| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Binomial expansion with reciprocals |
| Difficulty | Easy -1.2 This is a straightforward binomial expansion with n=4, requiring only mechanical application of the binomial theorem and simplification of powers of x. It's easier than average as it involves a small power, no coefficient manipulation beyond basic arithmetic, and the 'simplifying terms' just means combining x^a ยท x^(-b). A routine C2 exercise testing basic technique rather than problem-solving. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \(= x^4 + 8x^2 + 24 + \frac{32}{x^2} + \frac{16}{x^4}\) (or equiv) | M1*, M1*, A1dep*, A1, A1 5 | Attempt expansion, using powers of \(x\) and \(\frac{z}{x}\) (or the two terms in their bracket), to get at least 4 terms. Use binomial coefficients of 1, 4, 6, 4, 1. Obtain two correct, simplified, terms. Obtain a further one correct, simplified, term. Obtain a fully correct, simplified, expansion. |
| Answer | Marks |
|---|---|
| M1*, M1* | Attempt expansion using all four brackets. Obtain expansion containing the correct 5 powers only (could be unsimplified powers eg \(x^3 \cdot x^{-1}\)) |
| A1dep*, A1, A1 5 | Obtain two correct, simplified, terms. Obtain a further one correct, simplified, term. Obtain a fully correct, simplified, expansion. |
$(x + \frac{z}{x})^4 = x^4 + 4x^3(\frac{z}{x}) + 6x^2(\frac{z}{x})^2 + 4x(\frac{z}{x})^3 + (\frac{z}{x})^4$
$= x^4 + 8x^2 + 24 + \frac{32}{x^2} + \frac{16}{x^4}$ (or equiv) | M1*, M1*, A1dep*, A1, A1 5 | Attempt expansion, using powers of $x$ and $\frac{z}{x}$ (or the two terms in their bracket), to get at least 4 terms. Use binomial coefficients of 1, 4, 6, 4, 1. Obtain two correct, simplified, terms. Obtain a further one correct, simplified, term. Obtain a fully correct, simplified, expansion.
**OR**
| M1*, M1* | Attempt expansion using all four brackets. Obtain expansion containing the correct 5 powers only (could be unsimplified powers eg $x^3 \cdot x^{-1}$)
| A1dep*, A1, A1 5 | Obtain two correct, simplified, terms. Obtain a further one correct, simplified, term. Obtain a fully correct, simplified, expansion.2 Expand $\left( x + \frac { 2 } { x } \right) ^ { 4 }$ completely, simplifying the terms.
\hfill \mbox{\textit{OCR C2 2007 Q2 [5]}}