OCR C1 2005 June — Question 6 7 marks

Exam BoardOCR
ModuleC1 (Core Mathematics 1)
Year2005
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTangents, normals and gradients
TypeFind second derivative
DifficultyEasy -1.2 This is a straightforward C1 question requiring basic algebraic expansion followed by routine differentiation using the power rule twice. All steps are mechanical with no problem-solving or conceptual insight needed—significantly easier than average A-level questions.
Spec1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums
6 Given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 3 x - 4 )\),
  1. express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\),
  2. find \(\mathrm { f } ^ { \prime } ( x )\),
  3. find \(\mathrm { f } ^ { \prime \prime } ( x )\).
Question 6(i):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\((x^2 + 2x + 1)(3x - 4)\)M1 Expand 2 brackets to give expression of form \(ax^2 + bx + c\) (\(a \neq 0, b \neq 0, c \neq 0\)) and attempt to multiply by third bracket
\(= 3x^3 + 2x^2 - 5x - 4\)A1 3 correct simplified terms
A1 3Completely correct
Question 6(ii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(9x^2 + 4x - 5\)B1ft 1 term correct
B1ft 2Completely correct (3 terms). SR: (ii) \(3ax^2 + 2bx + c\) B1
Question 6(iii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(18x + 4\)M1 Attempt to differentiate their (ii)
A1ft 2\(18x + 4\) (2 terms). SR: (iii) \(6ax + 2b\) B1 
## Question 6(i):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $(x^2 + 2x + 1)(3x - 4)$ | M1 | Expand 2 brackets to give expression of form $ax^2 + bx + c$ ($a \neq 0, b \neq 0, c \neq 0$) and attempt to multiply by third bracket |
| $= 3x^3 + 2x^2 - 5x - 4$ | A1 | 3 correct simplified terms |
| | A1 **3** | Completely correct |

## Question 6(ii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $9x^2 + 4x - 5$ | B1ft | 1 term correct |
| | B1ft **2** | Completely correct (3 terms). SR: (ii) $3ax^2 + 2bx + c$ **B1** |

## Question 6(iii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $18x + 4$ | M1 | Attempt to differentiate their (ii) |
| | A1ft **2** | $18x + 4$ (2 terms). SR: (iii) $6ax + 2b$ **B1** |

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6 Given that $\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 3 x - 4 )$,\\
(i) express $\mathrm { f } ( x )$ in the form $a x ^ { 3 } + b x ^ { 2 } + c x + d$,\\
(ii) find $\mathrm { f } ^ { \prime } ( x )$,\\
(iii) find $\mathrm { f } ^ { \prime \prime } ( x )$.

\hfill \mbox{\textit{OCR C1 2005 Q6 [7]}}
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