The number of solution of
`sin{sin^(-1)(log_(1//2)x)}+2|cos{sin^(-1)(x/2-3/2)}|=0` is

Options

A. 1
B. 2
C. 3
D. none of these

Answer

To solve the equation

\[ \sin\left(\sin^{-1}(\log_{1/2} x)\right) + 2 \left| \cos\left(\sin^{-1}\left(\frac{x}{2} - \frac{3}{2}\right)\right) \right| = 0, \]

we will follow these steps:

### Step 1: Determine the range of \(\log_{1/2} x\)

The function \(\sin^{-1}(y)\) is defined for \(y\) in the range \([-1, 1]\). Therefore, we need to find the values of \(x\) such that:

\[ -1 \leq \log_{1/2} x \leq 1. \]

### Step 2: Solve the inequalities for \(\log_{1/2} x\)

The logarithm \(\log_{1/2} x\) can be rewritten using the change of base formula:

\[ \log_{1/2} x = -\log_2 x. \]

Thus, the inequalities become:

\[ -1 \leq -\log_2 x \leq 1. \]

This simplifies to:

\[ 1 \geq \log_2 x \geq -1. \]

### Step 3: Convert logarithmic inequalities to exponential form

Converting these inequalities back to exponential form gives us:

1. From \(\log_2 x \leq 1\), we have \(x \leq 2\).
2. From \(\log_2 x \geq -1\), we have \(x \geq \frac{1}{2}\).

Thus, we have:

\[ \frac{1}{2} \leq x \leq 2. \]

### Step 4: Determine the range of \(\frac{x}{2} - \frac{3}{2}\)

Next, we analyze the expression \(\frac{x}{2} - \frac{3}{2}\):

\[ -1 \leq \frac{x}{2} - \frac{3}{2} \leq 1. \]

### Step 5: Solve the inequalities for \(\frac{x}{2} - \frac{3}{2}\)

1. From \(\frac{x}{2} - \frac{3}{2} \geq -1\):

\[ \frac{x}{2} \geq \frac{1}{2} \implies x \geq 1. \]

2. From \(\frac{x}{2} - \frac{3}{2} \leq 1\):

\[ \frac{x}{2} \leq \frac{5}{2} \implies x \leq 5. \]

### Step 6: Combine the ranges

Now we combine the ranges obtained from both parts:

1. From \(\frac{1}{2} \leq x \leq 2\).
2. From \(1 \leq x \leq 5\).

The intersection of these ranges gives:

\[ 1 \leq x \leq 2. \]

### Step 7: Analyze the original equation

Now we analyze the original equation:

\[ \sin\left(\sin^{-1}(\log_{1/2} x)\right) + 2 \left| \cos\left(\sin^{-1}\left(\frac{x}{2} - \frac{3}{2}\right)\right) \right| = 0. \]

### Step 8: Simplify the equation

Using the property of inverse sine, we have:

\[ \sin\left(\sin^{-1}(\log_{1/2} x)\right) = \log_{1/2} x. \]

Thus, the equation simplifies to:

\[ \log_{1/2} x + 2 \left| \cos\left(\sin^{-1}\left(\frac{x}{2} - \frac{3}{2}\right)\right) \right| = 0. \]

### Step 9: Analyze the cosine term

The term \(\cos\left(\sin^{-1}\left(\frac{x}{2} - \frac{3}{2}\right)\right)\) can be expressed as:

\[ \cos\left(\sin^{-1}(y)\right) = \sqrt{1 - y^2}, \]

where \(y = \frac{x}{2} - \frac{3}{2}\).

### Step 10: Find the number of solutions

The left-hand side of the equation must equal zero. Therefore, we need to check if there exists an \(x\) in the range \(1 \leq x \leq 2\) such that:

\[ \log_{1/2} x + 2\sqrt{1 - \left(\frac{x}{2} - \frac{3}{2}\right)^2} = 0. \]

After analyzing the behavior of the functions involved, we find that there is exactly one solution in the interval \(1 \leq x \leq 2\).

### Final Answer

Thus, the number of solutions to the equation is:

\[ \boxed{1}. \]