Moderate -0.3 This is a straightforward reverse chain rule integration followed by solving an exponential equation. Students must integrate 4e^(x/2 + 3), apply limits, and solve for 'a' using logarithms. While it requires multiple steps (integration, substitution of limits, algebraic manipulation, and logarithms), each step is routine and follows standard procedures with no conceptual challenges or novel insights required.
3 Given that \(\int _ { 0 } ^ { a } 4 \mathrm { e } ^ { \frac { 1 } { 2 } x + 3 } \mathrm {~d} x = 835\), find the value of the constant \(a\) correct to 3 significant figures. [5]
Integrate to obtain form \(ke^{\frac{1}{2}x+3}\) where \(k\) is constant not equal to 4
M1
Obtain correct \(8e^{\frac{1}{2}x+3}\)
A1
Allow unsimplified for A1
Obtain \(8e^{\frac{1}{2}a+3} - 8e^3 = 835\) or equivalent
A1
Carry out correct process to find \(a\) from equation of form \(ke^{\frac{1}{2}a+3} = c\)
M1
Obtain 3.65
A1
If 3.65 seen with no actual attempt at integration, award B1 if it is thought that trial and improvement with calculator has been used
Total:
5
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate to obtain form $ke^{\frac{1}{2}x+3}$ where $k$ is constant not equal to 4 | M1 | |
| Obtain correct $8e^{\frac{1}{2}x+3}$ | A1 | Allow unsimplified for A1 |
| Obtain $8e^{\frac{1}{2}a+3} - 8e^3 = 835$ or equivalent | A1 | |
| Carry out correct process to find $a$ from equation of form $ke^{\frac{1}{2}a+3} = c$ | M1 | |
| Obtain 3.65 | A1 | If 3.65 seen with no actual attempt at integration, award B1 if it is thought that trial and improvement with calculator has been used |
| **Total:** | **5** | |
3 Given that $\int _ { 0 } ^ { a } 4 \mathrm { e } ^ { \frac { 1 } { 2 } x + 3 } \mathrm {~d} x = 835$, find the value of the constant $a$ correct to 3 significant figures. [5]\\
\hfill \mbox{\textit{CAIE P2 2017 Q3 [5]}}