Number of solutions of the equation
`2(sin^(-1)x^2)-sin^(-1)x-6=0` is

Options

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

Answer

To solve the equation \( 2(\sin^{-1} x^2) - \sin^{-1} x - 6 = 0 \), we will follow these steps:

### Step 1: Substitute \( y = \sin^{-1} x \)

Let \( y = \sin^{-1} x \). Then, we can express \( \sin^{-1} x^2 \) in terms of \( y \):
\[ \sin^{-1} x^2 = \sin^{-1} (\sin^2 y) = 2y \quad \text{(using the double angle identity)} \]

### Step 2: Rewrite the equation

Substituting \( y \) into the original equation gives us:
\[ 2(2y) - y - 6 = 0 \]
This simplifies to:
\[ 4y - y - 6 = 0 \]
or
\[ 3y - 6 = 0 \]

### Step 3: Solve for \( y \)

Now, we can solve for \( y \):
\[ 3y = 6 \implies y = 2 \]

### Step 4: Find \( x \)

Since \( y = \sin^{-1} x \), we have:
\[ \sin^{-1} x = 2 \]

### Step 5: Check the validity of \( y \)

The range of \( \sin^{-1} x \) is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). The value \( 2 \) is outside this range, which means there are no valid solutions for \( x \).

### Conclusion

Thus, the number of solutions to the equation \( 2(\sin^{-1} x^2) - \sin^{-1} x - 6 = 0 \) is:
\[ \boxed{0} \]