| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Derive equation from calculus condition |
| Difficulty | Standard +0.8 This is a substantial multi-part question requiring sign-change verification, tangent equations, trigonometric identity manipulation (expressing a linear combination of sin/cos as a single cosine with phase shift), and finding stationary points. Part (c) requires non-trivial algebraic manipulation of trig identities, which elevates this above routine C3 questions. The question integrates multiple techniques across calculus and trigonometry, making it moderately challenging but still within standard A-level scope. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.09b Sign change methods: understand failure cases |
| Answer | Marks |
|---|---|
| \(f(0.7) = -0.25, f(0.8) = 0.23\) | M1 |
| sign change, \(f(x)\) continuous \(\therefore\) root | A1 |
| Answer | Marks |
|---|---|
| \(f'(x) = 2 + \cos x + 3\sin x\) | M1 |
| \(x=0, y=-3\), grad \(= 3\) | A1 |
| \(\therefore y = 3x - 3\) | M1 A1 |
| Answer | Marks |
|---|---|
| \(\cos x + 3\sin x = b\cos x \cos c + b\sin x \sin c\) | |
| \(b\cos c = 1, b\sin c = 3\) | |
| \(\therefore b = \sqrt{1^2+3^2} = \sqrt{10}\) | M1 |
| \(\tan c = 3, c = 1.25\) (3sf) | M1 |
| \(\therefore a = 2, b = \sqrt{10}, c = 1.25\) | A2 |
| Answer | Marks | Guidance |
|---|---|---|
| SP: \(2 + \sqrt{10}\cos(x-1.249) = 0\) | ||
| \(\cos(x-1.249) = -\frac{2}{\sqrt{10}}\) | M1 | |
| \(x - 1.249 = \pi - 0.8861, \pi + 0.8861 = 2.256, 4.028\) | M1 | |
| \(x = 3.50, 5.28\) (2dp) | A2 | (14) |
| Answer | Marks | Guidance |
|---|---|---|
| Total | (75) |
**(a)**
$f(0.7) = -0.25, f(0.8) = 0.23$ | M1 |
sign change, $f(x)$ continuous $\therefore$ root | A1 |
**(b)**
$f'(x) = 2 + \cos x + 3\sin x$ | M1 |
$x=0, y=-3$, grad $= 3$ | A1 |
$\therefore y = 3x - 3$ | M1 A1 |
**(c)**
$\cos x + 3\sin x = b\cos x \cos c + b\sin x \sin c$ | |
$b\cos c = 1, b\sin c = 3$ | |
$\therefore b = \sqrt{1^2+3^2} = \sqrt{10}$ | M1 |
$\tan c = 3, c = 1.25$ (3sf) | M1 |
$\therefore a = 2, b = \sqrt{10}, c = 1.25$ | A2 |
**(d)**
SP: $2 + \sqrt{10}\cos(x-1.249) = 0$ | |
$\cos(x-1.249) = -\frac{2}{\sqrt{10}}$ | M1 |
$x - 1.249 = \pi - 0.8861, \pi + 0.8861 = 2.256, 4.028$ | M1 |
$x = 3.50, 5.28$ (2dp) | A2 | (14)
---
**Total** | | (75)8.
$$f ( x ) = 2 x + \sin x - 3 \cos x$$
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $\mathrm { f } ( x ) = 0$ has a root in the interval [0.7, 0.8].
\item Find an equation for the tangent to the curve $y = \mathrm { f } ( x )$ at the point where it crosses the $y$-axis.
\item Find the values of the constants $a , b$ and $c$, where $b > 0$ and $0 < c < \frac { \pi } { 2 }$, such that
$$f ^ { \prime } ( x ) = a + b \cos ( x - c )$$
\item Hence find the $x$-coordinates of the stationary points of the curve $y = \mathrm { f } ( x )$ in the interval $0 \leq x \leq 2 \pi$, giving your answers to 2 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q8 [14]}}