Challenging +1.2 This is a straightforward stationary point problem requiring differentiation of a standard function (tan), setting the derivative to zero, and solving numerically. The chain rule application is routine, and the iterative/numerical solution (likely using a calculator's solver) is a standard A-level technique. It's slightly above average difficulty due to the transcendental equation requiring numerical methods rather than algebraic manipulation, but the setup is mechanical and the interval constraint simplifies the search.
3 The curve with equation
$$y = 5 x - 2 \tan 2 x$$
has exactly one stationary point in the interval \(0 \leqslant x < \frac { 1 } { 4 } \pi\).
Find the coordinates of this stationary point, giving each coordinate correct to 3 significant figures.
Differentiate to obtain first derivative of form \(k_1+k_2\sec^2 2x\)
M1
Where \(k_1k_2\neq 0\)
Obtain correct \(5-4\sec^2 2x\)
A1
Equate first derivative of form \(k_1+k_2\sec^2 2x\) to zero and obtain value for \(\cos^2 2x\)
M1
OE Or value for \(\tan^2 2x\) which must be \(>0\). \(-1\leqslant\cos 2x\leqslant 1\)
Obtain \(\cos 2x=\frac{2}{\sqrt{5}}\) or decimal equivalent
A1
Or \(\tan 2x=\frac{1}{2}\)
Obtain \(x=0.232\)
A1
AWRT
Obtain \(y=0.159\)
A1
AWRT
Total
6
## Question 3:
| Answer | Mark | Guidance |
|--------|------|----------|
| Differentiate to obtain first derivative of form $k_1+k_2\sec^2 2x$ | M1 | Where $k_1k_2\neq 0$ |
| Obtain correct $5-4\sec^2 2x$ | A1 | |
| Equate first derivative of form $k_1+k_2\sec^2 2x$ to zero and obtain value for $\cos^2 2x$ | M1 | OE Or value for $\tan^2 2x$ which must be $>0$. $-1\leqslant\cos 2x\leqslant 1$ |
| Obtain $\cos 2x=\frac{2}{\sqrt{5}}$ or decimal equivalent | A1 | Or $\tan 2x=\frac{1}{2}$ |
| Obtain $x=0.232$ | A1 | AWRT |
| Obtain $y=0.159$ | A1 | AWRT |
| **Total** | **6** | |
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3 The curve with equation
$$y = 5 x - 2 \tan 2 x$$
has exactly one stationary point in the interval $0 \leqslant x < \frac { 1 } { 4 } \pi$.\\
Find the coordinates of this stationary point, giving each coordinate correct to 3 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2021 Q3 [6]}}