OCR H240/03 2018 December — Question 2 5 marks

Exam BoardOCR
ModuleH240/03 (Pure Mathematics and Mechanics)
Year2018
SessionDecember
Marks5
TopicTangents, normals and gradients
TypeIncreasing/decreasing intervals
DifficultyModerate -0.3 This is a straightforward calculus optimization problem requiring differentiation of a polynomial, solving a quadratic inequality, and expressing the solution in set notation. While it involves multiple steps (differentiate, set dy/dx > 0, solve quadratic inequality, test regions), each step uses standard A-level techniques with no conceptual challenges. The 5 marks reflect the working required rather than difficulty. Slightly easier than average due to the routine nature of all steps.
Spec1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation1.07i Differentiate x^n: for rational n and sums1.07o Increasing/decreasing: functions using sign of dy/dx
In this question you must show detailed reasoning. Find the values of \(x\) for which the gradient of the curve \(y = \frac{2}{3}x^3 + \frac{5}{2}x^2 - 3x + 7\) is positive. Give your answer in set notation. [5]
AnswerMarks Guidance
DR
\(\frac{dy}{dx} = 2x^2 + 5x - 3\)M1 Attempt to differentiate (all powers reduced by 1)
A1Correct differentiation of all terms
\(2x^2 + 5x - 3 > 0 \Rightarrow (2x - 1)(x + 3) > 0\)M1 Attempt to find critical values by any appropriate method (e.g. factorising, completing the square, quadratic formula)
\(x < -3 \text{ or } x > \frac{1}{2}\)M1 Choose 'outside region' for their critical values
\(\{x : x < -3\} \cup \{x : x > \frac{1}{2}\}\)A1 
**DR** | | 

$\frac{dy}{dx} = 2x^2 + 5x - 3$ | M1 | Attempt to differentiate (all powers reduced by 1)

| A1 | Correct differentiation of all terms

$2x^2 + 5x - 3 > 0 \Rightarrow (2x - 1)(x + 3) > 0$ | M1 | Attempt to find critical values by any appropriate method (e.g. factorising, completing the square, quadratic formula)

$x < -3 \text{ or } x > \frac{1}{2}$ | M1 | Choose 'outside region' for their critical values

$\{x : x < -3\} \cup \{x : x > \frac{1}{2}\}$ | A1 |
In this question you must show detailed reasoning.

Find the values of $x$ for which the gradient of the curve $y = \frac{2}{3}x^3 + \frac{5}{2}x^2 - 3x + 7$ is positive. Give your answer in set notation. [5]

\hfill \mbox{\textit{OCR H240/03 2018 Q2 [5]}}
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