| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Factorization then partial fractions |
| Difficulty | Standard +0.3 This is a standard A-level integration question combining polynomial factorization and partial fractions. Part (i) is routine factor verification using substitution or division. Part (ii) requires factorizing the cubic (given one factor), decomposing into partial fractions, and integrating—all well-practiced techniques in P3/C4. The numerator degree is less than denominator, avoiding the need for polynomial division. Slightly above average difficulty due to the multi-step nature and cubic factorization, but follows a predictable template without requiring novel insight. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks |
|---|---|
| Substitute \(x = -1\) and evaluate | M1 |
| Obtain 0 and conclude \(x + 1\) is a factor | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Divide by \(x + 1\) and obtain a constant remainder | M1 | |
| Obtain remainder = 0 and conclude \(x + 1\) is a factor | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt division, or equivalent, at least as far as quotient \(4x^2 + kx\) | M1 | |
| Obtain complete quotient \(4x^2 - 5x - 6\) | A1 | |
| State form \(\frac{A}{x + 1} + \frac{B}{x - 2} + \frac{C}{4x + 3}\) | A1 | |
| Use relevant method for finding at least one constant | M1 | |
| Obtain one of \(A = -2, B = 1, C = 8\) | A1 | |
| Obtain all three values | A1 | |
| Integrate to obtain three terms each involving natural logarithm of linear form | M1 | |
| Obtain \(-2\ln(x + 1) + \ln(x - 2) + 2\ln(4x + 3)\), condoning no use of modulus signs and absence of \(\ldots + c\) | A1 | [8] |
**(i)**
**Either**
Substitute $x = -1$ and evaluate | M1 |
Obtain 0 and conclude $x + 1$ is a factor | A1 |
**Or**
Divide by $x + 1$ and obtain a constant remainder | M1 |
Obtain remainder = 0 and conclude $x + 1$ is a factor | A1 | [2]
**(ii)**
Attempt division, or equivalent, at least as far as quotient $4x^2 + kx$ | M1 |
Obtain complete quotient $4x^2 - 5x - 6$ | A1 |
State form $\frac{A}{x + 1} + \frac{B}{x - 2} + \frac{C}{4x + 3}$ | A1 |
Use relevant method for finding at least one constant | M1 |
Obtain one of $A = -2, B = 1, C = 8$ | A1 |
Obtain all three values | A1 |
Integrate to obtain three terms each involving natural logarithm of linear form | M1 |
Obtain $-2\ln(x + 1) + \ln(x - 2) + 2\ln(4x + 3)$, condoning no use of modulus signs and absence of $\ldots + c$ | A1 | [8]7 (i) Show that ( $x + 1$ ) is a factor of $4 x ^ { 3 } - x ^ { 2 } - 11 x - 6$.\\
(ii) Find $\int \frac { 4 x ^ { 2 } + 9 x - 1 } { 4 x ^ { 3 } - x ^ { 2 } - 11 x - 6 } \mathrm {~d} x$.
\hfill \mbox{\textit{CAIE P3 2015 Q7 [10]}}