CAIE P2 2006 June — Question 7 11 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2006
SessionJune
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard Integrals and Reverse Chain Rule
TypePolynomial division before integration
DifficultyModerate -0.3 This is a structured multi-part question that guides students through standard techniques: differentiating ln(ax+b), recognizing reverse chain rule for integration, polynomial division, and combining results. While it requires multiple steps, each part is routine and heavily scaffolded, making it slightly easier than average for A-level Pure Mathematics 2.
Spec1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.07l Derivative of ln(x): and related functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits
7
  1. Differentiate \(\ln ( 2 x + 3 )\).
  2. Hence, or otherwise, show that $$\int _ { - 1 } ^ { 3 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x = \ln 3$$
  3. Find the quotient and remainder when \(4 x ^ { 2 } + 8 x\) is divided by \(2 x + 3\).
  4. Hence show that $$\int _ { - 1 } ^ { 3 } \frac { 4 x ^ { 2 } + 8 x } { 2 x + 3 } d x = 12 - 3 \ln 3$$
Question 7:
Part (i):
AnswerMarks Guidance
Answer/WorkingMark Guidance
Obtain derivative of the form \(\frac{k}{2x+3}\), where \(k = 2\) or \(k = 1\)M1
Obtain correct derivative \(\frac{2}{2x+3}\)A1 Total: 2
Part (ii):
AnswerMarks Guidance
Answer/WorkingMark Guidance
State indefinite integral of the form \(m\ln(2x+3)\)M1*
Use limits correctlyM1(dep*)
Obtain given answerA1 Total: 3
Part (iii):
AnswerMarks Guidance
Answer/WorkingMark Guidance
Carry out division method reaching a linear quotient and constant remainderM1
Obtain quotient \(2x + 1\)A1
Obtain remainder \(-3\)A1 Total: 3
Part (iv):
AnswerMarks Guidance
Answer/WorkingMark Guidance
Attempt integration of an integrand of the form \(ax + b + \frac{c}{2x+3}\)M1
Obtain indefinite integral \(x^2 + x - \frac{3}{2}\ln(2x+3)\)A1\(\sqrt{}\)
Substitute limits and obtain given answerA1 Total: 3
[The f.t. mark is also available if the indefinite integral of the third term is omitted but its definite integral is stated to be \(c\ln 3\)] 
## Question 7:

### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain derivative of the form $\frac{k}{2x+3}$, where $k = 2$ or $k = 1$ | M1 | |
| Obtain correct derivative $\frac{2}{2x+3}$ | A1 | Total: 2 |

### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State indefinite integral of the form $m\ln(2x+3)$ | M1* | |
| Use limits correctly | M1(dep*) | |
| Obtain given answer | A1 | Total: 3 |

### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Carry out division method reaching a linear quotient and constant remainder | M1 | |
| Obtain quotient $2x + 1$ | A1 | |
| Obtain remainder $-3$ | A1 | Total: 3 |

### Part (iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt integration of an integrand of the form $ax + b + \frac{c}{2x+3}$ | M1 | |
| Obtain indefinite integral $x^2 + x - \frac{3}{2}\ln(2x+3)$ | A1$\sqrt{}$ | |
| Substitute limits and obtain given answer | A1 | Total: 3 |
| | | [The f.t. mark is also available if the indefinite integral of the third term is omitted but its definite integral is stated to be $c\ln 3$] |
7 (i) Differentiate $\ln ( 2 x + 3 )$.\\
(ii) Hence, or otherwise, show that

$$\int _ { - 1 } ^ { 3 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x = \ln 3$$

(iii) Find the quotient and remainder when $4 x ^ { 2 } + 8 x$ is divided by $2 x + 3$.\\
(iv) Hence show that

$$\int _ { - 1 } ^ { 3 } \frac { 4 x ^ { 2 } + 8 x } { 2 x + 3 } d x = 12 - 3 \ln 3$$

\hfill \mbox{\textit{CAIE P2 2006 Q7 [11]}}
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Q1 3 Q2 5 Q3 7 Q4 7 Q5 8 Q6 9 Q7 11