Statement I If ` alpha , beta ` are roots of `6x^(2) + 11x + 3 = 0 ", then " cos^(-1) alpha ` exists but not `cot^(-1) beta ( alpha gt beta)`.
Statement II Domain of ` cos^(-1) x " is " [-1, 1]`.

Answer

To solve the problem, we need to analyze the two statements regarding the roots of the quadratic equation \(6x^2 + 11x + 3 = 0\).

### Step 1: Find the roots of the quadratic equation.
We start with the equation:
\[ 6x^2 + 11x + 3 = 0 \]
We can factor this equation. We look for two numbers that multiply to \(6 \cdot 3 = 18\) and add to \(11\). The numbers \(9\) and \(2\) satisfy this condition. Thus, we can rewrite the equation as:
\[ 6x^2 + 9x + 2x + 3 = 0 \]
Now, we can factor by grouping:
\[ (6x^2 + 9x) + (2x + 3) = 0 \]
Factoring out common terms:
\[ 3x(2x + 3) + 1(2x + 3) = 0 \]
This gives us:
\[ (2x + 3)(3x + 1) = 0 \]
Setting each factor to zero, we find the roots:
\[ 2x + 3 = 0 \quad \Rightarrow \quad x = -\frac{3}{2} \]
\[ 3x + 1 = 0 \quad \Rightarrow \quad x = -\frac{1}{3} \]

### Step 2: Identify alpha and beta.
According to the problem, we have:
\[ \alpha = -\frac{1}{3}, \quad \beta = -\frac{3}{2} \]
Since \(\alpha > \beta\), this assignment is correct.

### Step 3: Check if \( \cos^{-1} \alpha \) exists.
The domain of the function \( \cos^{-1} x \) is \([-1, 1]\). We check if \(\alpha = -\frac{1}{3}\) lies within this interval:
\[ -1 \leq -\frac{1}{3} \leq 1 \]
Since \(-\frac{1}{3}\) is within the domain, \( \cos^{-1} \alpha \) exists.

### Step 4: Check if \( \cot^{-1} \beta \) exists.
The function \( \cot^{-1} x \) is defined for all real numbers \(x\). However, we need to check if \(\beta = -\frac{3}{2}\) is valid for the context of the problem. Since \(\cot^{-1} x\) is defined for all \(x\), it exists. However, the statement claims it does not exist, which may be misleading.

### Conclusion:
- Statement I claims that \( \cos^{-1} \alpha \) exists (which is true) but \( \cot^{-1} \beta \) does not exist (which is false).
- Statement II correctly states that the domain of \( \cos^{-1} x \) is \([-1, 1]\).

Thus, the conclusion is that Statement I is false, while Statement II is true.