Standard +0.3 This is a straightforward stationary points question requiring differentiation of standard trigonometric functions, using the chain rule for cos(2x), then solving the resulting equation numerically. While it involves multiple trig functions and requires numerical methods, it's a standard textbook exercise with no conceptual challenges—slightly easier than average due to its routine nature.
4 The equation of a curve is
$$y = 3 \cos 2 x + 7 \sin x + 2$$
Find the \(x\)-coordinates of the stationary points in the interval \(0 \leqslant x \leqslant \pi\). Give each answer correct to 3 significant figures.
Differentiate to obtain form \(a\sin 2x + b\cos x\)
M1
Obtain correct \(-6\sin 2x + 7\cos x\)
A1
Use identity \(\sin 2x = 2\sin x \cos x\)
B1
Solve equation of form \(c\sin x \cos x + d\cos x = 0\) to find at least one value of \(x\)
M1
Obtain 0.623
A1
Obtain 2.52
A1
Obtain \(1.57\) or \(\frac{1}{2}\pi\) from equation of form \(c\sin x \cos x + d\cos x = 0\)
A1
Treat answers in degrees as MR – 1 situation
[7]
Differentiate to obtain form $a\sin 2x + b\cos x$ | M1 |
Obtain correct $-6\sin 2x + 7\cos x$ | A1 |
Use identity $\sin 2x = 2\sin x \cos x$ | B1 |
Solve equation of form $c\sin x \cos x + d\cos x = 0$ to find at least one value of $x$ | M1 |
Obtain 0.623 | A1 |
Obtain 2.52 | A1 |
Obtain $1.57$ or $\frac{1}{2}\pi$ from equation of form $c\sin x \cos x + d\cos x = 0$ | A1 |
Treat answers in degrees as MR – 1 situation | | [7]
4 The equation of a curve is
$$y = 3 \cos 2 x + 7 \sin x + 2$$
Find the $x$-coordinates of the stationary points in the interval $0 \leqslant x \leqslant \pi$. Give each answer correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P3 2015 Q4 [7]}}