Standard +0.3 This is a straightforward application of the quotient rule followed by a simple trigonometric identity to show the derivative is always positive. The algebraic manipulation is routine (using 1 + cos x = 2cos²(x/2) or similar), and the conclusion follows directly without requiring sophisticated reasoning or multiple complex steps.
2 The equation of a curve is \(y = \frac { \sin x } { 1 + \cos x }\), for \(- \pi < x < \pi\). Show that the gradient of the curve is positive for all \(x\) in the given interval.
Use correct quotient or product rule; obtain correct derivative in any form
M1, A1
Use Pythagoras to simplify the derivative to \(\frac{1}{1+\cos x}\), or equivalent
A1
Justify the given statement, \(-1 < \cos x < 1\) statement, or equivalent
A1
[4]
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct quotient or product rule; obtain correct derivative in any form | M1, A1 | |
| Use Pythagoras to simplify the derivative to $\frac{1}{1+\cos x}$, or equivalent | A1 | |
| Justify the given statement, $-1 < \cos x < 1$ statement, or equivalent | A1 | [4] |
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2 The equation of a curve is $y = \frac { \sin x } { 1 + \cos x }$, for $- \pi < x < \pi$. Show that the gradient of the curve is positive for all $x$ in the given interval.
\hfill \mbox{\textit{CAIE P3 2016 Q2 [4]}}