Moderate -0.3 This is a straightforward tangent line question requiring differentiation of standard trigonometric functions (sin x and cos 2x using chain rule), substitution of the given x-coordinate to find the gradient, then using point-slope form. While it involves multiple steps and the chain rule for cos 2x, these are routine A-level techniques with no problem-solving insight required, making it slightly easier than average.
3 The equation of a curve is
$$y = 6 \sin x - 2 \cos 2 x$$
Find the equation of the tangent to the curve at the point \(\left( \frac { 1 } { 6 } \pi , 2 \right)\). Give the answer in the form \(y = m x + c\), where the values of \(m\) and \(c\) are correct to 3 significant figures.
Differentiate to obtain form \(p \cos x + q \sin 2x\) or equivalent
M1
Obtain correct \(6\cos x + 4 \sin 2x\) or equivalent
A1
Substitute \(\frac{1}{6}\pi\) to obtain derivative equal to \(5\sqrt{3}\) or 8.66
A1
Form equation of tangent (not normal) using numerical value of gradient obtained by differentiation
M1
Obtain \(y = 8.66x - 2.53\) cao
A1
[5]
Differentiate to obtain form $p \cos x + q \sin 2x$ or equivalent | M1 |
Obtain correct $6\cos x + 4 \sin 2x$ or equivalent | A1 |
Substitute $\frac{1}{6}\pi$ to obtain derivative equal to $5\sqrt{3}$ or 8.66 | A1 |
Form equation of tangent (not normal) using numerical value of gradient obtained by differentiation | M1 |
Obtain $y = 8.66x - 2.53$ cao | A1 | [5]
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3 The equation of a curve is
$$y = 6 \sin x - 2 \cos 2 x$$
Find the equation of the tangent to the curve at the point $\left( \frac { 1 } { 6 } \pi , 2 \right)$. Give the answer in the form $y = m x + c$, where the values of $m$ and $c$ are correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P2 2015 Q3 [5]}}