| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Direct binomial expansion then integrate |
| Difficulty | Moderate -0.8 This is a straightforward two-part question requiring routine application of binomial expansion with n=3 (a small integer making it very manageable), followed by term-by-term integration of simple powers. Both parts are standard textbook exercises with no problem-solving or insight required, making it easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x^6 + 3x^3 + 3 + \frac{1}{x^3}\) | M1, A1, A1, A1 | For 4 term binomial attempt or equiv; For any one (unsimplified) term correct; For any other (unsimplified) term correct; For full, simplified expansion |
| (ii) \(\frac{1}{7}x^7 + \frac{3}{4}x^4 + 3x - \frac{1}{2}x^{-2} + c\) | M1, A1, M1, A1 | For any correct use of \(\frac{x^{n+1}}{n+1}\); For any two terms integrated correctly; For any correct use of \(x^{n+1}\) using a negative index; For all terms integrated correctly (must have at least 4 terms, including at least 1 negative index) [No penalty for omission of +c in this part] |
**(i)** $x^6 + 3x^3 + 3 + \frac{1}{x^3}$ | M1, A1, A1, A1 | For 4 term binomial attempt or equiv; For any one (unsimplified) term correct; For any other (unsimplified) term correct; For full, simplified expansion
**(ii)** $\frac{1}{7}x^7 + \frac{3}{4}x^4 + 3x - \frac{1}{2}x^{-2} + c$ | M1, A1, M1, A1 | For any correct use of $\frac{x^{n+1}}{n+1}$; For any two terms integrated correctly; For any correct use of $x^{n+1}$ using a negative index; For all terms integrated correctly (must have at least 4 terms, including at least 1 negative index) [No penalty for omission of +c in this part]
---6 (i) Find the binomial expansion of $\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 3 }$, simplifying the terms.\\
(ii) Hence find $\int \left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 3 } \mathrm {~d} x$.
\hfill \mbox{\textit{OCR C2 2005 Q6 [8]}}