First slide

Properties of ITF

Question

The number of solutions of
$\sin \{\sin^{- 1} (\log_{1 / 2} x)\} + 2 |\cos \{\sin^{- 1} (\frac{x}{2} - \frac{3}{2})\}| = 0$ , is

Moderate

A

1
B

2
C

3
D

none of these

Solution

The two terms on the LHS of the given equation
are meaningful, if
$\begin{array}{l} - 1 \leq \log_{1 / 2} x \leq 1 and - 1 \leq \frac{x}{2} - \frac{3}{2} \leq 1 \\ \Rightarrow 2 \geq x \geq \frac{1}{2} and 1 \leq x \leq 5 \Rightarrow 1 \leq x \leq 2 \end{array}$
Now,
$\begin{array}{l} 1 \leq x \leq 2 \\ \Rightarrow \frac{1}{2} \leq \frac{x}{2} \leq 1 \\ \Rightarrow - \frac{1}{2} \leq \frac{x}{2} - 1 \leq 0 \\ \Rightarrow - \frac{π}{6} \leq \sin^{- 1} (\frac{x}{2} - 1) \leq 0 \\ \Rightarrow \frac{\sqrt{3}}{2} \leq \cos \{\sin^{- 1} (\frac{x}{2} - 1)\} \leq 1 \\ \Rightarrow \sqrt{3} \leq 2 \cos \{\sin^{- 1} (\frac{x}{2} - 1)\} \leq 2 \\ \Rightarrow \sqrt{3} \leq 2 |\cos \{\sin^{- 1} (\frac{x}{2} - 1)\}| \leq 2 \end{array}$
Again,
$\begin{array}{l} 1 \leq x \leq 2 \\ \Rightarrow - 1 \leq \log_{1 / 2} x \leq 0 \\ \Rightarrow - \frac{π}{2} \leq \sin^{- 1} (\log_{1 / 2} x) \leq 0 \\ \Rightarrow - 1 \leq \sin \{\sin^{- 1} (\log_{1 / 2} x)\} \leq 0 \\ \Rightarrow \sqrt{3} - 1 \leq \sin \{\sin^{- 1} (\log_{1 / 2} x\} + |\cos \sin^{- 1} (\frac{x}{2} - \frac{3}{2})| \leq 2 \\ \Rightarrow \sin \{\sin^{- 1} (\log_{1 / 2} x)\} + 2 |\cos \{\sin^{- 1} (\frac{x}{2} - \frac{3}{2})\}| \end{array}$
$\neq 0$ for any $x$
Hence, the given equation has no solution.

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