Challenging +1.2 This question requires differentiating a product of exponential and trigonometric functions using the product rule, setting the derivative to zero, and solving a transcendental equation. While it involves multiple techniques (product rule, chain rule, trigonometric identities), the constraint that there's 'only one' stationary point provides a strong hint that guides the solution. The algebraic manipulation is moderately challenging but follows standard patterns for A-level, making this harder than average but not exceptionally difficult.
5 The curve with equation \(y = \mathrm { e } ^ { - a x } \tan x\), where \(a\) is a positive constant, has only one point in the interval \(0 < x < \frac { 1 } { 2 } \pi\) at which the tangent is parallel to the \(x\)-axis. Find the value of \(a\) and state the exact value of the \(x\)-coordinate of this point.
Equate derivative to zero, use Pythagoras and obtain a quadratic equation in \(\tan x\)
M1
Obtain \(\tan^2 x - a\tan x + 1 = 0\), or equivalent
A1
Use the condition for a quadratic to have only one root
M1
Obtain answer \(a = 2\)
A1
Obtain answer \(x = \frac{1}{4}\pi\)
A1
Total: 7
## Question 5:
| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero, use Pythagoras and obtain a quadratic equation in $\tan x$ | M1 | |
| Obtain $\tan^2 x - a\tan x + 1 = 0$, or equivalent | A1 | |
| Use the condition for a quadratic to have only one root | M1 | |
| Obtain answer $a = 2$ | A1 | |
| Obtain answer $x = \frac{1}{4}\pi$ | A1 | |
**Total: 7**
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5 The curve with equation $y = \mathrm { e } ^ { - a x } \tan x$, where $a$ is a positive constant, has only one point in the interval $0 < x < \frac { 1 } { 2 } \pi$ at which the tangent is parallel to the $x$-axis. Find the value of $a$ and state the exact value of the $x$-coordinate of this point.\\
\hfill \mbox{\textit{CAIE P3 2017 Q5 [7]}}