| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| 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 the binomial theorem with n=4 (a standard C2 topic), followed by term-by-term integration of the resulting polynomial. Both steps are mechanical with no problem-solving or insight required, making it easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((x^3)^4 + 4(x^3)^3(2x^{-2}) + 6(x^3)^2(2x^{-2})^2 + 4(x^3)(2x^{-2})^3 + (2x^{-2})^4\) | M1* | Must attempt at least 4 terms; each term must be an attempt at a product including binomial coeffs if used; allow M1 if no longer \(2x^{-2}\) due to index errors; allow M1 for no or incorrect binomial coeffs |
| \(= x^{12} + 8x^7 + 24x^2 + 32x^{-3} + 16x^{-8}\) | M1d* | Attempt to use correct binomial coeffs; at least 4 correct from 1, 4, 6, 4, 1 |
| Two correct simplified terms | A1 | Either linked by \(+\) or as part of a list; powers and coefficients must be simplified |
| A further two correct terms | A1 | Either linked by \(+\) or as part of a list |
| Fully correct expansion | A1 | Terms must be linked by \(+\) and not just commas; A0 if subsequent attempt to simplify indices (eg \(x\) by \(x^8\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt integration | M1* | Increase in power by 1 for at least three terms; allow if three terms include \(x^{-1}\) becoming \(k\ln x\) (but not \(x^0\)) |
| Obtain at least 3 correct terms following (i) | A1FT | Allow unsimplified coefficients |
| \(\frac{1}{13}x^{13} + x^8 + 8x^3 - 16x^{-2} - \frac{16}{7}x^{-7} + c\) | A1 | Coefficients must be fully simplified; inc \(x^8\) not \(1x^8\); isw subsequent errors eg \(16x^{-2}\) then being written with 16 as well as \(x^2\) in denominator of a fraction |
| \(+c\), and no \(dx\) or integral sign in answer | B1d* | Ignore notation on LHS such as \(\int = ...\), \(y = ...\), \(\frac{dy}{dx} = ...\) |
## Question 6:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(x^3)^4 + 4(x^3)^3(2x^{-2}) + 6(x^3)^2(2x^{-2})^2 + 4(x^3)(2x^{-2})^3 + (2x^{-2})^4$ | M1* | Must attempt at least 4 terms; each term must be an attempt at a product including binomial coeffs if used; allow M1 if no longer $2x^{-2}$ due to index errors; allow M1 for no or incorrect binomial coeffs |
| $= x^{12} + 8x^7 + 24x^2 + 32x^{-3} + 16x^{-8}$ | M1d* | Attempt to use correct binomial coeffs; at least 4 correct from 1, 4, 6, 4, 1 |
| Two correct simplified terms | A1 | Either linked by $+$ or as part of a list; powers and coefficients must be simplified |
| A further two correct terms | A1 | Either linked by $+$ or as part of a list |
| Fully correct expansion | A1 | Terms must be linked by $+$ and not just commas; A0 if subsequent attempt to simplify indices (eg $x$ by $x^8$) |
**SR:** M2 for attempt involving all 4 brackets resulting in a quartic with at most one term missing; A1, A1, A1 for correct simplified terms.
---
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt integration | M1* | Increase in power by 1 for at least three terms; allow if three terms include $x^{-1}$ becoming $k\ln x$ (but not $x^0$) |
| Obtain at least 3 correct terms following (i) | A1FT | Allow unsimplified coefficients |
| $\frac{1}{13}x^{13} + x^8 + 8x^3 - 16x^{-2} - \frac{16}{7}x^{-7} + c$ | A1 | Coefficients must be fully simplified; inc $x^8$ not $1x^8$; isw subsequent errors eg $16x^{-2}$ then being written with 16 as well as $x^2$ in denominator of a fraction |
| $+c$, and no $dx$ or integral sign in answer | B1d* | Ignore notation on LHS such as $\int = ...$, $y = ...$, $\frac{dy}{dx} = ...$ |
---6 (i) Find the binomial expansion of $\left( x ^ { 3 } + \frac { 2 } { x ^ { 2 } } \right) ^ { 4 }$, simplifying the terms.\\
(ii) Hence find $\int \left( x ^ { 3 } + \frac { 2 } { x ^ { 2 } } \right) ^ { 4 } \mathrm {~d} x$.
\hfill \mbox{\textit{OCR C2 2014 Q6 [9]}}