| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Sketching Polynomial Curves |
| Difficulty | Standard +0.3 This is a standard C1 polynomial question involving routine factor theorem verification, factorization, finding roots, and basic curve sketching. Part (v) requires equating polynomials and solving a quadratic—all straightforward techniques. While multi-part with several steps, each component is textbook-standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks |
|---|---|
| \(f(-3) = 0\) | B1 |
| Answer | Marks |
|---|---|
| Or: Trial and error \(\Rightarrow f(1) = 0, f(-4) = 0\) | M1, A1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Cuts axis when \(x = 1, -3, -4\) | B1 | Through points given |
| Curve shown | B1, B1 | Curve |
| Answer | Marks | Guidance |
|---|---|---|
| Points and labels shown | B1, B1 | Through points; Curves and labels |
| Answer | Marks |
|---|---|
| \(\Rightarrow A\) is when \(x = 1\); \(B\) is when \(x = -\frac{4}{3}\) | M1, A1, A1 |
**Part (i)**
$f(-3) = 0$ | B1 |
**Part (ii)**
Either: Divide out $\Rightarrow f(x) = (x + 3)(x^2 + 3x - 4) = (x + 3)(x + 4)(x - 1)$
Or: Trial and error $\Rightarrow f(1) = 0, f(-4) = 0$ | M1, A1, A1 |
**Part (iii)**
Cuts axis when $x = 1, -3, -4$ | B1 | Through points given
Curve shown | B1, B1 | Curve
**Part (iv)**
Points and labels shown | B1, B1 | Through points; Curves and labels
**Part (v)**
$x^3 + 6x^2 + 5x - 12 = x(x-1)(x-2) = x^3 - 3x^2 + 2x$
$\Rightarrow 9x^2 + 3x - 12 = 0 \Rightarrow 3x^2 + x - 4 = 0$
$\Rightarrow (x - 1)(3x + 4) = 0$
$\Rightarrow A$ is when $x = 1$; $B$ is when $x = -\frac{4}{3}$ | M1, A1, A1 |12 The function $\mathrm { f } ( x )$ is given by $\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5 x - 12$.\\
(i) Show that $( x + 3 )$ is a factor of $\mathrm { f } ( x )$.\\
(ii) Find the other factors of $\mathrm { f } ( x )$.\\
(iii) State the coordinates where the graph of $y = \mathrm { f } ( x )$ cuts the $x$ axis.
Hence sketch the graph of $y = \mathrm { f } ( x )$.\\
(iv) On the same graph sketch also $y = x ( x - 1 ) ( x - 2 )$ Label the two points of intersection of the two curves A and B .\\
(v) By equating the two curves, show that the $x$ coordinates of A and B satisfy the equation $3 x ^ { 2 } + x - 4 = 0$.\\
Solve this equation to find the $x$-coordinates of A and B .
\hfill \mbox{\textit{OCR MEI C1 Q12 [12]}}