| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Show that derivative equals expression |
| Difficulty | Standard +0.3 Part (a) is a straightforward chain rule application with standard verification. Part (b) requires algebraic manipulation to derive a cubic equation from the derivative condition, which is routine. Part (c) is mechanical iteration requiring only calculator work. This is a standard multi-part question slightly easier than average due to the guided structure and lack of conceptual challenges. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Equate answer to part (a) to 1 and obtain a quartic equation in \(t\) or \(\tan x\) | \*M1 | At least as far as \(\left(1+\tan^2 x\right)^2 = 4\tan x\) |
| Obtain correct answer, i.e. \(t^4 + 2t^2 - 4t + 1 = 0\) | A1 | Or equivalent horizontal form. |
| Commence division by \(t - 1\) | DM1 | As far as \(t^3 + t^2 + \ldots\) by long division or inspection. Allow verification by multiplying given answer by \(t - 1\). |
| Obtain the given answer | A1 | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative process correctly with the given formula at least once | M1 | Obtain one value and use that to obtain the next. Must be working in radians. |
| Obtain final answer \(a = 0.29\) | A1 | |
| Show sufficient iterations to 4 d.p. to justify \(0.29\) to 2 d.p., or show there is a sign change in \((0.285, 0.295)\) | A1 | e.g. \(0.3, 0.2854, 0.2894, 0.2883, \ldots\) \(0.4, 0.2436, 0.2984, 0.2841, 0.2883, 0.2871, \ldots\) \(0.5, 0.1776, 0.3103, 0.2805, 0.2893, 0.2868, \ldots\) |
| Total | 3 |
## Question 11(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Equate answer to part (a) to 1 and obtain a quartic equation in $t$ or $\tan x$ | \*M1 | At least as far as $\left(1+\tan^2 x\right)^2 = 4\tan x$ |
| Obtain correct answer, i.e. $t^4 + 2t^2 - 4t + 1 = 0$ | A1 | Or equivalent horizontal form. |
| Commence division by $t - 1$ | DM1 | As far as $t^3 + t^2 + \ldots$ by long division or inspection. Allow verification by multiplying given answer by $t - 1$. |
| Obtain the given answer | A1 | |
| **Total** | **4** | |
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## Question 11(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative process correctly with the given formula at least once | M1 | Obtain one value and use that to obtain the next. Must be working in radians. |
| Obtain final answer $a = 0.29$ | A1 | |
| Show sufficient iterations to 4 d.p. to justify $0.29$ to 2 d.p., or show there is a sign change in $(0.285, 0.295)$ | A1 | e.g. $0.3, 0.2854, 0.2894, 0.2883, \ldots$ $0.4, 0.2436, 0.2984, 0.2841, 0.2883, 0.2871, \ldots$ $0.5, 0.1776, 0.3103, 0.2805, 0.2893, 0.2868, \ldots$ |
| **Total** | **3** | |11 The equation of a curve is $y = \sqrt { \tan x }$, for $0 \leqslant x < \frac { 1 } { 2 } \pi$.
\begin{enumerate}[label=(\alph*)]
\item Express $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\tan x$, and verify that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1$ when $x = \frac { 1 } { 4 } \pi$.\\
The value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ is also 1 at another point on the curve where $x = a$, as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{87be326f-f638-43e9-a654-b7b53d5141ef-18_605_492_1493_822}
\item Show that $t ^ { 3 } + t ^ { 2 } + 3 t - 1 = 0$, where $t = \tan a$.
\item Use the iterative formula
$$a _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 3 } \left( 1 - \tan ^ { 2 } a _ { n } - \tan ^ { 3 } a _ { n } \right) \right)$$
to determine $a$ correct to 2 decimal places, giving the result of each iteration to 4 decimal places.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q11 [11]}}