`lambda cos x - 3 sinx=lambda +1` is solvabel for which value of λ .

Options

A. ` lambda in [0,5] `
B. `lambda in [4,5]`
C. lambda in (-oo,4]`
D. None of these

Answer

To solve the equation \( \lambda \cos x - 3 \sin x = \lambda + 1 \) for the values of \( \lambda \) for which it is solvable, we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ \lambda \cos x - 3 \sin x = \lambda + 1 \]
Rearranging gives us:
\[ \lambda \cos x - 3 \sin x - \lambda - 1 = 0 \]
This can be rewritten as:
\[ \lambda (\cos x - 1) - 3 \sin x - 1 = 0 \]

### Step 2: Identifying the Form
We can express the equation in the form \( a \sin x + b \cos x = p \). Here, we identify:
- \( a = -3 \)
- \( b = \lambda \)
- \( p = \lambda + 1 \)

### Step 3: Finding Maximum and Minimum Values
The maximum value of \( p \) can be determined using the formula:
\[ p_{\text{max}} = \sqrt{a^2 + b^2} \]
Calculating gives:
\[ p_{\text{max}} = \sqrt{(-3)^2 + \lambda^2} = \sqrt{9 + \lambda^2} \]
The minimum value of \( p \) is:
\[ p_{\text{min}} = -\sqrt{9 + \lambda^2} \]

### Step 4: Setting Up the Inequality
For the equation to be solvable, \( p \) must lie between the maximum and minimum values:
\[ -\sqrt{9 + \lambda^2} \leq \lambda + 1 \leq \sqrt{9 + \lambda^2} \]

### Step 5: Solving the Upper Bound
Starting with the upper bound:
\[ \lambda + 1 \leq \sqrt{9 + \lambda^2} \]
Squaring both sides:
\[ (\lambda + 1)^2 \leq 9 + \lambda^2 \]
Expanding gives:
\[ \lambda^2 + 2\lambda + 1 \leq 9 + \lambda^2 \]
Cancelling \( \lambda^2 \) from both sides:
\[ 2\lambda + 1 \leq 9 \]
This simplifies to:
\[ 2\lambda \leq 8 \quad \Rightarrow \quad \lambda \leq 4 \]

### Step 6: Solving the Lower Bound
Now for the lower bound:
\[ -\sqrt{9 + \lambda^2} \leq \lambda + 1 \]
Squaring both sides (noting that we must consider the sign):
\[ 9 + \lambda^2 \geq (\lambda + 1)^2 \]
Expanding gives:
\[ 9 + \lambda^2 \geq \lambda^2 + 2\lambda + 1 \]
Cancelling \( \lambda^2 \) from both sides:
\[ 9 \geq 2\lambda + 1 \]
This simplifies to:
\[ 8 \geq 2\lambda \quad \Rightarrow \quad 4 \geq \lambda \]

### Step 7: Conclusion
From both bounds, we find that for the equation to be solvable, \( \lambda \) must satisfy:
\[ \lambda \leq 4 \]
Thus, the final solution is:
\[ \lambda \in (-\infty, 4] \]