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Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

MathematicsInverse Trigonometric FunctionsProperties of ITFMedium2 minQB

MathematicsMediumsingle choice

Passage / Comprehension

$A = (tan^{- 1} x)^{3} + (cot^{- 1} x)^{3}$ for $x > 0; B = (cos^{- 1} t)^{2} + (sin^{- 1} t)^{2}$ for $0 \leq t \leq 1/ 2$ . Use $tan^{- 1} x + cot^{- 1}$ $x = π /2$ and $sin^{- 1} t + cos^{- 1} t = π /2$ on these domains.

Which interval contains exactly all attainable values of A?

Options:

Answer:

Solution:

Put $θ = tan^{- 1} x \in (0, π /2)$ . Then $A = θ^{3} + (π /2 - θ)^{3} = π^{3} /32 + (3 π /2) (θ - π /4)^{2}$ . The minimum $π^{3} /32$ is attained at $x = 1$ ; $π^{3} /8$ is approached, not attained, as $x \to 0 +$ or $\infty$ .

Stream:JEESubject:MathematicsTopic:Inverse Trigonometric FunctionsSubtopic:Properties of ITF

⏱ 2mℹ️ Source: QB

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