| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find stationary points and nature |
| Difficulty | Standard +0.3 This is a straightforward application of the quotient rule to find dy/dx, setting it to zero to find stationary points, then using the second derivative test. While it requires careful algebraic manipulation and trigonometric identities, it follows a standard procedure with no novel insights required. The interval restriction simplifies finding the solution, making it slightly easier than an average A-level question. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| Use correct product or quotient rule or rewrite as \(2\sec x - \tan x\) and differentiate | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate the derivative to zero and solve for \(x\) | M1 | |
| Obtain \(x = \frac{1}{6}\pi\) | A1 | |
| Obtain \(y = \sqrt{3}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| Carry out an appropriate method for determining the nature of a stationary point | M1 | |
| Show the point is a minimum point with no errors seen | A1 |
## Question 4(i):
| Answer | Mark | Notes |
|--------|------|-------|
| Use correct product or quotient rule or rewrite as $2\sec x - \tan x$ and differentiate | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate the derivative to zero and solve for $x$ | M1 | |
| Obtain $x = \frac{1}{6}\pi$ | A1 | |
| Obtain $y = \sqrt{3}$ | A1 | |
## Question 4(ii):
| Answer | Mark | Notes |
|--------|------|-------|
| Carry out an appropriate method for determining the nature of a stationary point | M1 | |
| Show the point is a minimum point with no errors seen | A1 | |4 The curve with equation $y = \frac { 2 - \sin x } { \cos x }$ has one stationary point in the interval $- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi$.\\
(i) Find the exact coordinates of this point.\\
(ii) Determine whether this point is a maximum or a minimum point.\\
\hfill \mbox{\textit{CAIE P3 2017 Q4 [7]}}