Moderate -0.3 This is a straightforward separable variables question requiring standard technique: separate variables, integrate both sides (using tan²x = sec²x - 1), apply initial condition, and evaluate at a specific point. The integration is routine for P3 level with no conceptual challenges, making it slightly easier than average but not trivial due to the algebraic manipulation needed.
5 The variables \(x\) and \(y\) satisfy the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - 2 y } \tan ^ { 2 } x$$
for \(0 \leqslant x < \frac { 1 } { 2 } \pi\), and it is given that \(y = 0\) when \(x = 0\). Solve the differential equation and calculate the value of \(y\) when \(x = \frac { 1 } { 4 } \pi\).
Separate variables and make reasonable attempt at integration of either integral
M1
Obtain term \(\frac{1}{2}e^{2y}\)
B1
Use Pythagoras
M1
Obtain terms \(\tan x - x\)
A1
Evaluate a constant or use \(x = 0, y = 0\) as limits in a solution containing terms \(ae^{2y}\) and \(b \tan x (ab \neq 0)\)
M1
Obtain correct solution in any form, e.g. \(\frac{1}{2}e^{2y} = \tan x - x + \frac{1}{2}\)
A1
Set \(x = \frac{1}{4}\pi\) and use correct method to solve an equation of the form \(e^{2y} = a\) or \(e^{-y} = a\), where \(a > 0\)
M1
Obtain answer \(y = 0.179\)
A1
[8]
Separate variables and make reasonable attempt at integration of either integral | M1 |
Obtain term $\frac{1}{2}e^{2y}$ | B1 |
Use Pythagoras | M1 |
Obtain terms $\tan x - x$ | A1 |
Evaluate a constant or use $x = 0, y = 0$ as limits in a solution containing terms $ae^{2y}$ and $b \tan x (ab \neq 0)$ | M1 |
Obtain correct solution in any form, e.g. $\frac{1}{2}e^{2y} = \tan x - x + \frac{1}{2}$ | A1 |
Set $x = \frac{1}{4}\pi$ and use correct method to solve an equation of the form $e^{2y} = a$ or $e^{-y} = a$, where $a > 0$ | M1 |
Obtain answer $y = 0.179$ | A1 | [8]
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5 The variables $x$ and $y$ satisfy the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - 2 y } \tan ^ { 2 } x$$
for $0 \leqslant x < \frac { 1 } { 2 } \pi$, and it is given that $y = 0$ when $x = 0$. Solve the differential equation and calculate the value of $y$ when $x = \frac { 1 } { 4 } \pi$.
\hfill \mbox{\textit{CAIE P3 2016 Q5 [8]}}