| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Trigonometric substitution equations |
| Difficulty | Standard +0.3 Part (i) is routine factor theorem application: verify f(1)=0, perform polynomial division to get quadratic, then solve. Part (ii) requires recognizing the substitution tan x = roots from (i), then finding angles in the given range. This is a standard C2 exercise combining polynomial factorization with trigonometric equations, slightly above average due to the substitution step and need for exact forms across multiple periods. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(1) = 1 - 1 - 3 + 3 = 0\) | B1 | Confirm \(f(1)=0\), or division with no remainder, or matching coefficients with \(R=0\) |
| \(f(x) = (x-1)(x^2-3)\) | M1, A1, A1 | Attempt complete division by \((x-1)\); obtain \(x^2+k\); obtain fully correct quotient |
| \(x^2 = 3\), \(x = \pm\sqrt{3}\) | M1, A1 6 | Attempt to solve \(x^2=3\); obtain \(x = \pm\sqrt{3}\) only |
| Answer | Marks | Guidance |
|---|---|---|
| \(\tan x = 1, \sqrt{3}, -\sqrt{3}\) | B1\(\sqrt{}\) | State or imply \(\tan x = 1\) or \(\tan x =\) at least one root from (i) |
| \(\tan x = \sqrt{3} \Rightarrow x = \frac{\pi}{3}, \frac{4\pi}{3}\) | M1 | Attempt to solve \(\tan x = k\) at least once |
| \(\tan x = -\sqrt{3} \Rightarrow x = \frac{2\pi}{3}, \frac{5\pi}{3}\) | A1 | Obtain at least 2 of \(\frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\) (allow degs/decimals) |
| \(\tan x = 1 \Rightarrow x = \frac{\pi}{4}, \frac{5\pi}{4}\) | A1, B1, B1 6 | Obtain all 4 of \(\frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\) (exact radians); obtain \(\frac{\pi}{4}\) (allow degs/decimals); obtain \(\frac{5\pi}{4}\) (exact radians only) |
| SR answer only is B1 per root, max of B4 if degs/decimals |
# Question 9:
## Part (i):
| $f(1) = 1 - 1 - 3 + 3 = 0$ | B1 | Confirm $f(1)=0$, or division with no remainder, or matching coefficients with $R=0$ |
|---|---|---|
| $f(x) = (x-1)(x^2-3)$ | M1, A1, A1 | Attempt complete division by $(x-1)$; obtain $x^2+k$; obtain fully correct quotient |
| $x^2 = 3$, $x = \pm\sqrt{3}$ | M1, A1 **6** | Attempt to solve $x^2=3$; obtain $x = \pm\sqrt{3}$ only |
## Part (ii):
| $\tan x = 1, \sqrt{3}, -\sqrt{3}$ | B1$\sqrt{}$ | State or imply $\tan x = 1$ or $\tan x =$ at least one root from (i) |
|---|---|---|
| $\tan x = \sqrt{3} \Rightarrow x = \frac{\pi}{3}, \frac{4\pi}{3}$ | M1 | Attempt to solve $\tan x = k$ at least once |
| $\tan x = -\sqrt{3} \Rightarrow x = \frac{2\pi}{3}, \frac{5\pi}{3}$ | A1 | Obtain at least 2 of $\frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}$ (allow degs/decimals) |
| $\tan x = 1 \Rightarrow x = \frac{\pi}{4}, \frac{5\pi}{4}$ | A1, B1, B1 **6** | Obtain all 4 of $\frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}$ (exact radians); obtain $\frac{\pi}{4}$ (allow degs/decimals); obtain $\frac{5\pi}{4}$ (exact radians only) |
| | | **SR** answer only is B1 per root, max of B4 if degs/decimals |9 (i) The polynomial $\mathrm { f } ( x )$ is defined by
$$\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 3 x + 3$$
Show that $x = 1$ is a root of the equation $\mathrm { f } ( x ) = 0$, and hence find the other two roots.\\
(ii) Hence solve the equation
$$\tan ^ { 3 } x - \tan ^ { 2 } x - 3 \tan x + 3 = 0$$
for $0 \leqslant x \leqslant 2 \pi$. Give each solution for $x$ in an exact form.
\hfill \mbox{\textit{OCR C2 2009 Q9 [12]}}