Standard +0.3 This is a straightforward stationary points question requiring differentiation of trigonometric functions, use of the double angle formula (sin 2x → 2sin x cos x), factorization to solve dy/dx = 0, and second derivative test. While it involves multiple steps, each is standard A-level technique with no novel insight required, making it slightly easier than average.
5 The equation of a curve is \(y = 2 \cos x + \sin 2 x\). Find the \(x\)-coordinates of the stationary points on the curve for which \(0 < x < \pi\), and determine the nature of each of these stationary points.
Obtain derivative \(\pm 2\sin x + k\cos 2x\) or \(\pm 2\sin x + k(\cos^2 x + \sin^2 x)\)
M1
Equate derivative to zero and use trig formula to obtain an equation involving only one trig function
M1
Obtain a correct equation of this type e.g. \(2\sin^2 x + \sin x - 1 = 0\) or \(\cos 2x = \cos\left(\frac{1}{3}\pi - x\right)\)
A1
Obtain value \(x = \frac{1}{6}\pi\) (allow \(0.524\) radians or \(30°\))
A1
Show by any method that the corresponding point is a maximum point
A1
Obtain second value \(x = \frac{5}{6}\pi\) (allow \(2.62\) radians or \(150°\)), and no others in range
A1
Determine that it corresponds to a minimum point
A1
Total: 7 marks
Obtain derivative $\pm 2\sin x + k\cos 2x$ or $\pm 2\sin x + k(\cos^2 x + \sin^2 x)$ | M1 |
Equate derivative to zero and use trig formula to obtain an equation involving only one trig function | M1 |
Obtain a correct equation of this type e.g. $2\sin^2 x + \sin x - 1 = 0$ or $\cos 2x = \cos\left(\frac{1}{3}\pi - x\right)$ | A1 |
Obtain value $x = \frac{1}{6}\pi$ (allow $0.524$ radians or $30°$) | A1 |
Show by any method that the corresponding point is a maximum point | A1 |
Obtain second value $x = \frac{5}{6}\pi$ (allow $2.62$ radians or $150°$), and no others in range | A1 |
Determine that it corresponds to a minimum point | A1 | **Total: 7 marks**
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5 The equation of a curve is $y = 2 \cos x + \sin 2 x$. Find the $x$-coordinates of the stationary points on the curve for which $0 < x < \pi$, and determine the nature of each of these stationary points.
\hfill \mbox{\textit{CAIE P3 2002 Q5 [7]}}