| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Sum/difference of two binomials simplification |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question requiring routine application of binomial expansion for n=3 (which students often know by pattern), simple algebraic manipulation by subtracting expansions, and basic differentiation to show no stationary points. Each part scaffolds the next with minimal problem-solving required—well below average difficulty for A-level. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n1.07a Derivative as gradient: of tangent to curve1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| \((2+x)^3 = (2^3) + 3(2^2)(x) + 3(2)(x^2) + (x^3)\); \(... = 8 + 12x + 6x^2 + x^3\) (*) | M1, A1, A1 | Any valid method; must contain all components; Accept \(a = 12\); Accept \(b = 6\) |
| Total for 6(a)(i): 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \((2-x)^3 = 8 - 12x + 6x^2 - x^3\) (**) | M1, A1√ | Clear \(x \to -x\) in (i) OE; ft numerical \(a\) and \(b\) |
| Total for 6(a)(ii): 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \((2+x)^3 - (2-x)^3 = (*) - (**)\); \(...... = 24x + 2x^3\) | M1, A1 | Subtracts the 2 expressions in (a); CSO AG (be convinced) |
| Total for 6(b): 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = 24 + 6x^2\); For st. pt. \(24 + 6x^2 = 0\); Not possible since \(24 + 6x^2 > 0\) | M1, A1, E1 | A power of \(x\) decreased by 1; Any valid explanation |
| Total for 6(c): 3 marks | ||
| Total for Question 6: 10 marks |
**6(a)(i)**
| $(2+x)^3 = (2^3) + 3(2^2)(x) + 3(2)(x^2) + (x^3)$; $... = 8 + 12x + 6x^2 + x^3$ (*) | M1, A1, A1 | Any valid method; must contain all components; Accept $a = 12$; Accept $b = 6$ |
| Total for 6(a)(i): 3 marks |
**6(a)(ii)**
| $(2-x)^3 = 8 - 12x + 6x^2 - x^3$ (**) | M1, A1√ | Clear $x \to -x$ in (i) OE; ft numerical $a$ and $b$ |
| Total for 6(a)(ii): 2 marks |
**6(b)**
| $(2+x)^3 - (2-x)^3 = (*) - (**)$; $...... = 24x + 2x^3$ | M1, A1 | Subtracts the 2 expressions in (a); CSO AG (be convinced) |
| Total for 6(b): 2 marks |
**6(c)**
| $\frac{dy}{dx} = 24 + 6x^2$; For st. pt. $24 + 6x^2 = 0$; Not possible since $24 + 6x^2 > 0$ | M1, A1, E1 | A power of $x$ decreased by 1; Any valid explanation |
| Total for 6(c): 3 marks |
| **Total for Question 6: 10 marks** |
---6
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Using the binomial expansion, or otherwise, express $( 2 + x ) ^ { 3 }$ in the form $8 + a x + b x ^ { 2 } + x ^ { 3 }$, where $a$ and $b$ are integers. (3 marks)
\item Write down the expansion of $( 2 - x ) ^ { 3 }$.
\end{enumerate}\item Hence show that $( 2 + x ) ^ { 3 } - ( 2 - x ) ^ { 3 } = 24 x + 2 x ^ { 3 }$.
\item Hence show that the curve with equation
$$y = ( 2 + x ) ^ { 3 } - ( 2 - x ) ^ { 3 }$$
has no stationary points.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2005 Q6 [10]}}