Statement 1 : If `cos^2\ pi/8` is a root of the equation `x^2+ax+b=0,` where `a,b in QQ,` then ordered pair `(a,b)` is `[-1, 1/8].` Statement 2: If `a + mb = 0` and m is irrational, then `a, b=0.`

Explanation

To solve the question step-by-step, we will analyze both statements provided and derive the necessary conclusions.

### Step 1: Analyze Statement 1
We are given that \( \cos^2 \frac{\pi}{8} \) is a root of the quadratic equation \( x^2 + ax + b = 0 \), where \( a, b \in \mathbb{Q} \).

### Step 2: Substitute the Root into the Equation
Substituting \( x = \cos^2 \frac{\pi}{8} \) into the equation gives us:
\[ \cos^4 \frac{\pi}{8} + a \cos^2 \frac{\pi}{8} + b = 0 \]

### Step 3: Use the Cosine Double Angle Identity
We can use the identity \( \cos 2\theta = 2\cos^2 \theta - 1 \) to express \( \cos^4 \frac{\pi}{8} \):
\[ \cos^2 \frac{\pi}{4} = 2\cos^2 \frac{\pi}{8} - 1 \]
Thus,
\[ \cos^4 \frac{\pi}{8} = \left(\cos^2 \frac{\pi}{8}\right)^2 = \left(\frac{1 + \cos \frac{\pi}{4}}{2}\right)^2 \]

### Step 4: Calculate \( \cos^2 \frac{\pi}{8} \)
Using the known value \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \):
\[ \cos^2 \frac{\pi}{8} = \frac{1 + \frac{\sqrt{2}}{2}}{2} = \frac{2 + \sqrt{2}}{4} \]

### Step 5: Substitute Back into the Quadratic Equation
Now substituting \( \cos^2 \frac{\pi}{8} \) back into the quadratic equation:
\[ \left(\frac{2 + \sqrt{2}}{4}\right)^2 + a \left(\frac{2 + \sqrt{2}}{4}\right) + b = 0 \]

### Step 6: Simplify the Expression
Calculating \( \left(\frac{2 + \sqrt{2}}{4}\right)^2 \):
\[ \frac{(2 + \sqrt{2})^2}{16} = \frac{4 + 4\sqrt{2} + 2}{16} = \frac{6 + 4\sqrt{2}}{16} = \frac{3 + 2\sqrt{2}}{8} \]

### Step 7: Formulate the Equation
Now we have:
\[ \frac{3 + 2\sqrt{2}}{8} + a \cdot \frac{2 + \sqrt{2}}{4} + b = 0 \]

### Step 8: Separate Rational and Irrational Parts
Since \( a \) and \( b \) are rational, we can separate the rational and irrational parts:
1. For the irrational part: \( 2a + 2 = 0 \) leading to \( a = -1 \)
2. For the rational part: \( \frac{3}{8} + b - \frac{2}{4} = 0 \) leading to \( b = \frac{1}{8} \)

### Conclusion for Statement 1
Thus, the ordered pair \( (a, b) = \left(-1, \frac{1}{8}\right) \) is correct.

### Step 9: Analyze Statement 2
Statement 2 states that if \( a + mb = 0 \) and \( m \) is irrational, then \( a = 0 \) and \( b = 0 \).

### Step 10: Solve for \( a \) and \( b \)
If \( a + mb = 0 \), we can rearrange it to:
\[ a = -mb \]
Since \( m \) is irrational, for the equation to hold true, both \( a \) and \( b \) must be zero; otherwise, if \( b \neq 0 \), \( a \) would be irrational, contradicting the assumption that \( a \) is rational.

### Conclusion for Statement 2
Thus, Statement 2 is also correct.

### Final Answer
Both statements are correct.