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Mathematics: AA HL

Paper 3

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Mathematics: AA HL

Available time 20 Minute(s)

This question asks you to investigate possible applications of the curve.

Paper 3

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Available time 20 Minute(s)

01. Question

Level Medium

Maximum Mark 26

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This question asks you to investigate possible applications of the curve $y = \frac{\ln x}{x}$ .

(a). For the curve $y = \frac{\ln x}{x}$ , find the
(i) $x$ -intercept(s)

[Marks: 3]

(ii) asymptote(s)

[Marks: 3]

(iii) coordinates of the stationary point(s) and their nature

[Marks: 3]

(b). Identify the curve $y = \frac{\ln x}{x}$ .

[Marks: 3]

(c). By considering two points on the curve $y = \frac{\ln x}{x}$ , show that $π^{e} < e^{π}$ .

[Marks: 3]

(d). Given that $a^{b} = b^{a}$ where $a, b \in ℝ$ and $0 < a < b$ , find the ranges of $a$ and $b$ .

[Marks: 5]

(e). Given that $\log_{a} b = \log_{b} a$ where $a, b \in ℝ$ and $a < b$ , find the ranges of $a$ and $b$ .

[Marks: 6]

End Test