If the equation `sin^(-1)(x^2+x +1)+cos^(-1)(lambda x+1)=pi/2` has exactly two solutions, then the value of `lambda` is

Options

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

Answer

To solve the equation \( \sin^{-1}(x^2 + x + 1) + \cos^{-1}(\lambda x + 1) = \frac{\pi}{2} \) and find the value of \( \lambda \) such that the equation has exactly two solutions, we can follow these steps:

### Step 1: Use the identity for inverse trigonometric functions
We know that:
\[ \sin^{-1}(a) + \cos^{-1}(a) = \frac{\pi}{2} \]
for any \( a \). Thus, we can equate:
\[ x^2 + x + 1 = \lambda x + 1 \]

### Step 2: Rearrange the equation
Rearranging the equation gives:
\[ x^2 + x + 1 - \lambda x - 1 = 0 \]
which simplifies to:
\[ x^2 + (1 - \lambda)x + 0 = 0 \]

### Step 3: Factor the equation
Factoring out \( x \):
\[ x(x + (1 - \lambda)) = 0 \]
This gives us two solutions:
\[ x = 0 \quad \text{or} \quad x = \lambda - 1 \]

### Step 4: Determine the conditions for exactly two solutions
For the equation to have exactly two solutions, the solutions must be distinct. Thus, we require:
\[ \lambda - 1 \neq 0 \implies \lambda \neq 1 \]

### Step 5: Check the range of \( x \)
Next, we need to ensure that both solutions fall within the valid range of the inverse sine function. The function \( \sin^{-1}(x^2 + x + 1) \) is defined when:
\[ -1 \leq x^2 + x + 1 \leq 1 \]

### Step 6: Analyze the inequality
1. **Lower Bound**:
\[ x^2 + x + 1 \geq -1 \implies x^2 + x + 2 \geq 0 \]
The discriminant of \( x^2 + x + 2 \) is:
\[ D = 1^2 - 4 \cdot 1 \cdot 2 = 1 - 8 = -7 < 0 \]
Thus, \( x^2 + x + 2 \) is always positive.

2. **Upper Bound**:
\[ x^2 + x + 1 \leq 1 \implies x^2 + x \leq 0 \]
Factoring gives:
\[ x(x + 1) \leq 0 \]
The critical points are \( x = 0 \) and \( x = -1 \). The solution to this inequality is:
\[ -1 \leq x \leq 0 \]

### Step 7: Find the range for \( \lambda \)
From the solutions \( x = 0 \) and \( x = \lambda - 1 \), we need:
\[ -1 \leq \lambda - 1 \leq 0 \]
Adding 1 to all parts:
\[ 0 \leq \lambda \leq 1 \]

### Conclusion
Thus, the value of \( \lambda \) can be any value in the interval \( [0, 1] \) excluding \( 1 \) (to ensure distinct solutions). Therefore, the possible values for \( \lambda \) are:
\[ \lambda = 0 \quad \text{or} \quad \lambda \text{ in } (0, 1) \]