| 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 | Moderate -0.3 This is a structured multi-part question covering standard C1 techniques: verifying a root by substitution, polynomial division, factorization, curve sketching, and solving a cubic equation. While it requires multiple steps, each part follows routine procedures with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(-2)\) used | M1 | Or M1 for division by \((x+2)\) attempted |
| \(-8 + 36 - 40 + 12 = 0\) | A1 | As far as \(x^3 + 2x^2\) then A1 for \(x^2 + 7x + 6\) with no remainder |
| Answer | Marks | Guidance |
|---|---|---|
| Division attempted as far as \(x^2 + 3x\) | M1 | Or inspection with \(b=3\) or \(c=2\) found |
| \(x^2 + 3x + 2\) or \((x+2)(x+1)\) | A1 | B2 for correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \((x+2)(x+6)(x+1)\) | 2 | Allow seen earlier; M1 for \((x+2)(x+1)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Sketch of cubic the right way up | G1 | With 2 turning pts; no 3rd tp |
| Through 12 marked on \(y\) axis | G1 | Curve must extend to \(x > 0\) |
| Intercepts \(-6, -2, -1\) on \(x\) axis | G1 | Condone no graph for \(x < -6\) |
| Answer | Marks | Guidance |
|---|---|---|
| \([x](x^2 + 9x + 20)\) | M1 | Or other partial factorisation |
| \([x](x+4)(x+5)\) | M1 | |
| \(x = 0, -4, -5\) | A1 | Or B1 for each root found e.g. using factor theorem |
## Question 3:
### Part (i)
| $f(-2)$ used | M1 | Or M1 for division by $(x+2)$ attempted |
|---|---|---|
| $-8 + 36 - 40 + 12 = 0$ | A1 | As far as $x^3 + 2x^2$ then A1 for $x^2 + 7x + 6$ with no remainder |
### Part (ii)
| Division attempted as far as $x^2 + 3x$ | M1 | Or inspection with $b=3$ or $c=2$ found |
|---|---|---|
| $x^2 + 3x + 2$ or $(x+2)(x+1)$ | A1 | B2 for correct answer |
### Part (iii)
| $(x+2)(x+6)(x+1)$ | 2 | Allow seen earlier; M1 for $(x+2)(x+1)$ |
### Part (iv)
| Sketch of cubic the right way up | G1 | With 2 turning pts; no 3rd tp |
|---|---|---|
| Through 12 marked on $y$ axis | G1 | Curve must extend to $x > 0$ |
| Intercepts $-6, -2, -1$ on $x$ axis | G1 | Condone no graph for $x < -6$ |
### Part (v)
| $[x](x^2 + 9x + 20)$ | M1 | Or other partial factorisation |
|---|---|---|
| $[x](x+4)(x+5)$ | M1 | |
| $x = 0, -4, -5$ | A1 | Or B1 for each root found e.g. using factor theorem |
---3 You are given that $\mathrm { f } ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 20 x + 12$.\\
(i) Show that $x = - 2$ is a root of $\mathrm { f } ( x ) = 0$.\\
(ii) Divide $\mathrm { f } ( x )$ by $x + 6$.\\
(iii) Express $\mathrm { f } ( x )$ in fully factorised form.\\
(iv) Sketch the graph of $y = \mathrm { f } ( x )$.\\
(v) Solve the equation $\mathrm { f } ( x ) = 12$.
\hfill \mbox{\textit{OCR MEI C1 Q3 [12]}}