| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Series expansion of rational function |
| Difficulty | Moderate -0.3 This is a straightforward application of the binomial series requiring students to rewrite the expression in the form (1+y)^n, expand using the standard formula, and simplify coefficients. It's slightly easier than average because it's a direct template question with clear instructions and no problem-solving element, though the algebraic manipulation and coefficient simplification require care. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((2+5x)^{-3}\) | M1 | Writes down \((2+5x)^{-3}\) or uses power of \(-3\) |
| \(2^{-3}\) or \(\frac{1}{8}\) | B1 | \(2^{-3}\) or \(\frac{1}{8}\) outside brackets |
| \(1+(-3)(kx)+\frac{(-3)(-4)}{2!}(kx)^2+\frac{(-3)(-4)(-5)}{3!}(kx)^3+...\) | M1 A1 | Expands \((...+kx)^{-3}\), any 2 terms out of 4; then all four terms correct with consistent \((kx)\) |
| \(\frac{1}{8}-\frac{15}{16}x+\frac{75}{16}x^2-\frac{625}{32}x^3+...\) | A1; A1 | \(\frac{1}{8}-\frac{15}{16}x\) (simplified); \(\frac{75}{16}x^2-\frac{625}{32}x^3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x)=(2+5x)^{-3}\) | M1 | Writes down \((2+5x)^{-3}\) or uses power of \(-3\) |
| \(f''(x)=300(2+5x)^{-5}\), \(f'''(x)=-7500(2+5x)^{-6}\) | B1 | Correct \(f''(x)\) and \(f'''(x)\) |
| \(f'(x)=-15(2+5x)^{-4}\) | M1, A1 oe | \(\pm a(2+5x)^{-4}\), \(a\neq\pm 1\); \(-15(2+5x)^{-4}\) |
| \(f(0)=\frac{1}{8}\), \(f'(0)=-\frac{15}{16}\), \(f''(0)=\frac{75}{8}\), \(f'''(0)=-\frac{1875}{16}\) | — | — |
| \(f(x)=\frac{1}{8}-\frac{15}{16}x+\frac{75}{16}x^2-\frac{625}{32}x^3+...\) | A1; A1 | Same as Way 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((2+5x)^{-3}\) | M1 | Same as Way 1 |
| \((2)^{-3}+(-3)(2)^{-4}(5x)+\frac{(-3)(-4)}{2!}(2)^{-5}(5x)^2+\frac{(-3)(-4)(-5)}{3!}(2)^{-6}(5x)^3\) | B1, M1, A1 | \(\frac{1}{8}\) outside; any two terms correct; all four terms correct |
| \(=\frac{1}{8}-\frac{15}{16}x+\frac{75}{16}x^2-\frac{625}{32}x^3+...\) | A1; A1 | Same as Way 1 |
| Answer | Marks |
|---|---|
| Note | Guidance |
| 1st M1 | Implied by constant term of \((2)^{-3}\) or \(\frac{1}{8}\) |
| B1 | \(2^{-3}\) or \(\frac{1}{8}\) outside brackets or \(\frac{1}{8}\) as candidate's constant term |
| 2nd M1 | Expands \((...+kx)^{-3}\), \(k\neq 1\), to give any 2 terms out of 4, simplified or un-simplified |
| 1st A1 | Correct simplified or un-simplified expansion with consistent \((kx)\), \(k\neq 1\) |
| 2nd A1 | For \(\frac{1}{8}-\frac{15}{16}x\) (simplified) or \(0.125-0.9375x\) |
| 3rd A1 | Accept only \(\frac{75}{16}x^2-\frac{625}{32}x^3\) or \(4\frac{11}{16}x^2-19\frac{17}{32}x^3\) or \(4.6875x^2-19.53125x^3\) |
| SC | If candidate would otherwise score 2nd A0, 3rd A0: allow Special Case 2nd A1 for un-multiplied bracket expressions with each term a simplified fraction or decimal |
