| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Prove root count with given polynomial |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard Factor/Remainder Theorem techniques: direct substitution for remainder, factorization given a known root, and discriminant analysis. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average but still requiring multiple techniques. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f(3) = 54 + 27 - 51 + 6\); Attempt \(f(3)\) | M1 | Allow equiv methods as long as remainder is attempted. A0 if answer subsequently stated as \(-36\) ie do not isw |
| \(= 36\); Obtain 36 | A1 | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Step | Mark | Guidance |
| State or imply that \((x-2)\) is a factor | B1 | Just stating this is enough for B1, even if not used. Could be implied by attempting division by \((x-2)\) |
| Attempt full division, or equiv, by \((x \pm 2)\) | M1 | Must be complete method – all three terms attempted. If long division must subtract lower line; if inspection then expansion must give correct first and last terms and also one of the two middle terms; if coefficient matching must be valid attempt at all 3 quadratic coeffs. Allow M1 for valid division attempt by \((x+2)\) |
| Obtain \(2x^2\) and at least one other correct term | A1 | If coeff matching then allow for stating values e.g. \(A = 2\) etc |
| Obtain \((x-2)(2x^2 + 7x - 3)\) | A1 [4] | Must be stated as a product |
| Answer | Marks | Guidance |
|---|---|---|
| Step | Mark | Guidance |
| Attempt explicit numerical calculation to find number of roots of quadratic | M1 | Could attempt discriminant (allow \(b^2 \pm 4ac\)), or could use full quadratic formula to attempt to find the roots themselves (implied by stating decimal roots); M0 for factorising unless their incorrect quotient could be factorised. M0 for '3 roots as positive discriminant' but no evidence |
| State 3 roots (\(\sqrt{}\) their quotient). Condone no explicit check for repeated roots | A1ft [2] | Sufficient working must be shown, and all values shown. Discriminant needs to be 73 (allow \(7^2 - 4(2)(-3)\)). Quadratic formula must be correct, though may not necessarily be simplified as far as \(\frac{1}{4}(-7 \pm \sqrt{73})\). Need to state no. of roots – just listing them is not enough. SR: if a conclusion is given in part (iii) then allow evidence from part (ii) e.g. finding actual roots |
## Question 5:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f(3) = 54 + 27 - 51 + 6$; Attempt $f(3)$ | M1 | Allow equiv methods as long as remainder is attempted. A0 if answer subsequently stated as $-36$ ie do not isw |
| $= 36$; Obtain 36 | A1 | |
| **[2]** | | |
## Question 5(ii):
**Answer:** $f(x) = (x-2)(2x^2 + 7x - 3)$
| Step | Mark | Guidance |
|------|------|----------|
| State or imply that $(x-2)$ is a factor | B1 | Just stating this is enough for B1, even if not used. Could be implied by attempting division by $(x-2)$ |
| Attempt full division, or equiv, by $(x \pm 2)$ | M1 | Must be complete method – all three terms attempted. If long division must subtract lower line; if inspection then expansion must give correct first and last terms and also one of the two middle terms; if coefficient matching must be valid attempt at all 3 quadratic coeffs. Allow M1 for valid division attempt by $(x+2)$ |
| Obtain $2x^2$ and at least one other correct term | A1 | If coeff matching then allow for stating values e.g. $A = 2$ etc |
| Obtain $(x-2)(2x^2 + 7x - 3)$ | A1 [4] | Must be stated as a product |
---
## Question 5(iii):
**Answer:** $b^2 - 4ac = 73 > 0$ hence 3 roots
| Step | Mark | Guidance |
|------|------|----------|
| Attempt explicit numerical calculation to find number of roots of quadratic | M1 | Could attempt discriminant (allow $b^2 \pm 4ac$), or could use full quadratic formula to attempt to find the roots themselves (implied by stating decimal roots); M0 for factorising unless their incorrect quotient could be factorised. M0 for '3 roots as positive discriminant' but no evidence |
| State 3 roots ($\sqrt{}$ their quotient). Condone no explicit check for repeated roots | A1ft [2] | Sufficient working must be shown, and all values shown. Discriminant needs to be 73 (allow $7^2 - 4(2)(-3)$). Quadratic formula must be correct, though may not necessarily be simplified as far as $\frac{1}{4}(-7 \pm \sqrt{73})$. Need to state no. of roots – just listing them is not enough. **SR:** if a conclusion is given in part (iii) then allow evidence from part (ii) e.g. finding actual roots |
---5 The cubic polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 17 x + 6$.\\
(i) Find the remainder when $\mathrm { f } ( x )$ is divided by $( x - 3 )$.\\
(ii) Given that $\mathrm { f } ( 2 ) = 0$, express $\mathrm { f } ( x )$ as the product of a linear factor and a quadratic factor.\\
(iii) Determine the number of real roots of the equation $\mathrm { f } ( x ) = 0$, giving a reason for your answer.
\hfill \mbox{\textit{OCR C2 2012 Q5 [8]}}