Standard +0.3 This question requires applying the binomial expansion formula to match coefficients, solving simultaneous equations for p and q, then stating the validity condition |qx| < 1. It's slightly above average difficulty due to the algebraic manipulation needed, but follows a standard pattern for C4 binomial expansion questions with multiple routine steps.
2 Given the binomial expansion \(( 1 + q x ) ^ { p } = 1 - x + 2 x ^ { 2 } + \ldots\), find the values of \(p\) and \(q\). Hence state the set of values of \(x\) for which the expansion is valid. [6]
\(pq = -1,\ q = -1/p\) and \(\frac{1}{2}p(p-1)q^2 = 2\)
M1
Eliminating \(q\) (or \(p\)) from simultaneous equations involving both variables; \(\frac{1}{2}\left(\frac{-1}{q}\right)\left(\frac{-1}{q}-1\right)q^2 = 2\), \(-1(-1-q)=4\), \(q=3\)
\(p(p-1)/2p^2 = (p-1)/2p = 2\)
\(\Rightarrow p-1 = 4p,\ p = -1/3\)
A1
\(p = -1/3\) www (or \(q=3\))
\(\Rightarrow q = 3\)
A1ft
\(q=3\) (or \(p=-1/3\)) for second value, ft their \(p\) or \(q\), provided only a single computational error in the method and correct initial equations
Valid for \(-1 < 3x < 1 \Rightarrow -\frac{1}{3} < x < \frac{1}{3}\)
B1
or \(
[6]
## Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(1+qx)^p = 1 + pqx + \frac{1}{2}p(p-1)q^2x^2 + \ldots$ | B1 | $(1)\ldots + pqx$ |
| | B1 | $\ldots + \frac{1}{2}p(p-1)q^2x^2$ |
| $pq = -1,\ q = -1/p$ and $\frac{1}{2}p(p-1)q^2 = 2$ | M1 | Eliminating $q$ (or $p$) from simultaneous equations involving both variables; $\frac{1}{2}\left(\frac{-1}{q}\right)\left(\frac{-1}{q}-1\right)q^2 = 2$, $-1(-1-q)=4$, $q=3$ |
| $p(p-1)/2p^2 = (p-1)/2p = 2$ | | |
| $\Rightarrow p-1 = 4p,\ p = -1/3$ | A1 | $p = -1/3$ www (or $q=3$) |
| $\Rightarrow q = 3$ | A1ft | $q=3$ (or $p=-1/3$) for second value, ft their $p$ or $q$, provided only a single computational error in the method and correct initial equations |
| Valid for $-1 < 3x < 1 \Rightarrow -\frac{1}{3} < x < \frac{1}{3}$ | B1 | or $|x| < 1/3$ www, allow $-1/3 \leq |x| < 1/3$ but not say $x < 1/3$ (actually $-1/3 < x \leq 1/3$ is correct) |
| **[6]** | | |
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2 Given the binomial expansion $( 1 + q x ) ^ { p } = 1 - x + 2 x ^ { 2 } + \ldots$, find the values of $p$ and $q$. Hence state the set of values of $x$ for which the expansion is valid. [6]
\hfill \mbox{\textit{OCR MEI C4 Q2 [6]}}