If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7),…

The correct answer is Option (4) → \( \pi \)

Note: The given answer is as per the official NTA Answer Key. However, the answer given by NTA appears to be incorrect, and the correct answer should be 0, which is not among the options, as explained below.

The direction ratios of the two lines are found from the coordinates of their end points:

    AB: \( (4-1,\ 5-2,\ 7-3) = (3,\ 3,\ 4) \)
CD: \( (2+4,\ 9-3,\ 2+6) = (6,\ 6,\ 8) = 2(3,\ 3,\ 4) \)

The direction ratios are proportional, so AB and CD are parallel. For the angle θ between the lines,

    \( \cos\theta = \displaystyle\frac{|3(6)+3(6)+4(8)|}{\sqrt{34}\,\sqrt{136}} = \displaystyle\frac{68}{68} = 1, \quad \theta = 0 \)

Option (1) \( \displaystyle\frac{\pi}{3} \) — Incorrect. That would need \( \cos\theta = \frac{1}{2} \), but \( \cos\theta = 1 \).

Option (2) \( \displaystyle\frac{\pi}{4} \) — Incorrect. That would need \( \cos\theta = \frac{1}{\sqrt2} \), but \( \cos\theta = 1 \).

Option (3) \( \displaystyle\frac{\pi}{6} \) — Incorrect. That would need \( \cos\theta = \frac{\sqrt3}{2} \), but \( \cos\theta = 1 \).

Option (4) \( \pi \) — The answer given by NTA, but it seems to be incorrect. The lines are parallel, so the angle between them is 0. The angle between two lines is never taken as π.

Advice for the exam: In the real exam, mark the answer that is mathematically correct. If NTA’s provisional answer key marks a different option, challenge it during the answer-key challenge window, giving proper reasoning. NTA does revise its key when a challenge is well justified, so a sound challenge can raise your marks.