| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Differentiation of reciprocal functions |
| Difficulty | Standard +0.3 This is a structured multi-part question requiring quotient rule differentiation (standard technique), equating derivatives (routine), and iterative numerical methods (standard P3 content). Each part is guided with clear instructions, requiring no novel insight—slightly easier than average due to scaffolding. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use correct quotient or chain rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Obtain the given answer correctly | A1 | [3] |
| (ii) State a correct equation, e.g. \(-e^{-a} = -\cos ec \, a \cot a\) | B1 | |
| Rearrange it correctly in the given form | B1 | [2] |
| (iii) Calculate 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] |
| (iv) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(1.317\) | A1 | |
| Show sufficient iterations to 5 d.p. to justify 1.317 to 3 d.p., or show there is a sign change in the interval (1.3165, 1.3175) | A1 | [3] |
**(i)** Use correct quotient or chain rule | M1 |
Obtain correct derivative in any form | A1 |
Obtain the given answer correctly | A1 | [3]
**(ii)** State a correct equation, e.g. $-e^{-a} = -\cos ec \, a \cot a$ | B1 |
Rearrange it correctly in the given form | B1 | [2]
**(iii)** Calculate 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]
**(iv)** Use the iterative formula correctly at least once | M1 |
Obtain final answer $1.317$ | A1 |
Show sufficient iterations to 5 d.p. to justify 1.317 to 3 d.p., or show there is a sign change in the interval (1.3165, 1.3175) | A1 | [3]8\\
\includegraphics[max width=\textwidth, alt={}, center]{9d3af34a-670f-425e-8156-0ad4d08fbdc0-3_499_552_258_792}
The diagram shows the curve $y = \operatorname { cosec } x$ for $0 < x < \pi$ and part of the curve $y = \mathrm { e } ^ { - x }$. When $x = a$, the tangents to the curves are parallel.\\
(i) By differentiating $\frac { 1 } { \sin x }$, show that if $y = \operatorname { cosec } x$ then $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } x \cot x$.\\
(ii) By equating the gradients of the curves at $x = a$, show that
$$a = \tan ^ { - 1 } \left( \frac { \mathrm { e } ^ { a } } { \sin a } \right)$$
(iii) Verify by calculation that $a$ lies between 1 and 1.5.\\
(iv) Use an iterative formula based on the equation in part (ii) to determine $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
\hfill \mbox{\textit{CAIE P3 2016 Q8 [10]}}