| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Show constant equals specific form |
| Difficulty | Moderate -0.3 This is a multi-part C2 question covering standard exponential function topics. Parts (a)-(c) are routine: sketching 2^x, applying trapezium rule with given ordinates, and identifying transformations. Part (d) requires substituting x=0, solving 2^k=5, and expressing as log₂5, which is straightforward algebraic manipulation. While it spans multiple techniques, each component is textbook-standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.06a Exponential function: a^x and e^x graphs and properties1.06c Logarithm definition: log_a(x) as inverse of a^x1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Shape with asymptotic behaviour in 2nd quadrant, below point of intersection with \(y\)-axis | B1 | Shape with some indication of asymptotic behaviour in \(2^{nd}\) quadrant below pt of intersection with \(y\)-axis |
| Only intersection is with \(y\)-axis at \((0, 1)\) stated/indicated | B1 | Accept 1 on \(y\)-axis as equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(h = 0.5\) | B1 | PI |
| \(I \approx \frac{h}{2}\{f(0)+f(2)+2[f(0.5)+f(1)+f(1.5)]\}\) | M1 | OE summing of areas of the 4 'trapezia' |
| \(\{\ldots\} = 1 + 4 + 2(\sqrt{2} + 2 + \sqrt{8}) = 5 + 2 \times 6.2426\ldots = 17.485\ldots\) | A1 | OE Accept 2dp (rounded or truncated) as evidence for surds |
| \(I \approx 4.3713\ldots = 4.37\) (to 3sf) | A1 | CAO Must be 4.37. SC for those who use 5 strips, max possible is B0M1A1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Increase the number of ordinates | E1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Translation | B1 | Accept 'translat…' as equivalent. [T or Tr is NOT sufficient] |
| \(\begin{bmatrix}-7\\3\end{bmatrix}\) | B1;B1 | B1 for each component of the vector. Condone if the equiv 2 vectors are given. Accept full equivalent to vector(s) in words provided linked to 'translation/move/shift' and correct directions. (No marks if different transformations) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(8 = 2^k + 3 \Rightarrow 2^k = 5\) | M1 | Correct subst. and an attempted rearrangement to \(2^k = N\). PI by \(k = \frac{\log 5}{\log 2}\) |
| \(k = \log_2 5\) | A1 | Accept \(m = 2\), \(n = 5\) |
## Question 6:
### Part 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Shape with asymptotic behaviour in 2nd quadrant, below point of intersection with $y$-axis | B1 | Shape with some indication of asymptotic behaviour in $2^{nd}$ quadrant below pt of intersection with $y$-axis |
| Only intersection is with $y$-axis at $(0, 1)$ stated/indicated | B1 | Accept 1 on $y$-axis as equivalent |
### Part 6(b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $h = 0.5$ | B1 | PI |
| $I \approx \frac{h}{2}\{f(0)+f(2)+2[f(0.5)+f(1)+f(1.5)]\}$ | M1 | OE summing of areas of the 4 'trapezia' |
| $\{\ldots\} = 1 + 4 + 2(\sqrt{2} + 2 + \sqrt{8}) = 5 + 2 \times 6.2426\ldots = 17.485\ldots$ | A1 | OE Accept 2dp (rounded or truncated) as evidence for surds |
| $I \approx 4.3713\ldots = 4.37$ (to 3sf) | A1 | CAO Must be 4.37. SC for those who use 5 strips, max possible is B0M1A1A0 |
### Part 6(b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Increase the number of ordinates | E1 | OE |
### Part 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Translation | B1 | Accept 'translat…' as equivalent. [T or Tr is NOT sufficient] |
| $\begin{bmatrix}-7\\3\end{bmatrix}$ | B1;B1 | B1 for each component of the vector. Condone if the equiv 2 vectors are given. Accept **full** equivalent to vector(s) in words provided linked to 'translation/move/shift' and **correct** directions. (No marks if **different** transformations) |
### Part 6(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $8 = 2^k + 3 \Rightarrow 2^k = 5$ | M1 | Correct subst. and an attempted rearrangement to $2^k = N$. PI by $k = \frac{\log 5}{\log 2}$ |
| $k = \log_2 5$ | A1 | Accept $m = 2$, $n = 5$ |
---6
\begin{enumerate}[label=(\alph*)]
\item Sketch the curve with equation $y = 2 ^ { x }$, indicating the coordinates of any point where the curve intersects the coordinate axes.
\item \begin{enumerate}[label=(\roman*)]
\item Use the trapezium rule with five ordinates (four strips) to find an approximate value for $\int _ { 0 } ^ { 2 } 2 ^ { x } \mathrm {~d} x$, giving your answer to three significant figures.
\item State how you could obtain a better approximation to the value of the integral using the trapezium rule.
\end{enumerate}\item Describe a geometrical transformation that maps the graph of $y = 2 ^ { x }$ onto the graph of $y = 2 ^ { x + 7 } + 3$.
\item The curve $y = 2 ^ { x + k } + 3$ intersects the $y$-axis at the point $A ( 0,8 )$.
Show that $k = \log _ { m } n$, where $m$ and $n$ are integers.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2010 Q6 [12]}}