| Exam Board | AQA |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Approximation for small x |
| Difficulty | Moderate -0.8 This is a straightforward AS-level binomial expansion question requiring routine application of the binomial theorem for a positive integer power (n=5), simple algebraic manipulation to combine two expansions, and substitution of a given value. All steps are standard textbook exercises with no problem-solving insight required, making it easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 + 5 \times (-2x) + \frac{5 \times 4}{2}(-2x)^2\) | M1 | Expands with at least first term and one other term correct |
| \(= 1 - 10x + 40x^2\) | A1 | Expands with all terms correctly simplified |
| Answer | Marks | Guidance |
|---|---|---|
| \((1 + 5x)^2 = 1 + 10x + 25x^2\) | M1 | Correctly expands \((1+5x)^2\) |
| \(2 + 65x^2\) | A1F | Obtains correct simplified expression for \((1-2x)^5 + (1+5x)^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Use \(x = 0.001\) | M1 | Obtains \(x = 0.001\) as the value to use |
| \(2 + 65 \times 0.001^2 = 2.000\,065\) | A1 | CSO |
## Question 4(a):
$1 + 5 \times (-2x) + \frac{5 \times 4}{2}(-2x)^2$ | M1 | Expands with at least first term and one other term correct |
$= 1 - 10x + 40x^2$ | A1 | Expands with all terms correctly simplified |
## Question 4(b):
$(1 + 5x)^2 = 1 + 10x + 25x^2$ | M1 | Correctly expands $(1+5x)^2$ |
$2 + 65x^2$ | A1F | Obtains correct simplified expression for $(1-2x)^5 + (1+5x)^2$ |
## Question 4(c):
Use $x = 0.001$ | M1 | Obtains $x = 0.001$ as the value to use |
$2 + 65 \times 0.001^2 = 2.000\,065$ | A1 | CSO |
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\begin{enumerate}[label=(\alph*)]
\item Find the binomial expansion of $( 1 - 2 x ) ^ { 5 }$ in ascending powers of $x$ up to and including the term in $x ^ { 2 }$
4
\item Find the first two non-zero terms in the expansion of
$$( 1 - 2 x ) ^ { 5 } + ( 1 + 5 x ) ^ { 2 }$$
in ascending powers of $x$.\\
4
\item Hence, use an appropriate value of $x$ to obtain an approximation for $0.998 ^ { 5 } + 1.005 ^ { 2 }$ [2 marks]\\
$5 A B C$ is a triangle. The point $D$ lies on $A C$.\\
$A B = 8 \mathrm {~cm} , B C = B D = 7 \mathrm {~cm}$ and angle $A = 60 ^ { \circ }$ as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-06_604_978_486_532}
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 2 2021 Q4 [6]}}