CAIE P1 2014 June — Question 8 8 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2014
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicDifferential equations
TypeFirst-order integration
DifficultyStandard +0.3 This is a straightforward second-order differential equation requiring two integrations with standard techniques. The minimum point condition provides both boundary conditions (dy/dx=0 and y=-10 at x=3), making it a routine exercise in integration and applying conditions. While it requires multiple steps, each is mechanical with no conceptual challenges beyond basic calculus.
Spec1.07d Second derivatives: d^2y/dx^2 notation1.07n Stationary points: find maxima, minima using derivatives1.08b Integrate x^n: where n != -1 and sums
8 The equation of a curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 1\). Given that the curve has a minimum point at \(( 3 , - 10 )\), find the coordinates of the maximum point.
\(\frac{d^2y}{dx^2} = 2x-1\)
AnswerMarks Guidance
→ \(\int \frac{dy}{dx} = x^2-x+c\)B1 Correct integration (ignore +c)
= 0 when x = 3 → \(c = -6\)M1 A1 Uses a constant of integration. co
\(x^2-x-6=0\) when x = −2 (or 3)A1 Puts dy/dx to 0
→ \(\int y = \frac{1}{3}x^3-\frac{1}{2}x^2-6x\) (+k)B1✓B1✓ ✓ first 2 terms, ✓ for cx.
= −10 when x = 3M1 Correct method for k
→ \(k = 3\frac{1}{2}\)
→ \(y = 10\frac{5}{6}\)A1 [8] Co − r 10.8 
$\frac{d^2y}{dx^2} = 2x-1$

→ $\int \frac{dy}{dx} = x^2-x+c$ | B1 | Correct integration (ignore +c)

= 0 when x = 3 → $c = -6$ | M1 A1 | Uses a constant of integration. co

$x^2-x-6=0$ when x = −2 (or 3) | A1 | Puts dy/dx to 0

→ $\int y = \frac{1}{3}x^3-\frac{1}{2}x^2-6x$ (+k) | B1✓B1✓ | ✓ first 2 terms, ✓ for cx.

= −10 when x = 3 | M1 | Correct method for k

→ $k = 3\frac{1}{2}$ | | |

→ $y = 10\frac{5}{6}$ | A1 [8] | Co − r 10.8
8 The equation of a curve is such that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 1$. Given that the curve has a minimum point at $( 3 , - 10 )$, find the coordinates of the maximum point.

\hfill \mbox{\textit{CAIE P1 2014 Q8 [8]}}
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