The number of real values of x satisfying the equation `3 sin^(-1)x +pi x-pi=0` is/are :

Options

A. 0
B. 1
C. 2
D. -3

Answer

To solve the equation \( 3 \sin^{-1} x + \pi x - \pi = 0 \) for the number of real values of \( x \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ 3 \sin^{-1} x + \pi x - \pi = 0 \]
Rearranging gives:
\[ 3 \sin^{-1} x = \pi - \pi x \]

### Step 2: Isolate \( \sin^{-1} x \)
Now, we can isolate \( \sin^{-1} x \):
\[ \sin^{-1} x = \frac{\pi - \pi x}{3} \]

### Step 3: Define the Range of \( \sin^{-1} x \)
The function \( \sin^{-1} x \) is defined for \( x \) in the interval \([-1, 1]\) and its range is \([- \frac{\pi}{2}, \frac{\pi}{2}]\).

### Step 4: Analyze the Right Side
The right side of the equation is:
\[ \frac{\pi - \pi x}{3} = \frac{\pi(1 - x)}{3} \]
This is a linear function of \( x \) with a slope of \(-\frac{\pi}{3}\) and a y-intercept of \(\frac{\pi}{3}\).

### Step 5: Determine the Intersection Points
We need to find the values of \( x \) where:
\[ \sin^{-1} x = \frac{\pi(1 - x)}{3} \]
Since \( \sin^{-1} x \) is increasing, we can analyze the graphs of \( y = \sin^{-1} x \) and \( y = \frac{\pi(1 - x)}{3} \).

### Step 6: Graphical Representation
- The graph of \( y = \sin^{-1} x \) will start from \( (-1, -\frac{\pi}{2}) \) and end at \( (1, \frac{\pi}{2}) \).
- The line \( y = \frac{\pi(1 - x)}{3} \) will intersect the y-axis at \( \frac{\pi}{3} \) when \( x = 0 \) and will decrease to \( 0 \) when \( x = 1 \).

### Step 7: Finding Intersection Points
To find the number of intersection points:
1. At \( x = 0 \):
\[ \sin^{-1}(0) = 0 \quad \text{and} \quad \frac{\pi(1 - 0)}{3} = \frac{\pi}{3} \]
(No intersection)

2. At \( x = 1 \):
\[ \sin^{-1}(1) = \frac{\pi}{2} \quad \text{and} \quad \frac{\pi(1 - 1)}{3} = 0 \]
(No intersection)

3. The line will intersect the curve somewhere between \( x = 0 \) and \( x = 1 \) since \( \sin^{-1} x \) is continuous and increasing.

### Conclusion
From the analysis, we can conclude that there is exactly **one intersection point** between the graphs of \( y = \sin^{-1} x \) and \( y = \frac{\pi(1 - x)}{3} \) in the interval \([-1, 1]\).

Thus, the number of real values of \( x \) satisfying the equation is:
\[ \boxed{1} \]