Maths · Inverse trigonometrical functions and their properties
Statement I: The equation ^{3}+\left(\cos ^{-1} x\right)^{3}-a \pi^{3}=0 \) has
Statement I: The equation \( \left(\sin ^{-1} x\right)^{3}+\left(\cos ^{-1} x\right)^{3}-a \pi^{3}=0 \) has solution for all \( a \geqslant \frac{1}{32} \) Statement II : For any \( \boldsymbol{x} \boldsymbol{\epsilon} \boldsymbol{R}, \boldsymbol{s} \boldsymbol{i n}^{-1} \boldsymbol{x}+ \) \( \cos ^{-1} x=\frac{\pi}{2} \) and \( 0 \leq\left(\sin ^{-1} x-\frac{\pi}{4}\right)^{2} \leq \) \( \frac{9 \pi^{2}}{16} \)
- A. Both statements I and II are true.
- B. Both statements I and II are true but I is not an explanation of I
- C. statement l is true and statement II is false
- D. Statement I is false and statement II is true.
Step-by-step solution
Statement I: The equation reduces to (sin^{-1}x)(cos^{-1}x) = (π^2/12)(1-8a). The range of (sin^{-1}x)(cos^{-1}x) is [-π^2/2, π^2/16]. This gives a ∈ [1/32, 7/8], so the statement for all a ≥ 1/32 is false. Statement II: The identity sin^{-1}x + cos^{-1}x = π/2 holds for x in [-1,1], and the inequality holds for all x in the domain. Despite the phrase 'for any x ∈ R', it is typically interpreted as the domain, making the statement true. Thus I is false, II is true.
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