| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Finding unknown power and constant |
| Difficulty | Standard +0.3 This is a straightforward application of the binomial expansion formula requiring students to equate coefficients and solve for a constant, then state validity conditions and write a quadratic approximation. While it involves the generalised binomial theorem (a Further Maths topic), the algebraic manipulation is routine and the question follows a standard template with no novel problem-solving required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1+(-3)(-ax)+\frac{(-3)(-4)}{2}(-ax)^2+...\) | M1 | Attempt to use the binomial expansion; allow sign errors, bracket errors, a slip |
| Equate coefficients \(3a=6a^2\) | M1 | Equating their coefficients; allow recovery from missing brackets; their equation should not involve \(x\) |
| \(a=\frac{1}{2}\) | A1 [3] | oe www |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Valid for \( | x | <2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left(1-\frac{1}{2}x\right)^{-3}\approx 1+\frac{3}{2}x+\frac{3}{2}x^2\) | B1 [1] | Cao |
## Question 3:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1+(-3)(-ax)+\frac{(-3)(-4)}{2}(-ax)^2+...$ | M1 | Attempt to use the binomial expansion; allow sign errors, bracket errors, a slip |
| Equate coefficients $3a=6a^2$ | M1 | Equating their coefficients; allow recovery from missing brackets; their equation should not involve $x$ |
| $a=\frac{1}{2}$ | A1 [3] | oe www |
### Part (b)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Valid for $|x|<2$ | B1 [1] | Accept $|x|<\frac{1}{|a|}$ for their $a$; do not accept $x<2$, $|\frac{1}{2}x|<1$ or $|x|\leq 2$ or similar |
### Part (b)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left(1-\frac{1}{2}x\right)^{-3}\approx 1+\frac{3}{2}x+\frac{3}{2}x^2$ | B1 [1] | Cao |
---3 The function $\mathrm { f } ( x )$ is given by $\mathrm { f } ( x ) = ( 1 - a x ) ^ { - 3 }$, where $a$ is a non-zero constant. In the binomial expansion of $\mathrm { f } ( x )$, the coefficients of $x$ and $x ^ { 2 }$ are equal.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$.
\item Using this value for $a$,
\begin{enumerate}[label=(\roman*)]
\item state the set of values of $x$ for which the binomial expansion is valid,
\item write down the quadratic function which approximates $\mathrm { f } ( x )$ when $x$ is small.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 1 2019 Q3 [8]}}