Standard +0.8 This requires applying the product rule to a composite function (e^(-5x) and tan²x), then the chain rule for tan²x, followed by solving a transcendental equation involving tan x. The algebraic manipulation and numerical solution of 2tan x = 5 elevates this above a routine differentiation exercise, though it remains within standard A-level scope.
8 The equation of a curve is \(y = e ^ { - 5 x } \tan ^ { 2 } x\) for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
Find the \(x\)-coordinates of the stationary points of the curve. Give your answers correct to 3 decimal places where appropriate.
Obtain \(\frac{dy}{dx} = -5e^{-5x}\tan^2 x + 2e^{-5x}\tan x\sec^2 x\)
A1
OE.
Equate *their* derivative to zero, use \(\sec^2 x = 1+\tan^2 x\) and obtain an equation in \(\tan x\)
M1
Obtain \(2\tan^2 x - 5\tan x + 2 = 0\)
A1
Allow \(2\tan^3 x - 5\tan^2 x + 2\tan x = 0\)
State answer \(x=0\)
B1
From correct derivative.
Solve a 3 term quadratic in \(\tan x\) and obtain a value of \(x\)
M1
Must be in radians
Obtain answer, e.g. \(0.464\)
A1
Must be 3 d.p. as specified in the question. Allow A1A0 if both values given to 2 d.p. or \(>\) 3 d.p.
Obtain second non-zero answer, e.g. \(1.107\) and no other in the given interval
A1
Alternative method:
Use correct product (or quotient) rule
M1
At least 3 of 4 terms correct
Obtain \(\frac{dy}{dx} = -5e^{-5x}\tan^2 x + 2e^{-5x}\tan x\sec^2 x\)
A1
OE
Equate *their* derivative to zero and obtain an equation in \(\sin x\) and \(\cos x\)
M1
Obtain \(5\cos x\sin x = 2\)
A1
Or simplified equivalent (i.e. cancelled)
State answer \(x=0\)
B1
From correct derivative.
Use double angle formula or square both sides and solve for \(x\)
M1
Or equivalent method. Must be in radians.
Obtain answer, e.g. \(0.464\)
A1
Must be 3 d.p. as specified in the question. Allow A1A0 if both values given to 2 d.p. or \(>\) 3 d.p.
Obtain second non-zero answer, e.g. \(1.107\) and no other in the given interval
A1
Total
8
## Question 8:
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct product (or quotient) rule | M1 | At least 3 of 4 terms correct |
| Obtain $\frac{dy}{dx} = -5e^{-5x}\tan^2 x + 2e^{-5x}\tan x\sec^2 x$ | A1 | OE. |
| Equate *their* derivative to zero, use $\sec^2 x = 1+\tan^2 x$ and obtain an equation in $\tan x$ | M1 | |
| Obtain $2\tan^2 x - 5\tan x + 2 = 0$ | A1 | Allow $2\tan^3 x - 5\tan^2 x + 2\tan x = 0$ |
| State answer $x=0$ | B1 | From correct derivative. |
| Solve a 3 term quadratic in $\tan x$ and obtain a value of $x$ | M1 | Must be in radians |
| Obtain answer, e.g. $0.464$ | A1 | Must be 3 d.p. as specified in the question. Allow A1A0 if both values given to 2 d.p. or $>$ 3 d.p. |
| Obtain second non-zero answer, e.g. $1.107$ and no other in the given interval | A1 | |
| **Alternative method:** | | |
| Use correct product (or quotient) rule | M1 | At least 3 of 4 terms correct |
| Obtain $\frac{dy}{dx} = -5e^{-5x}\tan^2 x + 2e^{-5x}\tan x\sec^2 x$ | A1 | OE |
| Equate *their* derivative to zero and obtain an equation in $\sin x$ and $\cos x$ | M1 | |
| Obtain $5\cos x\sin x = 2$ | A1 | Or simplified equivalent (i.e. cancelled) |
| State answer $x=0$ | B1 | From correct derivative. |
| Use double angle formula or square both sides and solve for $x$ | M1 | Or equivalent method. Must be in radians. |
| Obtain answer, e.g. $0.464$ | A1 | Must be 3 d.p. as specified in the question. Allow A1A0 if both values given to 2 d.p. or $>$ 3 d.p. |
| Obtain second non-zero answer, e.g. $1.107$ and no other in the given interval | A1 | |
| **Total** | **8** | |
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8 The equation of a curve is $y = e ^ { - 5 x } \tan ^ { 2 } x$ for $- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi$.\\
Find the $x$-coordinates of the stationary points of the curve. Give your answers correct to 3 decimal places where appropriate.\\
\hfill \mbox{\textit{CAIE P3 2021 Q8 [8]}}