| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Moderate -0.3 This is a straightforward binomial theorem question requiring recall of the general term formula and basic algebraic manipulation. Part (a) involves setting up one equation to find k, part (b) is verification using the same formula, and part (c) requires simple multiplication of polynomials. All steps are routine C2-level techniques with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \((1+kx)^7 = \ldots + \binom{7}{2}(kx)^2 + \ldots\) | B1 | |
| \(\therefore \frac{7 \times 6}{2} \times k^2 = 525\) | ||
| \(k^2 = \frac{525}{21} = 25\) | M1 | |
| \(k > 0 \therefore k = 5\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \((1+5x)^7 = \ldots + \binom{7}{3}(5x)^3 + \ldots\) | ||
| \(\therefore\) coeff. of \(x^3 = \frac{7 \times 6 \times 5}{3 \times 2} \times 125 = 4375\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \((1+5x)^7 = 1 + 35x + 525x^2 + \ldots\) | B1 | |
| \((2-x)(1+5x)^7 = (2-x)(1 + 35x + 525x^2 + \ldots)\) | ||
| \(= 2 + 70x + 1050x^2 - x - 35x^2 + \ldots\) | M1 | |
| \(= 2 + 69x + 1015x^2 + \ldots\) | A1 | (8) |
## Question 4:
**(a)**
| Answer/Working | Marks | Notes |
|---|---|---|
| $(1+kx)^7 = \ldots + \binom{7}{2}(kx)^2 + \ldots$ | B1 | |
| $\therefore \frac{7 \times 6}{2} \times k^2 = 525$ | | |
| $k^2 = \frac{525}{21} = 25$ | M1 | |
| $k > 0 \therefore k = 5$ | A1 | |
**(b)**
| Answer/Working | Marks | Notes |
|---|---|---|
| $(1+5x)^7 = \ldots + \binom{7}{3}(5x)^3 + \ldots$ | | |
| $\therefore$ coeff. of $x^3 = \frac{7 \times 6 \times 5}{3 \times 2} \times 125 = 4375$ | M1 A1 | |
**(c)**
| Answer/Working | Marks | Notes |
|---|---|---|
| $(1+5x)^7 = 1 + 35x + 525x^2 + \ldots$ | B1 | |
| $(2-x)(1+5x)^7 = (2-x)(1 + 35x + 525x^2 + \ldots)$ | | |
| $= 2 + 70x + 1050x^2 - x - 35x^2 + \ldots$ | M1 | |
| $= 2 + 69x + 1015x^2 + \ldots$ | A1 | **(8)** |
---4. The coefficient of $x ^ { 2 }$ in the binomial expansion of $( 1 + k x ) ^ { 7 }$, where $k$ is a positive constant, is 525.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $k$.
Using this value of $k$,
\item show that the coefficient of $x ^ { 3 }$ in the expansion is 4375,
\item find the first three terms in the expansion in ascending powers of $x$ of
$$( 2 - x ) ( 1 + k x ) ^ { 7 }$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q4 [8]}}