| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Find intersection of exponential curves |
| Difficulty | Moderate -0.8 This is a straightforward C2 question testing basic exponential function skills: substitution (part a), trapezium rule application (b), solving a simple exponential equation using logarithms (c), and reflection transformation (d). All parts are routine textbook exercises requiring standard techniques with no problem-solving insight needed, making it easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b1.09f Trapezium rule: numerical integration |
| Answer | Marks |
|---|---|
| {y-coordinate of \(A\) is} 2 | B1 |
| Total for 8(a): 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| \(h = 0.25\); Integral \(= \frac{h}{2}\{...\}\); \(\{...\} = f(0) + 2[f(\frac{1}{4}) + f(\frac{1}{2}) + f(\frac{3}{4})] + f(1)\); \(\{...\} = \frac{2 + 4 + 2[(2.316... + 2.732... + 3.279(5...))]}{[= 6 + 2 \times 8.3276...]} = \{= 22.65(5...).\}\); Integral \(= 0.125 \times 22.655... = 2.8319..\); Integral \(= 2.83\) to 3 sf | M1, A1√, A1 | Condone one numerical slip; Accept values to 3sf (rnd or trunc); ft answer from (a) if not "2"; CAO Must be 2.83 (NMS scores 0/4) |
| Total for 8(b)(i): 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Relevant trapeza drawn on a copy of given graph; {Approximation is an}overestimate | M1, A1 | Accept relevant single trapezium with its sloping side above the curve; |
| Total for 8(b)(ii): 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(5 = 3^x + 1 \Rightarrow 3^x = 4\); \(\log_{10}3^x = \log_{10}4\); \(x\log_{10}3 = \log_{10}4\); \(x = \frac{\lg 4}{\lg 3} = 1.26185... = 1.2619\) to 4dp | B1, M1, m1, A1 | Takes ln or \(\log_{10}\) on both sides of \(3^x = k\), where \(k > 0\); Use of log \(3^x = x\log 3\); Accept 4dp or better; [If using T&I a full justification is required; else M0m0A0] |
| Total for 8(c): 4 marks |
| Answer | Marks |
|---|---|
| \(f(x) = 3^{-x} + 1\) | B1 |
| Total for Question 8: 12 marks |
**8(a)**
| {y-coordinate of $A$ is} 2 | B1 | |
| Total for 8(a): 1 mark |
**8(b)(i)**
| $h = 0.25$; Integral $= \frac{h}{2}\{...\}$; $\{...\} = f(0) + 2[f(\frac{1}{4}) + f(\frac{1}{2}) + f(\frac{3}{4})] + f(1)$; $\{...\} = \frac{2 + 4 + 2[(2.316... + 2.732... + 3.279(5...))]}{[= 6 + 2 \times 8.3276...]} = \{= 22.65(5...).\}$; Integral $= 0.125 \times 22.655... = 2.8319..$; Integral $= 2.83$ to 3 sf | M1, A1√, A1 | Condone one numerical slip; Accept values to 3sf (rnd or trunc); ft answer from (a) if not "2"; CAO Must be 2.83 (NMS scores 0/4) |
| Total for 8(b)(i): 4 marks |
**8(b)(ii)**
| Relevant trapeza drawn on a copy of given graph; {Approximation is an}overestimate | M1, A1 | Accept relevant single trapezium with its sloping side above the curve; |
| Total for 8(b)(ii): 2 marks |
**8(c)**
| $5 = 3^x + 1 \Rightarrow 3^x = 4$; $\log_{10}3^x = \log_{10}4$; $x\log_{10}3 = \log_{10}4$; $x = \frac{\lg 4}{\lg 3} = 1.26185... = 1.2619$ to 4dp | B1, M1, m1, A1 | Takes ln or $\log_{10}$ on both sides of $3^x = k$, where $k > 0$; Use of log $3^x = x\log 3$; Accept 4dp or better; [If using T&I a full justification is required; else M0m0A0] |
| Total for 8(c): 4 marks |
**8(d)**
| $f(x) = 3^{-x} + 1$ | B1 | |
| **Total for Question 8: 12 marks** |
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## **TOTAL FOR PAPER: 75 marks**8 The diagram shows a sketch of the curve with equation $y = 3 ^ { x } + 1$.\\
\includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-5_535_1011_411_513}
The curve intersects the $y$-axis at the point $A$.
\begin{enumerate}[label=(\alph*)]
\item Write down the $y$-coordinate of point $A$.
\item \begin{enumerate}[label=(\roman*)]
\item Use the trapezium rule with five ordinates (four strips) to find an approximation for $\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x$, giving your answer to three significant figures.\\
(4 marks)
\item By considering the graph of $y = 3 ^ { x } + 1$, explain with the aid of a diagram whether your approximation will be an overestimate or an underestimate of the true value of $\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x$.\\
(2 marks)
\end{enumerate}\item The line $y = 5$ intersects the curve $y = 3 ^ { x } + 1$ at the point $P$. By solving a suitable equation, find the $x$-coordinate of the point $P$. Give your answer to four decimal places.\\
(4 marks)
\item The curve $y = 3 ^ { x } + 1$ is reflected in the $y$-axis to give the curve with equation $y = \mathrm { f } ( x )$. Write down an expression for $\mathrm { f } ( x )$.\\
(1 mark)
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2005 Q8 [12]}}