| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Composite substitution expansion |
| Difficulty | Moderate -0.3 This is a straightforward two-part binomial expansion question requiring standard application of the formula for (1-x)^(1/2) followed by a substitution x = 2y - 4y². While part (ii) requires recognizing the composite substitution and careful algebraic manipulation to collect terms, this is a routine C4 technique with no novel problem-solving required. Slightly easier than average due to being a standard textbook exercise type. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - \frac{1}{2}x\ldots\) | B1 | |
| Third term = \(\frac{1-(-1)}{2}[(-x)^2 \text{ or } x^2 \text{ or } -x^2]\) | M1 | |
| \(= -\frac{1}{8}x^2\) | A1 3 | \(-\frac{1}{8}x^2\) without work \(\to\) M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to replace \(x\) by \(2y - 4y^2\) or \(2y + 4y^2\) | M1 | or write as \(1 - (2y - 4y^2 \text{ or } 2y + 4y^2)\) |
| First two terms are \(1 - y\) | B1 | |
| Third term = \(+\frac{3}{2}y^2\) or \(\sqrt{(4b+2)y^2}\) | A1∇ 3 | where \(b = \text{cf}[x^2]\) in part (i) |
**(i)**
$1 - \frac{1}{2}x\ldots$ | B1 |
Third term = $\frac{1-(-1)}{2}[(-x)^2 \text{ or } x^2 \text{ or } -x^2]$ | M1 |
$= -\frac{1}{8}x^2$ | A1 3 | $-\frac{1}{8}x^2$ without work $\to$ M1 A1
**(ii)**
Attempt to replace $x$ by $2y - 4y^2$ or $2y + 4y^2$ | M1 | or write as $1 - (2y - 4y^2 \text{ or } 2y + 4y^2)$
First two terms are $1 - y$ | B1 |
Third term = $+\frac{3}{2}y^2$ or $\sqrt{(4b+2)y^2}$ | A1∇ 3 | where $b = \text{cf}[x^2]$ in part (i)
---1 (i) Expand $( 1 - x ) ^ { \frac { 1 } { 2 } }$ in ascending powers of $x$ as far as the term in $x ^ { 2 }$.\\
(ii) Hence expand $\left( 1 - 2 y + 4 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }$ in ascending powers of $y$ as far as the term in $y ^ { 2 }$.
\hfill \mbox{\textit{OCR C4 2011 Q1 [6]}}