To analyze the three statements provided in the question, we will evaluate each statement one by one.
### Statement 1:
**Statement:** If \( l, m, n \) are direction ratios of a straight line, then the minimum value of \( l^2 + m^2 + n^2 \) will be 1.
**Solution:**
1. Direction ratios \( l, m, n \) are not necessarily normalized. They can be scaled by any non-zero constant.
2. The direction cosines \( \cos \alpha, \cos \beta, \cos \gamma \) satisfy the equation \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \).
3. If we take \( l = k \cos \alpha, m = k \cos \beta, n = k \cos \gamma \) for some constant \( k \), we have:
\[
l^2 + m^2 + n^2 = k^2 (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) = k^2 \cdot 1 = k^2
\]
4. The minimum value of \( l^2 + m^2 + n^2 \) can be made arbitrarily small by choosing \( k \) close to 0. Thus, the minimum value is not necessarily 1, contradicting the statement.
**Conclusion:** Statement 1 is **False**.
### Statement 2:
**Statement:** If the line joining the origin to the point (0, -1, 9) makes angles \( \alpha, \beta, \gamma \) with the positive direction of the axes, then the value of \( \cos 2\alpha + \cos 2\beta + \cos 2\gamma \) is -1.
**Solution:**
1. The direction cosines corresponding to the point (0, -1, 9) can be calculated as follows:
\[
\cos \alpha = \frac{0}{\sqrt{0^2 + (-1)^2 + 9^2}} = 0, \quad \cos \beta = \frac{-1}{\sqrt{0^2 + (-1)^2 + 9^2}} = \frac{-1}{\sqrt{82}}, \quad \cos \gamma = \frac{9}{\sqrt{0^2 + (-1)^2 + 9^2}} = \frac{9}{\sqrt{82}}
\]
2. Now, using the identity \( \cos 2\theta = 2\cos^2 \theta - 1 \):
\[
\cos 2\alpha = 2(0^2) - 1 = -1
\]
\[
\cos 2\beta = 2\left(-\frac{1}{\sqrt{82}}\right)^2 - 1 = 2\left(\frac{1}{82}\right) - 1 = \frac{2}{82} - 1 = \frac{2 - 82}{82} = \frac{-80}{82} = -\frac{40}{41}
\]
\[
\cos 2\gamma = 2\left(\frac{9}{\sqrt{82}}\right)^2 - 1 = 2\left(\frac{81}{82}\right) - 1 = \frac{162}{82} - 1 = \frac{162 - 82}{82} = \frac{80}{82} = \frac{40}{41}
\]
3. Now, summing them up:
\[
\cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1 - \frac{40}{41} + \frac{40}{41} = -1
\]
**Conclusion:** Statement 2 is **True**.
### Statement 3:
**Statement:** The angle between two lines that have the direction ratios (1, 2, 3) and (3, -2, 1) is \( \cos^{-1}\left(\frac{1}{7}\right) \).
**Solution:**
1. Let the direction ratios of the two lines be \( (l_1, m_1, n_1) = (1, 2, 3) \) and \( (l_2, m_2, n_2) = (3, -2, 1) \).
2. The cosine of the angle \( \theta \) between the two lines is given by:
\[
\cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \sqrt{l_2^2 + m_2^2 + n_2^2}}
\]
3. Calculating the numerator:
\[
l_1 l_2 + m_1 m_2 + n_1 n_2 = (1)(3) + (2)(-2) + (3)(1) = 3 - 4 + 3 = 2
\]
4. Calculating the denominators:
\[
\sqrt{l_1^2 + m_1^2 + n_1^2} = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}
\]
\[
\sqrt{l_2^2 + m_2^2 + n_2^2} = \sqrt{3^2 + (-2)^2 + 1^2} = \sqrt{9 + 4 + 1} = \sqrt{14}
\]
5. Thus, we have:
\[
\cos \theta = \frac{2}{\sqrt{14} \cdot \sqrt{14}} = \frac{2}{14} = \frac{1}{7}
\]
**Conclusion:** Statement 3 is **True**.
### Final Conclusion:
- Statement 1: False
- Statement 2: True
- Statement 3: True
