| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Sketch exponential graphs |
| Difficulty | Easy -1.2 This is a straightforward C2 question testing basic exponential graph sketching, calculator use for evaluating powers, and direct application of the trapezium rule formula. All parts are routine recall and mechanical application with no problem-solving required, making it easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 |
| \(3 ^ { x }\) | 1.246 | 1.552 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Shape: curve must be smooth (not a straight line). Curve must extend to the left of the \(y\)-axis and must be increasing. Curve can't 'touch' the \(x\)-axis but must not go below it. | B1, B1 | The B1 for (0, 1) is independent of the sketch. (2 marks) |
| Otherwise, be generous in cases of doubt. | ||
| (b) Missing values: \(1.933, 2.408\) | B1, B1 | (Accept awrt) (2 marks) |
| (c) \(\frac{1}{2} \times 0.2 \cdot \{(1 + 3) + 2(1.246 + 1.552 + 1.933 + 2.408)\} = 1.8278\) (awrt 1.83) | B1, M1, A1ft, A1 | (4 marks) |
| (8 marks) | Beware the order of marks! (a) Must be a curve (not a straight line). Curve must extend to the left of the \(y\)-axis, and must be increasing. Curve can't 'touch' the \(x\)-axis but must not go below it. Otherwise, be generous in cases of doubt. The B1 for (0, 1) is independent of the sketch. (c) Bracketing mistake: i.e. \(\frac{1}{2} \times 0.2(1 + 3) + 2(1.246 + 1.552 + 1.933 + 2.408)\) scores B1 M1 A0 A0 unless the final answer implies that the calculation has been done correctly (then full marks can be given). |
**(a)** Shape: curve must be smooth (not a straight line). Curve must extend to the left of the $y$-axis and must be increasing. Curve can't 'touch' the $x$-axis but must not go below it. | B1, B1 | The B1 for (0, 1) is independent of the sketch. (2 marks) |
| --- | --- | --- |
| Otherwise, be generous in cases of doubt. | | |
| **(b)** Missing values: $1.933, 2.408$ | B1, B1 | (Accept awrt) (2 marks) |
| **(c)** $\frac{1}{2} \times 0.2 \cdot \{(1 + 3) + 2(1.246 + 1.552 + 1.933 + 2.408)\} = 1.8278$ (awrt 1.83) | B1, M1, A1ft, A1 | (4 marks) |
| | (8 marks) | **Beware the order of marks!** **(a)** Must be a curve (not a straight line). Curve must extend to the left of the $y$-axis, and must be increasing. Curve can't 'touch' the $x$-axis but must not go below it. Otherwise, be generous in cases of doubt. The B1 for (0, 1) is independent of the sketch. **(c)** Bracketing mistake: i.e. $\frac{1}{2} \times 0.2(1 + 3) + 2(1.246 + 1.552 + 1.933 + 2.408)$ scores B1 M1 A0 A0 unless the final answer implies that the calculation has been done correctly (then full marks can be given). |5. (a) In the space provided, sketch the graph of $y = 3 ^ { x } , x \in \mathbb { R }$, showing the coordinates of the point at which the graph meets the $y$-axis.\\
(b) Complete the table, giving the values of $3 ^ { x }$ to 3 decimal places.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$x$ & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \\
\hline
$3 ^ { x }$ & & 1.246 & 1.552 & & & 3 \\
\hline
\end{tabular}
\end{center}
(c) Use the trapezium rule, with all the values from your table, to find an approximation for the value of $\int _ { 0 } ^ { 1 } 3 ^ { x } \mathrm {~d} x$.\\
\hfill \mbox{\textit{Edexcel C2 2006 Q5 [8]}}