OCR C2 2009 June — Question 6 8 marks

Exam BoardOCR
ModuleC2 (Core Mathematics 2)
Year2009
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicApplied differentiation
TypeFind curve equation from derivative
DifficultyModerate -0.8 This is a straightforward integration problem requiring students to integrate a simple polynomial, apply two boundary conditions to find two constants (a and the constant of integration), then write the final equation. It's more routine than average A-level questions since it involves only basic integration and simultaneous equations with no conceptual challenges or problem-solving insight required.
Spec1.07b Gradient as rate of change: dy/dx notation1.08a Fundamental theorem of calculus: integration as reverse of differentiation
6 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + a\), where \(a\) is a constant. The curve passes through the points \(( - 1,2 )\) and \(( 2,17 )\). Find the equation of the curve.
Question 6:
AnswerMarks Guidance
\(\int(3x^2 + a)\,dx = x^3 + ax + c\)M1 Attempt to integrate
A1Obtain at least one correct term, allow unsimplified
A1Obtain \(x^3 + ax\)
\((-1, 2) \Rightarrow -1 - a + c = 2\)M1 Substitute at least one of \((-1, 2)\) or \((2, 17)\) into integration attempt involving \(a\) and \(c\)
\((2, 17) \Rightarrow 8 + 2a + c = 17\)A1 Obtain two correct equations, allow unsimplified
M1Attempt to eliminate one variable from two equations in \(a\) and \(c\)
\(a = 2,\ c = 5\)A1 Obtain \(a = 2\), \(c = 5\), from correct equations
Hence \(y = x^3 + 2x + 5\)A1 (8) State \(y = x^3 + 2x + 5\) 
## Question 6:

| $\int(3x^2 + a)\,dx = x^3 + ax + c$ | M1 | Attempt to integrate |
|---|---|---|
| | A1 | Obtain at least one correct term, allow unsimplified |
| | A1 | Obtain $x^3 + ax$ |
| $(-1, 2) \Rightarrow -1 - a + c = 2$ | M1 | Substitute at least one of $(-1, 2)$ or $(2, 17)$ into integration attempt involving $a$ and $c$ |
| $(2, 17) \Rightarrow 8 + 2a + c = 17$ | A1 | Obtain two correct equations, allow unsimplified |
| | M1 | Attempt to eliminate one variable from two equations in $a$ and $c$ |
| $a = 2,\ c = 5$ | A1 | Obtain $a = 2$, $c = 5$, from correct equations |
| Hence $y = x^3 + 2x + 5$ | A1 **(8)** | State $y = x^3 + 2x + 5$ |

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6 The gradient of a curve is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + a$, where $a$ is a constant. The curve passes through the points $( - 1,2 )$ and $( 2,17 )$. Find the equation of the curve.

\hfill \mbox{\textit{OCR C2 2009 Q6 [8]}}
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