CAIE P1 2019 November — Question 3 4 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2019
SessionNovember
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCurve Sketching
TypeDomain restrictions from monotonicity
DifficultyStandard +0.8 This question requires finding stationary points by differentiation, solving a quadratic inequality to determine where the derivative is non-zero, and interpreting the result to find the boundary values. It combines calculus with algebraic manipulation and requires understanding of monotonicity, making it moderately challenging but still within standard A-level scope.
Spec1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives
3 The equation of a curve is \(y = x ^ { 3 } + x ^ { 2 } - 8 x + 7\). The curve has no stationary points in the interval \(a < x < b\). Find the least possible value of \(a\) and the greatest possible value of \(b\).
Question 3:
AnswerMarks Guidance
\(\frac{dy}{dx} = 3x^2 + 2x - 8\)B1
Set to zero (SOI) and solveM1
(Min) \(a = -2\), (Max) \(b = \frac{4}{3}\) — in terms of \(a\) and \(b\)A1 Accept \(a \geqslant -2\), \(b \leqslant \frac{4}{3}\)
A1SC: A1 for \(a > -2\), \(b < \frac{4}{3}\) or for \(-2 < x < \frac{4}{3}\) 
## Question 3:

$\frac{dy}{dx} = 3x^2 + 2x - 8$ | **B1** |

Set to zero (SOI) and solve | **M1** |

(Min) $a = -2$, (Max) $b = \frac{4}{3}$ — in terms of $a$ and $b$ | **A1** | Accept $a \geqslant -2$, $b \leqslant \frac{4}{3}$
| **A1** | **SC:** A1 for $a > -2$, $b < \frac{4}{3}$ or for $-2 < x < \frac{4}{3}$

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3 The equation of a curve is $y = x ^ { 3 } + x ^ { 2 } - 8 x + 7$. The curve has no stationary points in the interval $a < x < b$. Find the least possible value of $a$ and the greatest possible value of $b$.\\

\hfill \mbox{\textit{CAIE P1 2019 Q3 [4]}}
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Q1 4 Q2 5 Q3 4 Q4 5 Q5 7 Q6 7 Q7 7 Q8 8 Q9 8 Q10 9