Number of integral values of `lambda` such that the equation `cos^(-1)x+cot^(-1)x=lambda` possesses solution is :

Options

A. 2
B. 8
C. 5
D. 10

Answer

To solve the equation \( \cos^{-1}x + \cot^{-1}x = \lambda \) for the number of integral values of \( \lambda \), we will follow these steps:

### Step 1: Determine the ranges of \( \cos^{-1}x \) and \( \cot^{-1}x \)

The function \( \cos^{-1}x \) is defined for \( x \) in the interval \([-1, 1]\) and its range is:
\[ \cos^{-1}x \in [0, \pi] \]

The function \( \cot^{-1}x \) is defined for all real numbers \( x \) and its range is:
\[ \cot^{-1}x \in (0, \pi) \]

### Step 2: Find the combined range of \( \cos^{-1}x + \cot^{-1}x \)

Now, we need to find the range of the sum \( \cos^{-1}x + \cot^{-1}x \):
- The minimum value occurs when \( x = 1 \):
\[ \cos^{-1}(1) + \cot^{-1}(1) = 0 + \frac{\pi}{4} = \frac{\pi}{4} \]

- The maximum value occurs when \( x = -1 \):
\[ \cos^{-1}(-1) + \cot^{-1}(-1) = \pi + \frac{3\pi}{4} = \frac{7\pi}{4} \]

### Step 3: Identify the effective range of \( \cos^{-1}x + \cot^{-1}x \)

Thus, the effective range of \( \cos^{-1}x + \cot^{-1}x \) is:
\[ \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \]

### Step 4: Convert the range to numerical values

Calculating the numerical values:
- \( \frac{\pi}{4} \approx 0.785 \)
- \( \frac{7\pi}{4} \approx 5.497 \)

So, the range of \( \lambda \) is approximately:
\[ [0.785, 5.497] \]

### Step 5: Determine the integral values of \( \lambda \)

The integral values within this range are:
- \( 1, 2, 3, 4, 5 \)

### Step 6: Count the integral values

Counting these values gives us:
- Total integral values = 5

### Conclusion

Thus, the number of integral values of \( \lambda \) such that the equation \( \cos^{-1}x + \cot^{-1}x = \lambda \) possesses solutions is \( 5 \).