| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Combined expansions then integrate |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring routine binomial expansion of small powers (n=3,4), basic algebraic manipulation to combine terms, and substitution integration. All steps are standard C2 techniques with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((1-x)^3 = 1 - 3x + 3x^2 - x^3\) | B1 M1 A1 | M1 for attempt at expansion; A1 fully correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((1+y)^4 = 1 + 4y + 6y^2 + 4y^3 + y^4\) | B1 | Correct expansion |
| \((1-y)^3 = 1 - 3y + 3y^2 - y^3\) | B1 | Using part (a) |
| Subtracting: \(7y + py^2 + qy^3 + y^4\) where \(p = 3\), \(q = 5\) | M1 A1 | M1 for subtraction; A1 both correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(y = \sqrt{x} = x^{1/2}\) | M1 | |
| \(\int\left[7x^{1/2} + 3x + 5x^{3/2} + x^2\right]dx\) | A1 | |
| \(= \frac{14}{3}x^{3/2} + \frac{3}{2}x^2 + 2x^{5/2} + \frac{1}{3}x^3 + c\) | M1 A1 | M1 for integration attempt; A1 all correct |
# Question 5:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(1-x)^3 = 1 - 3x + 3x^2 - x^3$ | B1 M1 A1 | M1 for attempt at expansion; A1 fully correct |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(1+y)^4 = 1 + 4y + 6y^2 + 4y^3 + y^4$ | B1 | Correct expansion |
| $(1-y)^3 = 1 - 3y + 3y^2 - y^3$ | B1 | Using part (a) |
| Subtracting: $7y + py^2 + qy^3 + y^4$ where $p = 3$, $q = 5$ | M1 A1 | M1 for subtraction; A1 both correct |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $y = \sqrt{x} = x^{1/2}$ | M1 | |
| $\int\left[7x^{1/2} + 3x + 5x^{3/2} + x^2\right]dx$ | A1 | |
| $= \frac{14}{3}x^{3/2} + \frac{3}{2}x^2 + 2x^{5/2} + \frac{1}{3}x^3 + c$ | M1 A1 | M1 for integration attempt; A1 all correct |5
\begin{enumerate}[label=(\alph*)]
\item Using the binomial expansion, or otherwise, express $( 1 - x ) ^ { 3 }$ in ascending powers of $x$.
\item Show that the expansion of
$$( 1 + y ) ^ { 4 } - ( 1 - y ) ^ { 3 }$$
is
$$7 y + p y ^ { 2 } + q y ^ { 3 } + y ^ { 4 }$$
where $p$ and $q$ are constants to be found.
\item Hence find $\int \left[ ( 1 + \sqrt { x } ) ^ { 4 } - ( 1 - \sqrt { x } ) ^ { 3 } \right] \mathrm { d } x$, expressing each coefficient in its simplest form.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2011 Q5 [10]}}