| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Session | Specimen |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Expand and state validity |
| Difficulty | Moderate -0.8 This is a straightforward application of the binomial expansion formula for negative/fractional powers with a standard validity condition. It requires direct substitution into the formula and simple algebraic manipulation, making it easier than average but not trivial since students must handle fractional powers and remember the validity criterion |x| < 1/2. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((1-2x)^{-1} = 1 + (-\frac{1}{2})(-2x) + \frac{(-\frac{1}{2})(-\frac{3}{2})}{2}(-2x)^2 + \frac{(-\frac{1}{2})(-\frac{3}{2})(-\frac{5}{2})}{3!}(-2x)^3 + \ldots = 1 + x + \frac{3}{2}x^2 + \frac{5}{2}x^3\) | M1 | For 2nd, 3rd or 4th term OK (unsimplified) |
| A1 | For \(1+x\) correct | |
| A1 | For \(+\frac{3}{2}x^2\) correct | |
| A1 | For \(+\frac{5}{2}x^3\) correct | |
| 4 | ||
| (ii) Valid for \( | t | < 1\) |
| 1 |
**(i)** $(1-2x)^{-1} = 1 + (-\frac{1}{2})(-2x) + \frac{(-\frac{1}{2})(-\frac{3}{2})}{2}(-2x)^2 + \frac{(-\frac{1}{2})(-\frac{3}{2})(-\frac{5}{2})}{3!}(-2x)^3 + \ldots = 1 + x + \frac{3}{2}x^2 + \frac{5}{2}x^3$ | M1 | For 2nd, 3rd or 4th term OK (unsimplified)
| A1 | For $1+x$ correct
| A1 | For $+\frac{3}{2}x^2$ correct
| A1 | For $+\frac{5}{2}x^3$ correct
| **4** |
**(ii)** Valid for $|t| < 1$ | B1 | For any correct expression(s)
| **1** |2 (i) Expand $( 1 - 2 x ) ^ { - \frac { 1 } { 2 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.\\
(ii) State the set of values for which the expansion in part (i) is valid.
\hfill \mbox{\textit{OCR C4 Q2 [5]}}