The total number of ordered pairs of (x,y) satisfying the equation `13+12[tan^-1x]=24[In x]+8[e^x]+6[cos^-1y]` is/are:

Options

A. 0
B. 1
C. 2
D. Infinitely many

Answer

To solve the equation \( 13 + 12[\tan^{-1}x] = 24[\ln x] + 8[e^x] + 6[\cos^{-1}y] \) for the total number of ordered pairs \((x,y)\), we will analyze the components of the equation step by step.

### Step-by-Step Solution:

1. **Understanding the Components**:
- The equation involves the greatest integer function (denoted by square brackets).
- The left-hand side (LHS) is \( 13 + 12[\tan^{-1}x] \).
- The right-hand side (RHS) is \( 24[\ln x] + 8[e^x] + 6[\cos^{-1}y] \).

2. **Identifying the Nature of the Terms**:
- The term \( 12[\tan^{-1}x] \) will yield an integer since it is multiplied by 12.
- The term \( 24[\ln x] \) will also yield an integer since it is multiplied by 24.
- The term \( 8[e^x] \) is an integer since \( e^x \) is always positive and can be treated as a real number.
- The term \( 6[\cos^{-1}y] \) will yield an integer since it is multiplied by 6.

3. **Analyzing the LHS**:
- The LHS can be expressed as \( 13 + \text{even integer} \) (since \( 12[\tan^{-1}x] \) is even).
- Since 13 is an odd number, the LHS will always be odd.

4. **Analyzing the RHS**:
- The RHS can be expressed as \( \text{even integer} + \text{even integer} + \text{even integer} \).
- The sum of even integers is always even.

5. **Conclusion on Parity**:
- The LHS is odd, and the RHS is even.
- An odd number cannot equal an even number.

6. **Determining the Number of Solutions**:
- Since the LHS cannot equal the RHS for any values of \( x \) and \( y \), there are no ordered pairs \((x,y)\) that satisfy the equation.

### Final Answer:
The total number of ordered pairs \((x,y)\) satisfying the equation is **0**.