| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with exponentials |
| Difficulty | Moderate -0.8 This is a straightforward P2 question testing routine integration of exponentials and basic trapezium rule understanding. Part (i) requires direct application of standard integral formulas with no problem-solving, part (ii) is a simple sketch, and part (iii) tests recall of convexity properties. All parts are textbook exercises requiring minimal insight. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Obtain integral of form \(k_1 e^{\frac{1}{2}x} + k_2 x\) | M1 | Allow \(k_1 = 4\) |
| Obtain correct \(8e^{\frac{1}{2}x} + 3x\) oe | A1 | |
| Use limits correctly to confirm \(8e - 2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Draw increasing curve in first quadrant | M1 | If incorrect \(y\) intercept used then M1 A0 |
| Draw more or less accurate sketch with correct curvature, gradient at \(x = 0\) must be \(> 0\) | A1 | Allow if no intercept stated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State more and refer to top(s) of trapezium(s) above curve | B1 | Can be shown using a diagram. Reference to a trapezium must be made |
## Question 3(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain integral of form $k_1 e^{\frac{1}{2}x} + k_2 x$ | M1 | Allow $k_1 = 4$ |
| Obtain correct $8e^{\frac{1}{2}x} + 3x$ oe | A1 | |
| Use limits correctly to confirm $8e - 2$ | A1 | |
## Question 3(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Draw increasing curve in first quadrant | M1 | If incorrect $y$ intercept used then M1 A0 |
| Draw more or less accurate sketch with correct curvature, gradient at $x = 0$ must be $> 0$ | A1 | Allow if no intercept stated |
## Question 3(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State more and refer to top(s) of trapezium(s) above curve | B1 | Can be shown using a diagram. Reference to a trapezium must be made |
---3 The definite integral $I$ is defined by $I = \int _ { 0 } ^ { 2 } \left( 4 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 3 \right) \mathrm { d } x$.\\
(i) Show that $I = 8 \mathrm { e } - 2$.\\
(ii) Sketch the curve $y = 4 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 3$ for $0 \leqslant x \leqslant 2$.\\
(iii) State whether an estimate of $I$ obtained by using the trapezium rule will be more than or less than $8 \mathrm { e } - 2$. Justify your answer.
\hfill \mbox{\textit{CAIE P2 2016 Q3 [6]}}