| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Solve p(trig(θ)) = 0 |
| Difficulty | Moderate -0.3 This is a standard C2 polynomial question testing factor theorem, algebraic division/factorisation, and trigonometric substitution. Part (a) is routine verification, (b) requires polynomial division then factorising a quadratic, (c) is immediate from (b), and (d) applies the substitution sin θ = x with standard angle solutions. All techniques are core C2 curriculum with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks |
|---|---|
| (a) \(f(2) = 16 - 20 + 2 + 2 = 0\) \(\therefore (x - 2)\) is a factor | M1 A1 |
| Answer | Marks |
|---|---|
| \[\begin{array}{c | cc} |
| Answer | Marks | Guidance |
|---|---|---|
| \end{array}\] | M1 A1 | |
| (c) \(f(x) = (x - 2)(2x^2 - x - 1) = (x - 2)(2x + 1)(x - 1)\) | M1 A1 | |
| \(x = -\frac{1}{2}, 1, 2\) | B1 | |
| (d) \(\sin \theta = 2\) (no solutions), \(-\frac{1}{2}\) or \(1\) | ||
| \(\theta = \pi + \frac{\pi}{6}, 2\pi - \frac{\pi}{6}\) or \(\frac{\pi}{3}\) | M1 B1 | |
| \(\theta = \frac{\pi}{3}, \frac{7\pi}{6}, \frac{11\pi}{6}\) | A2 | (11) |
**(a)** $f(2) = 16 - 20 + 2 + 2 = 0$ $\therefore (x - 2)$ is a factor | M1 A1 |
**(b)** Long division shown:
$$\begin{array}{c|cc}
& 2x^2 & -x & -1 \\
\hline
x - 2 & 2x^3 & -5x^2 & +x & +2 \\
& 2x^3 & -4x^2 & & \\
\hline
& & -x^2 & +x & \\
& & -x^2 & +2x & \\
\hline
& & & -x & +2 \\
& & & -x & +2 \\
\hline
\end{array}$$ | M1 A1 |
**(c)** $f(x) = (x - 2)(2x^2 - x - 1) = (x - 2)(2x + 1)(x - 1)$ | M1 A1 |
$x = -\frac{1}{2}, 1, 2$ | B1 |
**(d)** $\sin \theta = 2$ (no solutions), $-\frac{1}{2}$ or $1$ | |
$\theta = \pi + \frac{\pi}{6}, 2\pi - \frac{\pi}{6}$ or $\frac{\pi}{3}$ | M1 B1 |
$\theta = \frac{\pi}{3}, \frac{7\pi}{6}, \frac{11\pi}{6}$ | A2 | (11)$$f(x) = 2x^3 - 5x^2 + x + 2.$$
\begin{enumerate}[label=(\alph*)]
\item Show that $(x - 2)$ is a factor of $f(x)$. [2]
\item Fully factorise $f(x)$. [4]
\item Solve the equation $f(x) = 0$. [1]
\item Find the values of $\theta$ in the interval $0 \leq \theta \leq 2\pi$ for which
$$2\sin^3 \theta - 5\sin^2 \theta + \sin \theta + 2 = 0,$$
giving your answers in terms of $\pi$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q7 [11]}}