| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | March |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show root in interval |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on fixed point iteration and differentiation. Part (a) requires simple substitution to verify a root location (routine). Part (b) applies a standard iterative formula with no complications. Part (c) involves quotient rule differentiation and solving an equation, both standard techniques. While it requires multiple steps and careful algebra, it demands no novel insight or problem-solving beyond applying well-practiced methods. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks |
|---|---|
| 9(a) | Calculate the values of a relevant expression or pair of expressions at x = 1 |
| and x = 1.5 | M1 |
| Complete the argument correctly with correct calculated values | A1 |
| Answer | Marks |
|---|---|
| 9(b) | e2x +1 |
| Answer | Marks |
|---|---|
| once | M1 |
| Obtain final answer 1.20 | A1 |
| Answer | Marks |
|---|---|
| sign change in the interval (1.195,1.205) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 9(c) | Use quotient rule | M1 |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to – 8 and obtain a quadratic in e2x | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain 2 e2x −5e2x +2=0 | A1 | OE |
| Solve a 3-term quadratic in e2x for x | M1 |
| Answer | Marks |
|---|---|
| 2 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Use quotient rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to – 8, take square roots and obtain a quadratic in ex | M1 | |
| Obtain 2e2x −ex − 2=0 | A1 | OE |
| Solve a 3-term quadratic in ex for x | M1 |
| Answer | Marks |
|---|---|
| 2 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 9:
--- 9(a) ---
9(a) | Calculate the values of a relevant expression or pair of expressions at x = 1
and x = 1.5 | M1
Complete the argument correctly with correct calculated values | A1
2
--- 9(b) ---
9(b) | e2x +1
Use the iterative formula x = n , or equivalent, correctly at least
n+1 e2x −1
n
once | M1
Obtain final answer 1.20 | A1
Show sufficient iterations to 4 dp to justify 1.20 to 2 dp, or show there is a
sign change in the interval (1.195,1.205) | A1
3
Question | Answer | Marks | Guidance
--- 9(c) ---
9(c) | Use quotient rule | M1
Obtain correct derivative in any form | A1
Equate derivative to – 8 and obtain a quadratic in e2x | M1
( )2
Obtain 2 e2x −5e2x +2=0 | A1 | OE
Solve a 3-term quadratic in e2x for x | M1
1
Obtain answer x = ln2 , or exact equivalent, only
2 | A1
Alternative method for question 9(c)
Use quotient rule | M1
Obtain correct derivative in any form | A1
Equate derivative to – 8, take square roots and obtain a quadratic in ex | M1
Obtain 2e2x −ex − 2=0 | A1 | OE
Solve a 3-term quadratic in ex for x | M1
1
Obtain answer x = ln2, or exact equivalent, only
2 | A1
6
Question | Answer | Marks | GuidanceLet $\text{f}(x) = \frac{e^{2x} + 1}{e^{2x} - 1}$, for $x > 0$.
\begin{enumerate}[label=(\alph*)]
\item The equation $x = \text{f}(x)$ has one root, denoted by $a$.
Verify by calculation that $a$ lies between 1 and 1.5. [2]
\item Use an iterative formula based on the equation in part (a) to determine $a$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
\item Find f$'(x)$. Hence find the exact value of $x$ for which f$'(x) = -8$. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q9 [11]}}