STATEMENT-1 : If a line making an angle `pi//4` with x-axis, `pi//4` with y-axis then it must be perpendicular to
z-axis
and
STATEMENT-2 : If direction ratios of two lines are `l_(1), m_(1), n_(1) and l_(2), m_(2), n_(2)` then the angle between them is
given by
`theta = cos ^(-1)(l_(1)l_(2)+m_(2)m_(2)+n_(1)n_(2))`

Options

A. Statement-1 is True, Statement-2 is true, Statement- is a correct explanation for Statement -1
B. Statement-1 is True, Statement-2 is true, Statement- is NOT a correct explanation for Statement -1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we need to analyze both statements and verify their correctness step by step.

### Step 1: Analyzing Statement 1
**Statement 1**: If a line makes an angle of \(\frac{\pi}{4}\) with the x-axis and \(\frac{\pi}{4}\) with the y-axis, then it must be perpendicular to the z-axis.

1. Let the direction ratios of the line be \(L\), \(M\), and \(N\).
2. The direction cosines of the angles made with the axes are:
- \(L = \cos(\alpha)\)
- \(M = \cos(\beta)\)
- \(N = \cos(\gamma)\)
3. Given that \(\alpha = \frac{\pi}{4}\) and \(\beta = \frac{\pi}{4}\):
- \(L = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\)
- \(M = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\)
4. According to the formula for direction cosines:
\[ L^2 + M^2 + N^2 = 1 \]
5. Substituting the values of \(L\) and \(M\):
\[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + N^2 = 1 \]
\[ \frac{1}{2} + \frac{1}{2} + N^2 = 1 \]
\[ 1 + N^2 = 1 \]
\[ N^2 = 0 \implies N = 0 \]
6. Since \(N = 0\), it indicates that the line is perpendicular to the z-axis.

**Conclusion for Statement 1**: True.

### Step 2: Analyzing Statement 2
**Statement 2**: If the direction ratios of two lines are \(l_1, m_1, n_1\) and \(l_2, m_2, n_2\), then the angle \(\theta\) between them is given by:
\[ \theta = \cos^{-1}(l_1 l_2 + m_1 m_2 + n_1 n_2) \]

1. The angle between two lines can be calculated using the formula:
\[ \cos(\theta) = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \sqrt{l_2^2 + m_2^2 + n_2^2}} \]
2. Rearranging gives:
\[ \theta = \cos^{-1}\left(\frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \sqrt{l_2^2 + m_2^2 + n_2^2}}\right) \]
3. The statement provided does not include the normalization by the magnitudes of the direction ratios, thus it is not entirely accurate.

**Conclusion for Statement 2**: True, but it is incomplete as it lacks the normalization factor.

### Final Conclusion
- **Statement 1** is true.
- **Statement 2** is true but does not provide a complete explanation for the angle between two lines.

### Answer
The correct option is: Statement 1 is true, Statement 2 is true but Statement 2 is not the correct explanation of Statement 1.

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