| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2017 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Find constants from coefficient conditions on terms |
| Difficulty | Moderate -0.8 This is a straightforward application of the binomial theorem requiring students to match coefficients to find parameters and then calculate a specific term. The method is routine: equate the coefficient of x to find a, then use the binomial coefficient formula for x² and x⁴. No problem-solving insight needed, just direct application of standard formulas. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((1+ax)^{20} = 1^{20} + {}^{20}C_1 1^{19}(ax)^1 + {}^{20}C_2 1^{18}(ax)^2 + \ldots\) | ||
| \({}^{20}C_1 1^{19}(ax)^1 = 4x \Rightarrow 20a = 4 \Rightarrow a = 0.2\) | M1A1 | M1: uses \({}^{20}C_1(1^{19})(ax)^1 = 4x\) or \(20a=4\) to obtain \(a\); A1: \(a=0.2\) or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \({}^{20}C_2 1^{18}(ax)^2 = px^2 \Rightarrow \frac{20\times19}{2}\times(0.2)^2 = p\) | M1 | Uses \({}^{20}C_2(1^{18})(ax)^2 = px^2\) and their value of \(a\) to find \(p\); condone use of \(a\) rather than \(a^2\) |
| \(p = 7.6\) | A1 | Accept equivalents such as \(\frac{38}{5}, \frac{190}{25}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Term is \({}^{20}C_4 1^{16}(ax)^4 \Rightarrow q = \ldots\) | M1 | Identifies correct term and uses value of \(a\) to find \(q\); condone use of \(a\) rather than \(a^4\); must attempt \({}^{20}C_4 a^4\) or \({}^{20}C_4 a\) or \({}^{20}C_{16}a^4\) or \({}^{20}C_{16}a\) |
| \(q = {}^{20}C_4 \times 0.2^4 = \frac{969}{125}\ (= 7.752)\) | A1 | \(q = \frac{969}{125}\) or exact equivalent; note \(q = \frac{969}{125}x^4\) scores A0 but \(qx^4 = \frac{969}{125}x^4\) scores A1 |
## Question 10:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(1+ax)^{20} = 1^{20} + {}^{20}C_1 1^{19}(ax)^1 + {}^{20}C_2 1^{18}(ax)^2 + \ldots$ | | |
| ${}^{20}C_1 1^{19}(ax)^1 = 4x \Rightarrow 20a = 4 \Rightarrow a = 0.2$ | M1A1 | M1: uses ${}^{20}C_1(1^{19})(ax)^1 = 4x$ or $20a=4$ to obtain $a$; A1: $a=0.2$ or equivalent |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| ${}^{20}C_2 1^{18}(ax)^2 = px^2 \Rightarrow \frac{20\times19}{2}\times(0.2)^2 = p$ | M1 | Uses ${}^{20}C_2(1^{18})(ax)^2 = px^2$ and their value of $a$ to find $p$; condone use of $a$ rather than $a^2$ |
| $p = 7.6$ | A1 | Accept equivalents such as $\frac{38}{5}, \frac{190}{25}$ |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Term is ${}^{20}C_4 1^{16}(ax)^4 \Rightarrow q = \ldots$ | M1 | Identifies correct term and uses value of $a$ to find $q$; condone use of $a$ rather than $a^4$; must attempt ${}^{20}C_4 a^4$ or ${}^{20}C_4 a$ or ${}^{20}C_{16}a^4$ or ${}^{20}C_{16}a$ |
| $q = {}^{20}C_4 \times 0.2^4 = \frac{969}{125}\ (= 7.752)$ | A1 | $q = \frac{969}{125}$ or exact equivalent; note $q = \frac{969}{125}x^4$ scores A0 but $qx^4 = \frac{969}{125}x^4$ scores A1 |
---10. The first 3 terms, in ascending powers of $x$, in the binomial expansion of $( 1 + a x ) ^ { 20 }$ are given by
$$1 + 4 x + p x ^ { 2 }$$
where $a$ and $p$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$.
\item Find the value of $p$.
One of the terms in the binomial expansion of $( 1 + a x ) ^ { 20 }$ is $q x ^ { 4 }$, where $q$ is a constant.
\item Find the value of $q$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2017 Q10 [6]}}