First slide

Introduction to ITF

Question

If $[\cot^{- 1} x] + [\cos^{- 1} x] = 0$ where $x$ is non – negative real number and [.] denotes the greatest integer function, then complete set of value of $x$ is

Difficult

A

$(\cos 1, 1]$
B

$(C o s 1, c o t 1)$
C

$(c o t 1, 1]$
D

$(0, \cos 1)$

Solution

$0 < \cot^{- 1} x < π$ and $0 \leq \cos^{- 1} x \leq π$
$a n d [\cot^{- 1} x] + [\cos^{- 1} x] = 0 \Rightarrow [\cot^{- 1} x] = 0 a n d [\cos^{- 1} x] = 0 \Rightarrow 0 \leq \cot^{- 1} x < 1 a n d 0 \leq \cos^{- 1} x < 1$
$\Rightarrow \cot 1 < x < \infty$ and cos $1 \leq x \leq 1$
$\Rightarrow x \in (\cot 1, 1]$ $(∵ \cos 1 < \cot 1)$

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