Standard +0.3 This is a straightforward application of the product rule to find dy/dx, setting it equal to zero, and solving a trigonometric equation. While it requires using a double angle formula and solving sin 3x = sin x, these are standard P3 techniques with no novel insight required. Slightly above average due to the algebraic manipulation needed, but still a routine exam question.
3 The equation of a curve is \(y = \sin x \sin 2 x\). The curve has a stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinate of this point, giving your answer correct to 3 significant figures.
Use correct double angle formulae to express derivative in terms of \(\sin x\) and \(\cos x\)
*M1
Equate derivative to zero and obtain an equation in one trig variable
DM1
Dependent on the 2 previous M marks
Obtain \(3\sin^2 x = 2\), \(3\cos^2 x = 1\) or \(\tan^2 x = 2\)
A1
OE
Solve and obtain \(x = 0.955\)
A1
3 sf only. Final answer in degrees is A0. Ignore any attempt to find the corresponding value of \(y\)
Alternative method (first three marks): Use correct double angle formula to obtain \(y = 2\cos x - 2\cos^3 x\)
*M1
or \(y = 2\sin^2 x \cos x\)
Use chain rule and/or product rule
*M1
Obtain derivative \(y' = -2\sin x + 6\sin x \cos^2 x\)
A1
\(y' = -2\sin^3 x + 4\sin x \cos^2 x\)
Alternative method (second and third M marks): Equate derivative to zero and obtain an equation in \(\tan x\) and \(\tan 2x\)
*M1
Use correct double angle formula to obtain an equation in \(\tan x\)
DM1
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use correct product rule on given expression | *M1 | |
| Obtain correct derivative in any form | A1 | e.g. $\cos x \sin 2x + 2\sin x \cos 2x$ |
| Use correct double angle formulae to express derivative in terms of $\sin x$ and $\cos x$ | *M1 | |
| Equate derivative to zero and obtain an equation in one trig variable | DM1 | Dependent on the 2 previous M marks |
| Obtain $3\sin^2 x = 2$, $3\cos^2 x = 1$ or $\tan^2 x = 2$ | A1 | OE |
| Solve and obtain $x = 0.955$ | A1 | 3 sf only. Final answer in degrees is A0. Ignore any attempt to find the corresponding value of $y$ |
| **Alternative method (first three marks):** Use correct double angle formula to obtain $y = 2\cos x - 2\cos^3 x$ | *M1 | or $y = 2\sin^2 x \cos x$ |
| Use chain rule and/or product rule | *M1 | |
| Obtain derivative $y' = -2\sin x + 6\sin x \cos^2 x$ | A1 | $y' = -2\sin^3 x + 4\sin x \cos^2 x$ |
| **Alternative method (second and third M marks):** Equate derivative to zero and obtain an equation in $\tan x$ and $\tan 2x$ | *M1 | |
| Use correct double angle formula to obtain an equation in $\tan x$ | DM1 | |
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3 The equation of a curve is $y = \sin x \sin 2 x$. The curve has a stationary point in the interval $0 < x < \frac { 1 } { 2 } \pi$. Find the $x$-coordinate of this point, giving your answer correct to 3 significant figures.\\
\hfill \mbox{\textit{CAIE P3 2022 Q3 [6]}}