| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Find intersection of exponential curves |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on exponential functions requiring standard techniques: sketching y-intercept, solving exponential equations using logarithms, applying trapezium rule formula, and solving a simple equation. All parts are routine C2-level exercises with clear methods and no novel problem-solving required, making it slightly easier than average. |
| Spec | 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 |
|---|---|---|
| Sketch graph showing exponential growth (both quadrants) | B1 | Sketch graph showing exponential growth (both quadrants) |
| B1 (2) | State or imply \((0, 4)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(4k^x = 20k^2\) | M1 | Equate \(4k^x\) to \(20k^2\) and take logs (any, or no, base) |
| \(k^x = 5k^2\) | M1 | Use \(\log ab = \log a + \log b\) |
| \(x = \log_k 5 + \log_k k^2\) | M1 | Use \(\log a^b = b\log a\) |
| \(x = 2\log_k k + \log_k 5\) | ||
| \(x = 2 + \log_k 5\) AG | A1 (4) | Show given answer correctly |
| OR: \(k^{x-2} = 5\), \(x - 2 = \log_k 5\), \(x = 2 + \log_k 5\) AG | M1, A1, M1, A1 | Attempt to rewrite as single index; obtain \(k^{x-2} = 5\); take logs; show given answer |
| Answer | Marks | Guidance |
|---|---|---|
| area \(\approx \frac{1}{2} \times \frac{1}{2} \times \left(4k^0 + 8k^{\frac{1}{2}} + 4k^1\right)\) | M1 | Attempt \(y\)-values at \(x = 0\), \(\frac{1}{2}\) and \(1\), and no others |
| M1 | Attempt to use correct trapezium rule, 3 \(y\)-values, \(h = \frac{1}{2}\) | |
| \(\approx 1 + 2k^{\frac{1}{2}} + k\) | A1 (3) | Obtain a correct expression, allow unsimplified |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 + 2k^{\frac{1}{2}} + k = 16\) | M1 | Equate attempt at area to \(16\) |
| \(\left(k^{\frac{1}{2}} + 1\right)^2 = 16\) | M1 | Attempt to solve 'disguised' 3 term quadratic |
| \(k^{\frac{1}{2}} = 3\) | ||
| \(k = 9\) | A1 (3) | Obtain \(k = 9\) only |
## Question 9:
### Part (i):
| Sketch graph showing exponential growth (both quadrants) | B1 | Sketch graph showing exponential growth (both quadrants) |
|---|---|---|
| | B1 **(2)** | State or imply $(0, 4)$ |
### Part (ii):
| $4k^x = 20k^2$ | M1 | Equate $4k^x$ to $20k^2$ and take logs (any, or no, base) |
|---|---|---|
| $k^x = 5k^2$ | M1 | Use $\log ab = \log a + \log b$ |
| $x = \log_k 5 + \log_k k^2$ | M1 | Use $\log a^b = b\log a$ |
| $x = 2\log_k k + \log_k 5$ | | |
| $x = 2 + \log_k 5$ AG | A1 **(4)** | Show given answer correctly |
| OR: $k^{x-2} = 5$, $x - 2 = \log_k 5$, $x = 2 + \log_k 5$ AG | M1, A1, M1, A1 | Attempt to rewrite as single index; obtain $k^{x-2} = 5$; take logs; show given answer |
### Part (iii)(a):
| area $\approx \frac{1}{2} \times \frac{1}{2} \times \left(4k^0 + 8k^{\frac{1}{2}} + 4k^1\right)$ | M1 | Attempt $y$-values at $x = 0$, $\frac{1}{2}$ and $1$, and no others |
|---|---|---|
| | M1 | Attempt to use correct trapezium rule, 3 $y$-values, $h = \frac{1}{2}$ |
| $\approx 1 + 2k^{\frac{1}{2}} + k$ | A1 **(3)** | Obtain a correct expression, allow unsimplified |
### Part (iii)(b):
| $1 + 2k^{\frac{1}{2}} + k = 16$ | M1 | Equate attempt at area to $16$ |
|---|---|---|
| $\left(k^{\frac{1}{2}} + 1\right)^2 = 16$ | M1 | Attempt to solve 'disguised' 3 term quadratic |
| $k^{\frac{1}{2}} = 3$ | | |
| $k = 9$ | A1 **(3)** | Obtain $k = 9$ only |9
\begin{enumerate}[label=(\roman*)]
\item Sketch the graph of $y = 4 k ^ { x }$, where $k$ is a constant such that $k > 1$. State the coordinates of any points of intersection with the axes.
\item The point $P$ on the curve $y = 4 k ^ { x }$ has its $y$-coordinate equal to $20 k ^ { 2 }$. Show that the $x$-coordinate of $P$ may be written as $2 + \log _ { k } 5$.
\item (a) Use the trapezium rule, with two strips each of width $\frac { 1 } { 2 }$, to find an expression for the approximate value of
$$\int _ { 0 } ^ { 1 } 4 k ^ { x } \mathrm {~d} x$$
(b) Given that this approximate value is equal to 16 , find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2009 Q9 [12]}}