AQA Paper 3 2019 June — Question 7 8 marks

Exam BoardAQA
ModulePaper 3 (Paper 3)
Year2019
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPartial Fractions
TypeRepeated linear factor with distinct linear factor – decompose and integrate
DifficultyStandard +0.3 This is a standard partial fractions question with a repeated linear factor, followed by routine integration. Part (a) is straightforward algebraic manipulation (cover-up method works directly), and part (b) requires integrating two simple terms and evaluating limits—all textbook techniques with no novel insight required. Slightly easier than average due to the repeated factor making partial fractions simpler than distinct factors.
Spec1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions
  1. Express \(\frac{4x + 3}{(x - 1)^2}\) in the form \(\frac{A}{x - 1} + \frac{B}{(x - 1)^2}\) [3 marks]
  2. Show that $$\int_3^4 \frac{4x + 3}{(x - 1)^2} \, dx = p + \ln q$$ where \(p\) and \(q\) are rational numbers. [5 marks]
Question 7:
AnswerMarks Guidance
7(a)Forms 4x+3≡ A ( x−1 )+B 1.1b
≡ +
( 𝑥𝑥 −+1 )≡² (𝑥𝑥 −−1 1)+B (𝑥𝑥 −1)²
4 x 3 A x
Let x = 1 hence B = 7
Let x = 0 then 3 = B – A and
hence
A = 4
A = 4 and B = 7
Uses substitution or comparison
of coefficients to find their A or B
(must have degree of LHS =
AnswerMarks Guidance
degree of RHS)1.1a M1
Obtains correct A and B1.1b A1
AnswerMarks Guidance
7(b)Integrates their expression, at
least one term correct3.1a M1
4 7
� � + �𝑑𝑑𝑥𝑥
3 𝑥𝑥−1 (𝑥𝑥−1)²
=
4
7
�4ln(𝑥𝑥−1)−𝑥𝑥−1�
3
= −
7 7
�4ln3−3� �4ln2−2�
=4ln +
3 7
2 6
=ln +
4
3 7
4
2 6
= ln +
81 7
16 6
Integrates their expression fully
correctly
Must be of the form
B
Aln ( x−1 ) −
x−1
OE
AnswerMarks Guidance
FT their A and B1.1b A1F
Substitutes limits correctly into
AnswerMarks Guidance
their integrated expression1.1a M1
Uses at least one law of logs
AnswerMarks Guidance
correctly1.1a M1
Completes argument to obtain
correct exact answer in correct
form or stating p = and q =
7 81
No subsequent incorrect
AnswerMarks Guidance
working 6 162.1 R1
Total8
QMarking instructions AO 
Question 7:
--- 7(a) ---
7(a) | Forms 4x+3≡ A ( x−1 )+B | 1.1b | B1 | 4𝑥𝑥+3 𝐴𝐴 𝐵𝐵
≡ +
( 𝑥𝑥 −+1 )≡² (𝑥𝑥 −−1 1)+B (𝑥𝑥 −1)²
4 x 3 A x
Let x = 1 hence B = 7
Let x = 0 then 3 = B – A and
hence
A = 4
A = 4 and B = 7
Uses substitution or comparison
of coefficients to find their A or B
(must have degree of LHS =
degree of RHS) | 1.1a | M1
Obtains correct A and B | 1.1b | A1
--- 7(b) ---
7(b) | Integrates their expression, at
least one term correct | 3.1a | M1 | 4
4 7
� � + �𝑑𝑑𝑥𝑥
3 𝑥𝑥−1 (𝑥𝑥−1)²
=
4
7
�4ln(𝑥𝑥−1)−𝑥𝑥−1�
3
= −
7 7
�4ln3−3� �4ln2−2�
=4ln +
3 7
2 6
=ln +
4
3 7
4
2 6
= ln +
81 7
16 6
Integrates their expression fully
correctly
Must be of the form
B
Aln ( x−1 ) −
x−1
OE
FT their A and B | 1.1b | A1F
Substitutes limits correctly into
their integrated expression | 1.1a | M1
Uses at least one law of logs
correctly | 1.1a | M1
Completes argument to obtain
correct exact answer in correct
form or stating p = and q =
7 81
No subsequent incorrect
working 6 16 | 2.1 | R1
Total | 8
Q | Marking instructions | AO | Mark | Typical solution
\begin{enumerate}[label=(\alph*)]
\item Express $\frac{4x + 3}{(x - 1)^2}$ in the form $\frac{A}{x - 1} + \frac{B}{(x - 1)^2}$ [3 marks]

\item Show that
$$\int_3^4 \frac{4x + 3}{(x - 1)^2} \, dx = p + \ln q$$
where $p$ and $q$ are rational numbers. [5 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 3 2019 Q7 [8]}}
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