| Exam Board | AQA |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2019 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Expansion of (a+bx^m)^n |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard A-level techniques: binomial expansion (routine application with fractional power), validity conditions (direct recall), and using the expansion to approximate an integral. Part (c) requires recognizing cos x ≈ 1 - x²/2 and substituting, but this is a guided application rather than requiring novel insight. Part (d) tests understanding of validity ranges. Slightly above average due to the integration step and need to justify, but all components are standard textbook material. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions1.08h Integration by substitution |
| Answer | Marks |
|---|---|
| 9(a) | Write in a form to which the |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| AG | 2.1 | R1 |
| Answer | Marks |
|---|---|
| 9(b) | x2 |
| Answer | Marks | Guidance |
|---|---|---|
| PI by −2x2 <4 | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| ACF | 1.1b | A1 |
| Answer | Marks |
|---|---|
| 9(c) | Explains that as 0.4 radians is |
| Answer | Marks | Guidance |
|---|---|---|
| angle approximation for cos x | 2.4 | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| from 9(a) as the integrand | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| at least one term correct | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| CAO | 1.1b | A1 |
| Answer | Marks |
|---|---|
| 9(d) | States that 1.4 radians is not a |
| Answer | Marks | Guidance |
|---|---|---|
| approximation to be valid | 2.4 | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| Total | 9 | |
| Q | Marking Instructions | AO |
Question 9:
--- 9(a) ---
9(a) | Write in a form to which the
binomial expansion can be
applied
1
x2 2
Must be of forma1−
2 | 3.1a | M1 | 1
x2 2
4−2x2 =21−
2
1 x2
≈21+ −
2 2
x2
≈2−
2
Completes rigorous argument to
obtain correct expansion
AG | 2.1 | R1
--- 9(b) ---
9(b) | x2
Compares their to 1
2
Condone incorrect inequality
PI by −2x2 <4 | 1.1a | M1 | x2
− <1
2
⇒ x < 2
Obtains correct range of values
ACF | 1.1b | A1
--- 9(c) ---
9(c) | Explains that as 0.4 radians is
x2
small therefore cosx≈1−
2
Must refer to 0.4 and small
angle approximation for cos x | 2.4 | E1 | As 0.4 is small
x2
cosx≈1−
2
0.4 0.4 x2
∫ ∫
cosxdx≈ 1− dx
2
0 0
1 0.4 x2
∫
≈ 2− dx
2 2
0
0.4 x2
∫
≈ 1− dx
4
0
0.4
x3
≈ x−
12
0
0.43
≈0.4−
12
≈0.39467
Uses half of their expansion
from 9(a) as the integrand | 1.1a | M1
Integrates their expression with
at least one term correct | 1.1a | M1
Obtains correct value must be at
least five decimal places
148
Condone
375
CAO | 1.1b | A1
--- 9(d) ---
9(d) | States that 1.4 radians is not a
small angle so the
approximation is not valid
Must refer to small angle
approximation and 1.4
or
State invalid as 1.4 is bigger
than 0.664
NB 0.664 is the limiting value for
approximation to be valid | 2.4 | E1 | Since 1.4 is not a small angle the
approximation is not suitable
Total | 9
Q | Marking Instructions | AO | Marks | Typical Solution\begin{enumerate}[label=(\alph*)]
\item Show that the first two terms of the binomial expansion of $\sqrt{4 - 2x^2}$ are
$$2 - \frac{x^2}{2}$$ [2 marks]
\item State the range of values of $x$ for which the expansion found in part (a) is valid. [2 marks]
\item Hence, find an approximation for
$$\int_0^{0.4} \sqrt{\cos x} \, dx$$
giving your answer to five decimal places.
Fully justify your answer. [4 marks]
\item A student decides to use this method to find an approximation for
$$\int_0^{1.4} \sqrt{\cos x} \, dx$$
Explain why this may not be a suitable method. [1 mark]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 2 2019 Q9 [9]}}