| Exam Board | AQA |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Sketch exponential graphs |
| Difficulty | Standard +0.3 This is a multi-part question that combines standard differentiation/integration of exponentials (parts a(i) and a(ii) are routine recall), geometric sequence summation (standard technique), and a limit calculation. While part (b)(iii) requires recognizing that the Riemann sum limit equals the definite integral, this is a well-practiced concept. The geometric sequence work is straightforward once the setup is understood. Overall, slightly easier than average due to heavy scaffolding and standard techniques throughout. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<11.07j Differentiate exponentials: e^(kx) and a^(kx)1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\dfrac{dy}{dx} = 2^x \ln 2\) | B1 | Obtains \(2^x \ln 2\), or \(\ln 2\, e^{x\ln 2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(k \cdot 2^x,\ k \neq 1\) or \(0\) | M1 | Integrates to obtain \(k2^x\) |
| \(\displaystyle\int 2^x\, dx = \dfrac{2^x}{\ln 2} + c\) | R1 | Deduces result; must include \(+c\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\dfrac{1}{2} \times \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{4}\) | M1 | Obtains \(2^{-\frac{1}{2}}\); exact value ACF |
| \(\dfrac{\sqrt{2}}{4}\) | R1 | Writes product \(0.5 \times 2^{-\frac{1}{2}}\) in exact form ACF; condone \(-0.5 \times 2^{-\frac{1}{2}}\) if reason given for rejecting negative sign |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Uses \(S_n = \frac{a(1-r^n)}{1-r}\) | M1 (1.1a) | With at least two of \(a = \frac{\sqrt{2}}{4}\), \(r = \frac{\sqrt{2}}{2}\) and \(n = 8\) correct; OR at least two of \(a = \frac{1}{32}\), \(r = \sqrt{2}\) and \(n = 8\) correct; OR forms sum of 8 rectangles with at least 4 correct terms |
| \(\frac{\frac{\sqrt{2}}{4}\left(1-\left(\frac{\sqrt{2}}{2}\right)^8\right)}{1-\frac{\sqrt{2}}{2}} = \frac{15+15\sqrt{2}}{32}\) | A1 (1.1b) | Obtains correct expression, can be left unsimplified |
| \(\frac{15(1+\sqrt{2})}{32}\) or \(\frac{15}{32}(1+\sqrt{2})\) | R1 (2.1) | Do not award if value of \(k\) is just stated without either of these answers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_{-4}^{0} 2^x \, dx\) | M1 (3.1a) | Forms the definite integral; PI by \(\frac{1}{\ln 2}\left[2^x\right]_{-4}^{0}\); condone swapped limits and missing \(dx\); PI by AWRT \(\pm 1.35\) |
| \(\int_{-4}^{0} 2^x \, dx = \frac{1}{\ln 2}\left[2^x\right]_{-4}^{0} = \frac{1}{\ln 2}(2^0 - 2^{-4})\) | A1 (1.1b) | Substitutes 0 and \(-4\) correctly into the correct integrated expression; OR obtains AWRT 1.35 |
| \(= \frac{15}{16\ln 2}\) | A1 (1.1b) | Obtains correct exact value (ACF) |
## Question 14(a)(i):
$\dfrac{dy}{dx} = 2^x \ln 2$ | B1 | Obtains $2^x \ln 2$, or $\ln 2\, e^{x\ln 2}$
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## Question 14(a)(ii):
$k \cdot 2^x,\ k \neq 1$ or $0$ | M1 | Integrates to obtain $k2^x$
$\displaystyle\int 2^x\, dx = \dfrac{2^x}{\ln 2} + c$ | R1 | Deduces result; must include $+c$
---
## Question 14(b)(i):
$\dfrac{1}{2} \times \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{4}$ | M1 | Obtains $2^{-\frac{1}{2}}$; exact value ACF
$\dfrac{\sqrt{2}}{4}$ | R1 | Writes product $0.5 \times 2^{-\frac{1}{2}}$ in exact form ACF; condone $-0.5 \times 2^{-\frac{1}{2}}$ if reason given for rejecting negative sign
## Question 14(b)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Uses $S_n = \frac{a(1-r^n)}{1-r}$ | M1 (1.1a) | With at least two of $a = \frac{\sqrt{2}}{4}$, $r = \frac{\sqrt{2}}{2}$ and $n = 8$ correct; OR at least two of $a = \frac{1}{32}$, $r = \sqrt{2}$ and $n = 8$ correct; OR forms sum of 8 rectangles with at least 4 correct terms |
| $\frac{\frac{\sqrt{2}}{4}\left(1-\left(\frac{\sqrt{2}}{2}\right)^8\right)}{1-\frac{\sqrt{2}}{2}} = \frac{15+15\sqrt{2}}{32}$ | A1 (1.1b) | Obtains correct expression, can be left unsimplified |
| $\frac{15(1+\sqrt{2})}{32}$ or $\frac{15}{32}(1+\sqrt{2})$ | R1 (2.1) | Do not award if value of $k$ is just stated without either of these answers |
**Subtotal: 3 marks**
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## Question 14(b)(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_{-4}^{0} 2^x \, dx$ | M1 (3.1a) | Forms the definite integral; PI by $\frac{1}{\ln 2}\left[2^x\right]_{-4}^{0}$; condone swapped limits and missing $dx$; PI by AWRT $\pm 1.35$ |
| $\int_{-4}^{0} 2^x \, dx = \frac{1}{\ln 2}\left[2^x\right]_{-4}^{0} = \frac{1}{\ln 2}(2^0 - 2^{-4})$ | A1 (1.1b) | Substitutes 0 and $-4$ correctly into the correct integrated expression; OR obtains AWRT 1.35 |
| $= \frac{15}{16\ln 2}$ | A1 (1.1b) | Obtains correct exact value (ACF) |
**Subtotal: 3 marks**
**Question 14 Total: 11 marks**
---14
\begin{enumerate}[label=(\alph*)]
\item (i) Given that
$$y = 2 ^ { x }$$
write down $\frac { \mathrm { d } y } { \mathrm {~d} x }$
14 (a) (ii) Hence find
$$\int 2 ^ { x } \mathrm {~d} x$$
14
\item The area, $A$, bounded by the curve with equation $y = 2 ^ { x }$, the $x$-axis, the $y$-axis and the line $x = - 4$ is approximated using eight rectangles of equal width as shown in the diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-23_1319_978_450_532}
14 (b) (i) Show that the exact area of the largest rectangle is $\frac { \sqrt { 2 } } { 4 }$\\
14 (b) (ii) The areas of these rectangles form a geometric sequence with common ratio $\frac { \sqrt { 2 } } { 2 }$\\
Find the exact value of the total area of the eight rectangles.\\
Give your answer in the form $k ( 1 + \sqrt { 2 } )$ where $k$ is a rational number.\\[0pt]
[3 marks]\\
14 (b) (iii) More accurate approximations for $A$ can be found by increasing the number, $n$, of rectangles used.
Find the exact value of the limit of the approximations for $A$ as $n \rightarrow \infty$
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 1 2023 Q14 [13]}}