OCR Further Pure Core 1 2022 June — Question 7 10 marks

Exam BoardOCR
ModuleFurther Pure Core 1 (Further Pure Core 1)
Year2022
SessionJune
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration with Partial Fractions
TypeImproper integrals with partial fractions (infinite limit)
DifficultyChallenging +1.2 This is a Further Maths question combining partial fractions with improper integrals. Part (a) is routine algebraic manipulation with repeated and irreducible quadratic factors. Part (b) requires evaluating limits at infinity and applying arctan integration, which are standard Further Pure techniques. While more demanding than typical A-level questions due to the improper integral and Further Maths content, it follows a predictable template without requiring novel insight.
Spec4.05c Partial fractions: extended to quadratic denominators4.08c Improper integrals: infinite limits or discontinuous integrands
7
  1. Determine the values of \(A , B , C\) and \(D\) such that \(\frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \equiv \frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { C x + D } { x ^ { 2 } + 9 }\).
  2. In this question you must show detailed reasoning. Hence determine the exact value of \(\int _ { 3 } ^ { \infty } \frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \mathrm { d } x\).
Question 7:
AnswerMarks Guidance
7(a) x2 +18≡ Ax ( x2 +9 ) +B ( x2 +9 ) +( Cx+D ) x2
e.g.x=0⇒9B=18⇒ B=2
x=1⇒10A+10B+C+D=19
x=−1⇒−10A+10B−C+D=19
⇒10B+D=19⇒ D=−1
x=3i⇒−9 ( D+3Ci )=9⇒C =0
⇒10A+20−1=19⇒ A=0
AnswerMarks
i.e. A = 0, B = 2, C = 0, D = −1B1
M1
A1
A1
AnswerMarks
[4]1.1
1.1
1.1
AnswerMarks
1.1Correct multiplying out of fractions
Any substitutions to get a set of (at least) four simultaneous
equations solvable for and .
Or equating coefficients which gives
𝐴𝐴,𝐵𝐵,𝐶𝐶 𝐷𝐷
.
𝐴𝐴+𝐶𝐶 = 0,𝐵𝐵 +𝐷𝐷 = 1,
9𝐴𝐴 = 0,9𝐵𝐵 = 18
Any two coefficients correct.
All four coefficients correct.
SC B1 after M0 if one or more coefficients are correct.
AnswerMarks
(b)DR
 2 1  2 1 x
∫  − dx=− − tan−1 (+c)
x2 x2+9 x 3 3
∞ 2 1 
⇒∫  − dx
x2 x2+9
3
 2 2 1 k 
=lim− − − tan−1 −tan−11
k→∞ k 3 3 3 
2 2 π 1  k
= −lim + − limtan−1 
3 k→∞k 12 3k→∞ 3
2 π 1 π
= −0+ − ×
3 12 3 2
2 π
= −
AnswerMarks
3 12M1
A1
M1
A1
A1
A1
AnswerMarks
[6]1.1
1.1
1.1
2.1
2.1
AnswerMarks
1.1integration including a tan-1 term
ft their part (a)
Use of limiting process on their integrated function.
Ignore notation for limits
for , or as , , A0 for eg. .
1 1 1
𝑘𝑘li→m∞�𝑘𝑘� = 0 , o𝑘𝑘r a→s ∞ 𝑘𝑘 →, 0 ∞, A=0 0for
−1𝑘𝑘 1 −1𝑘𝑘 1
𝑘𝑘 elig →m. ∞�tan 𝑐𝑐� = 2 . 𝜋𝜋 In both 𝑘𝑘ca→ses∞ mtuasnt see 𝑐𝑐 s→om 2 e𝜋𝜋 evidence of
the lim−it1ing pro 1 cess.
tan ∞ = 2𝜋𝜋
Question 7:
7 | (a) | x2 +18≡ Ax ( x2 +9 ) +B ( x2 +9 ) +( Cx+D ) x2
e.g.x=0⇒9B=18⇒ B=2
x=1⇒10A+10B+C+D=19
x=−1⇒−10A+10B−C+D=19
⇒10B+D=19⇒ D=−1
x=3i⇒−9 ( D+3Ci )=9⇒C =0
⇒10A+20−1=19⇒ A=0
i.e. A = 0, B = 2, C = 0, D = −1 | B1
M1
A1
A1
[4] | 1.1
1.1
1.1
1.1 | Correct multiplying out of fractions
Any substitutions to get a set of (at least) four simultaneous
equations solvable for and .
Or equating coefficients which gives
𝐴𝐴,𝐵𝐵,𝐶𝐶 𝐷𝐷
.
𝐴𝐴+𝐶𝐶 = 0,𝐵𝐵 +𝐷𝐷 = 1,
9𝐴𝐴 = 0,9𝐵𝐵 = 18
Any two coefficients correct.
All four coefficients correct.
SC B1 after M0 if one or more coefficients are correct.
(b) | DR
 2 1  2 1 x
∫  − dx=− − tan−1 (+c)
x2 x2+9 x 3 3
∞ 2 1 
⇒∫  − dx
x2 x2+9
3
 2 2 1 k 
=lim− − − tan−1 −tan−11
k→∞ k 3 3 3 
2 2 π 1  k
= −lim + − limtan−1 
3 k→∞k 12 3k→∞ 3
2 π 1 π
= −0+ − ×
3 12 3 2
2 π
= −
3 12 | M1
A1
M1
A1
A1
A1
[6] | 1.1
1.1
1.1
2.1
2.1
1.1 | integration including a tan-1 term
ft their part (a)
Use of limiting process on their integrated function.
Ignore notation for limits
for , or as , , A0 for eg. .
1 1 1
𝑘𝑘li→m∞�𝑘𝑘� = 0 , o𝑘𝑘r a→s ∞ 𝑘𝑘 →, 0 ∞, A=0 0for
−1𝑘𝑘 1 −1𝑘𝑘 1
𝑘𝑘 elig →m. ∞�tan 𝑐𝑐� = 2 . 𝜋𝜋 In both 𝑘𝑘ca→ses∞ mtuasnt see 𝑐𝑐 s→om 2 e𝜋𝜋 evidence of
the lim−it1ing pro 1 cess.
tan ∞ = 2𝜋𝜋
7
\begin{enumerate}[label=(\alph*)]
\item Determine the values of $A , B , C$ and $D$ such that $\frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \equiv \frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { C x + D } { x ^ { 2 } + 9 }$.
\item In this question you must show detailed reasoning.

Hence determine the exact value of $\int _ { 3 } ^ { \infty } \frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \mathrm { d } x$.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 1 2022 Q7 [10]}}
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