| Note | Ignore extra terms beyond \(x^3\); ignore subsequent working following a correct answer |
# Question 1:
## Way 1
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(2+5x)^{-3}$ | M1 | Writes down $(2+5x)^{-3}$ or uses power of $-3$ |
| $2^{-3}$ or $\frac{1}{8}$ | B1 | $2^{-3}$ or $\frac{1}{8}$ outside brackets |
| $1+(-3)(kx)+\frac{(-3)(-4)}{2!}(kx)^2+\frac{(-3)(-4)(-5)}{3!}(kx)^3+...$ | M1 A1 | Expands $(...+kx)^{-3}$, any 2 terms out of 4; then all four terms correct with consistent $(kx)$ |
| $\frac{1}{8}-\frac{15}{16}x+\frac{75}{16}x^2-\frac{625}{32}x^3+...$ | A1; A1 | $\frac{1}{8}-\frac{15}{16}x$ (simplified); $\frac{75}{16}x^2-\frac{625}{32}x^3$ |
## Way 2
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x)=(2+5x)^{-3}$ | M1 | Writes down $(2+5x)^{-3}$ or uses power of $-3$ |
| $f''(x)=300(2+5x)^{-5}$, $f'''(x)=-7500(2+5x)^{-6}$ | B1 | Correct $f''(x)$ and $f'''(x)$ |
| $f'(x)=-15(2+5x)^{-4}$ | M1, A1 oe | $\pm a(2+5x)^{-4}$, $a\neq\pm 1$; $-15(2+5x)^{-4}$ |
| $f(0)=\frac{1}{8}$, $f'(0)=-\frac{15}{16}$, $f''(0)=\frac{75}{8}$, $f'''(0)=-\frac{1875}{16}$ | — | — |
| $f(x)=\frac{1}{8}-\frac{15}{16}x+\frac{75}{16}x^2-\frac{625}{32}x^3+...$ | A1; A1 | Same as Way 1 |
## Way 3
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(2+5x)^{-3}$ | M1 | Same as Way 1 |
| $(2)^{-3}+(-3)(2)^{-4}(5x)+\frac{(-3)(-4)}{2!}(2)^{-5}(5x)^2+\frac{(-3)(-4)(-5)}{3!}(2)^{-6}(5x)^3$ | B1, M1, A1 | $\frac{1}{8}$ outside; any two terms correct; all four terms correct |
| $=\frac{1}{8}-\frac{15}{16}x+\frac{75}{16}x^2-\frac{625}{32}x^3+...$ | A1; A1 | Same as Way 1 |
## Question 1 Notes
| Note | Guidance |
|---|---|
| 1st M1 | Implied by constant term of $(2)^{-3}$ or $\frac{1}{8}$ |
| B1 | $2^{-3}$ or $\frac{1}{8}$ outside brackets or $\frac{1}{8}$ as candidate's constant term |
| 2nd M1 | Expands $(...+kx)^{-3}$, $k\neq 1$, to give any 2 terms out of 4, simplified or un-simplified |
| 1st A1 | Correct simplified or un-simplified expansion with consistent $(kx)$, $k\neq 1$ |
| 2nd A1 | For $\frac{1}{8}-\frac{15}{16}x$ **(simplified)** or $0.125-0.9375x$ |
| 3rd A1 | Accept only $\frac{75}{16}x^2-\frac{625}{32}x^3$ or $4\frac{11}{16}x^2-19\frac{17}{32}x^3$ or $4.6875x^2-19.53125x^3$ |
| SC | If candidate would otherwise score 2nd A0, 3rd A0: allow Special Case 2nd A1 for un-multiplied bracket expressions with each term a simplified fraction or decimal |
| Note | Ignore extra terms beyond $x^3$; ignore subsequent working following a correct answer |
---\begin{enumerate}
\item Use the binomial series to find the expansion of
\end{enumerate}
$$\frac { 1 } { ( 2 + 5 x ) ^ { 3 } } , \quad | x | < \frac { 2 } { 5 }$$
in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.\\
Give each coefficient as a fraction in its simplest form.\\
(6)\\
\hfill \mbox{\textit{Edexcel C4 2016 Q1 [6]}}