, D, E.]
5
AnswerMarks Guidance
QuestionAnswer Marks 
Question 8:
--- 8(i) ---
8(i) | A B C
State or imply the form + +
( )2
1−2x 2−x 2−x | B1
Use a correct method for finding a constant
M1 is available following a single slip in working from their form
but no A marks (even if a constant is “correct”) | M1 | 7= A+2B
−15=−4A−5B−2C
8=4A+2B+C
Obtain one of A = 1, B = 3, C = – 2 | A1
Obtain a second value | A1
Obtain the third value | A1
A Dx+E
[Mark the form + , where A = 1, D = – 3 and
( )2
1−2x 2−x
E = 4, B1M1A1A1A1 as above.]
5
Question | Answer | Marks | Guidance
--- 8(ii) ---
8(ii) | Use a correct method to find the first two terms of the expansion of
−1 −2
( )−1 ( )−1  1  ( )−2  1 
1−2x , 2−x , 1− x , 2−x or 1− x
 2   2  | M1 | Symbolic coefficients are not sufficient for the M1
Obtain correct unsimplified expansions up to the term in x2of each
partial fraction | A3ft | 1+2x+4x2
The ft is on A, B, C. 3 + 3x+ 3x2
2 4 8
−1 − 1x−3x2
2 2 8
9
Obtain final answer 2+ x+4x2
4 | A1
[For the A, D, E form of fractions give M1A2ft for the expanded
partial fractions, then, if D ≠ 0, M1 for multiplying out fully, and
A1 for the final answer.] | [The ft is on A, D, E.]
5
Question | Answer | Marks | Guidance
Let $f(x) = \frac{7x^2 - 15x + 8}{(1 - 2x)(2 - x)^2}$.

\begin{enumerate}[label=(\roman*)]
\item Express $f(x)$ in partial fractions. [5]

\item Hence obtain the expansion of $f(x)$ in ascending powers of $x$, up to and including the term in $x^2$. [5]
\end{enumerate}

\hfill \mbox{\textit{CAIE P3 2018 Q8 [10]}}
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