CAIE P3 2012 June — Question 9 10 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2012
SessionJune
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration with Partial Fractions
TypeImproper algebraic form then partial fractions
DifficultyStandard +0.8 This question requires recognizing improper rational form (degree of numerator ≥ denominator), performing polynomial long division, factoring a quadratic, decomposing into partial fractions, integrating logarithmic terms, and evaluating definite integrals with exact simplification. While systematic, it combines multiple techniques in sequence and requires careful algebraic manipulation throughout, making it moderately challenging but still within standard A-level scope.
Spec1.02y Partial fractions: decompose rational functions1.08d Evaluate definite integrals: between limits1.08j Integration using partial fractions
9 By first expressing \(\frac { 4 x ^ { 2 } + 5 x + 3 } { 2 x ^ { 2 } + 5 x + 2 }\) in partial fractions, show that $$\int _ { 0 } ^ { 4 } \frac { 4 x ^ { 2 } + 5 x + 3 } { 2 x ^ { 2 } + 5 x + 2 } \mathrm {~d} x = 8 - \ln 9$$
Question 9:
AnswerMarks Guidance
Answer/WorkingMark Guidance
State or imply form \(A + \frac{B}{2x+1} + \frac{C}{x+2}\)B1
State or obtain \(A = 2\)B1
Use correct method for finding \(B\) or \(C\)M1
Obtain \(B = 1\)A1
Obtain \(C = -3\)A1
Obtain \(2x + \frac{1}{2}\ln(2x+1) - 3\ln(x+2)\)B3✓ Deduct B1✓ for each error or omission
Substitute limits in expression containing \(a\ln(2x+1) + b\ln(x+2)\)M1
Show full and exact working to confirm that \(8 + \frac{1}{2}\ln 9 - 3\ln 6 + 3\ln 2\), or an equivalent expression, simplifies to given result \(8 - \ln 9\)A1 [10]
If \(A\) omitted from form of fractions, give B0B0M1A0A0 in (i); B0✓B1✓B1✓M1A0 in (ii) SR
For solution starting with \(\frac{M}{2x+1} + \frac{Nx}{x+2}\) or \(\frac{Px}{2x+1} + \frac{Q}{x+2}\), give B0B0M1A0A0 in (i); B1✓B1✓B1✓, if recover correct form, M1A0 in (ii) SR
For solution starting with \(\frac{B}{2x+1} + \frac{Dx+E}{x+2}\), give M1A1 for one of \(B=1, D=2, E=1\) and A1 for other two constants; then give B1B1 for \(A=2, C=-3\) SR
For solution starting with \(\frac{Fx+G}{2x+1} + \frac{C}{x+2}\), give M1A1 for one of \(C=-3, F=4, G=3\) and A1 for other constants; then give B1B1 for \(A=2, B=1\) SR 
# Question 9:

| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply form $A + \frac{B}{2x+1} + \frac{C}{x+2}$ | B1 | |
| State or obtain $A = 2$ | B1 | |
| Use correct method for finding $B$ or $C$ | M1 | |
| Obtain $B = 1$ | A1 | |
| Obtain $C = -3$ | A1 | |
| Obtain $2x + \frac{1}{2}\ln(2x+1) - 3\ln(x+2)$ | B3✓ | Deduct B1✓ for each error or omission |
| Substitute limits in expression containing $a\ln(2x+1) + b\ln(x+2)$ | M1 | |
| Show full and exact working to confirm that $8 + \frac{1}{2}\ln 9 - 3\ln 6 + 3\ln 2$, or an equivalent expression, simplifies to given result $8 - \ln 9$ | A1 | [10] |
| If $A$ omitted from form of fractions, give B0B0M1A0A0 in (i); B0✓B1✓B1✓M1A0 in (ii) | | SR |
| For solution starting with $\frac{M}{2x+1} + \frac{Nx}{x+2}$ or $\frac{Px}{2x+1} + \frac{Q}{x+2}$, give B0B0M1A0A0 in (i); B1✓B1✓B1✓, if recover correct form, M1A0 in (ii) | | SR |
| For solution starting with $\frac{B}{2x+1} + \frac{Dx+E}{x+2}$, give M1A1 for one of $B=1, D=2, E=1$ and A1 for other two constants; then give B1B1 for $A=2, C=-3$ | | SR |
| For solution starting with $\frac{Fx+G}{2x+1} + \frac{C}{x+2}$, give M1A1 for one of $C=-3, F=4, G=3$ and A1 for other constants; then give B1B1 for $A=2, B=1$ | | SR |
9 By first expressing $\frac { 4 x ^ { 2 } + 5 x + 3 } { 2 x ^ { 2 } + 5 x + 2 }$ in partial fractions, show that

$$\int _ { 0 } ^ { 4 } \frac { 4 x ^ { 2 } + 5 x + 3 } { 2 x ^ { 2 } + 5 x + 2 } \mathrm {~d} x = 8 - \ln 9$$

\hfill \mbox{\textit{CAIE P3 2012 Q9 [10]}}
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