| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Integration or area using factorised polynomial |
| Difficulty | Moderate -0.3 This is a straightforward multi-part C2 question combining routine algebraic factorisation (using factor theorem with a given root), standard polynomial integration, and interpretation of definite integrals with sign changes. All techniques are textbook-standard with no novel problem-solving required, making it slightly easier than average but not trivial due to the multi-step nature and the conceptual understanding needed in part (iii). |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply that \((x-2)\) is a factor | B1 | Could be stated explicitly, or implied by using it in an attempt at the quotient or a factorisation attempt. Could also give \((2-x)\) as the factor |
| Attempt complete division, or equiv | M1 | Must be dividing by \((x-2)\), or by one of the two other correct factors. Must be complete method - ie all 3 terms attempted. Long division - must subtract lower line (allow one slip). Synthetic division expects to see: \(2 \mid 1 \quad 0 \quad -19 \quad 30\) with \(1 \quad 2 \quad -15\) |
| Obtain correct quotient of \(x^2 + 2x - 15\) CWO | A1 | Or correct quotient for their factor. Implied by \(A=1, B=2, C=-15\) |
| \(f(x) = (x-2)(x+5)(x-3)\) | A1 | Must be written as product of three linear factors. Allow equiv eg \((2-x)(x+5)(3-x)\). SR: fully correct factorisation from division by \((x+5)\) or \((x-3)\) can still get full credit |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left[\frac{1}{4}x^4 - \frac{19}{2}x^2 + 30x\right]_{-5}^{3}\) | M1* | Attempt integration. Increase in power by 1 for at least 2 terms |
| Obtain correct integral | A1 | Could also have \(+c\) present; condone \(dx\) or \(\int\) still present |
| \(= 24.75-(-231.25)\) | M1d* | Attempt correct use of limits. Must be \(F(3)-F(-5)\) |
| \(= 256\) | A1 | A0 for \(256+c\). Answer only is 0/4 - need to see evidence of integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch positive cubic with 3 distinct roots | B1 | Must be a positive cubic. No need for roots to be labelled, but need one negative and two positive roots. Graph must be sketched for at least \(-5 \leq x \leq 3\) |
| Some of the area is below the \(x\)-axis which will make negative contribution to the total | B1 | Need to mention 'negative' and identify the relevant area. eg 'below \(x\)-axis' or \(2 \leq x \leq 3\) or clear shading. B0 for statements indicating that some area is ignored |
## Question 6:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply that $(x-2)$ is a factor | B1 | Could be stated explicitly, or implied by using it in an attempt at the quotient or a factorisation attempt. Could also give $(2-x)$ as the factor |
| Attempt complete division, or equiv | M1 | Must be dividing by $(x-2)$, or by one of the two other correct factors. Must be complete method - ie all 3 terms attempted. Long division - must subtract lower line (allow one slip). Synthetic division expects to see: $2 \mid 1 \quad 0 \quad -19 \quad 30$ with $1 \quad 2 \quad -15$ |
| Obtain correct quotient of $x^2 + 2x - 15$ CWO | A1 | Or correct quotient for their factor. Implied by $A=1, B=2, C=-15$ |
| $f(x) = (x-2)(x+5)(x-3)$ | A1 | Must be written as product of three linear factors. Allow equiv eg $(2-x)(x+5)(3-x)$. SR: fully correct factorisation from division by $(x+5)$ or $(x-3)$ can still get full credit |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left[\frac{1}{4}x^4 - \frac{19}{2}x^2 + 30x\right]_{-5}^{3}$ | M1* | Attempt integration. Increase in power by 1 for at least 2 terms |
| Obtain correct integral | A1 | Could also have $+c$ present; condone $dx$ or $\int$ still present |
| $= 24.75-(-231.25)$ | M1d* | Attempt correct use of limits. Must be $F(3)-F(-5)$ |
| $= 256$ | A1 | A0 for $256+c$. Answer only is 0/4 - need to see evidence of integration |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch positive cubic with 3 distinct roots | B1 | Must be a positive cubic. No need for roots to be labelled, but need one negative and two positive roots. Graph must be sketched for at least $-5 \leq x \leq 3$ |
| Some of the area is below the $x$-axis which will make negative contribution to the total | B1 | Need to mention 'negative' and identify the relevant area. eg 'below $x$-axis' or $2 \leq x \leq 3$ or clear shading. B0 for statements indicating that some area is ignored |
---6 The cubic polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = x ^ { 3 } - 19 x + 30$.\\
(i) Given that $x = 2$ is a root of the equation $\mathrm { f } ( x ) = 0$, express $\mathrm { f } ( x )$ as the product of 3 linear factors.\\
(ii) Use integration to find the exact value of $\int _ { - 5 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x$.\\
(iii) Explain with the aid of a sketch why the answer to part (ii) does not give the area enclosed by the curve $y = \mathrm { f } ( x )$ and the $x$-axis for $- 5 \leqslant x \leqslant 3$.
\hfill \mbox{\textit{OCR C2 2015 Q6 [10]}}