| 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 with clear guidance at each step. Part (i) involves expanding brackets (routine verification) and checking the discriminant of a quadratic. Part (ii) is substitution followed by factoring a cubic with one known root. Part (iii) is standard curve sketching using previously found information. All techniques are standard C1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2x^3 + 5x^2 + 4x - 6x^2 - 15x - 12\) | 1 | for correct interim step; allow correct long division of \(f(x)\) by \((x-3)\) to obtain \(2x^2 + 5x + 4\) with no remainder |
| \(3\) is root | B1 | allow \(f(3) = 0\) shown |
| use of \(b^2 - 4ac\) | M1 | or equivalents for M1 and A1 using formula or completing square |
| \(5^2 - 4 \times 2 \times 4 = -7\) and [negative] implies no real root | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| divn of \(f(x) + 22\) by \(x - 2\) as far as \(2x^3 - 4x^2\) used | M1 | or inspection eg \((x-2)(2x^2\ldots -5)\) |
| \(2x^2 + 3x - 5\) obtained | A1 | |
| \((2x+5)(x-1)\) | M1 | attempt at factorising/quad. formula/compl. sq. |
| \(1\) and \(-2.5\) o.e. | A1+A | |
| or \(2 \times 2^3 - 2^2 - 11 \times 2 - 12\) | M1 | or equivs using \(f(x) + 22\) |
| \(16 - 4 - 22 - 12\) | A1 | |
| \(x = 1\) is a root obtained by factor thm | B1 | not just stated |
| \(x = -2.5\) obtained as root | B2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| cubic right way up | G1 | must have turning points |
| crossing \(x\) axis only once | G1 | must have max and min below \(x\) axis at intns with axes or in working (indep of cubic shape); ignore other intns |
| \((3, 0)\) and \((0, -12)\) shown | G1 |
## Question 5(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2x^3 + 5x^2 + 4x - 6x^2 - 15x - 12$ | 1 | for correct interim step; allow correct long division of $f(x)$ by $(x-3)$ to obtain $2x^2 + 5x + 4$ with no remainder |
| $3$ is root | B1 | allow $f(3) = 0$ shown |
| use of $b^2 - 4ac$ | M1 | or equivalents for M1 and A1 using formula or completing square |
| $5^2 - 4 \times 2 \times 4 = -7$ and [negative] implies no real root | A1 | |
## Question 5(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| divn of $f(x) + 22$ by $x - 2$ as far as $2x^3 - 4x^2$ used | M1 | or inspection eg $(x-2)(2x^2\ldots -5)$ |
| $2x^2 + 3x - 5$ obtained | A1 | |
| $(2x+5)(x-1)$ | M1 | attempt at factorising/quad. formula/compl. sq. |
| $1$ and $-2.5$ o.e. | A1+A | |
| **or** $2 \times 2^3 - 2^2 - 11 \times 2 - 12$ | M1 | or equivs using $f(x) + 22$ |
| $16 - 4 - 22 - 12$ | A1 | |
| $x = 1$ is a root obtained by factor thm | B1 | not just stated |
| $x = -2.5$ obtained as root | B2 | |
## Question 5(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| cubic right way up | G1 | must have turning points |
| crossing $x$ axis only once | G1 | must have max and min below $x$ axis at intns with axes or in working (indep of cubic shape); ignore other intns |
| $(3, 0)$ and $(0, -12)$ shown | G1 | |5 A cubic polynomial is given by $\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12$.\\
(i) Show that $( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12$.
Hence show that $\mathrm { f } ( x ) = 0$ has exactly one real root.\\
(ii) Show that $x = 2$ is a root of the equation $\mathrm { f } ( x ) = - 22$ and find the other roots of this equation.\\
(iii) Using the results from the previous parts, sketch the graph of $y = \mathrm { f } ( x )$.
\hfill \mbox{\textit{OCR MEI C1 Q5 [12]}}