Consider two lines l_1 and l_2 with cartesian equations displaystylex…

The correct answer is Option (4) → (A) only

Direction ratios are read from the standard form \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\), in which x, y and z each have coefficient 1 in the numerators.

For \(l_1\), \(\frac{1 - y}{-2} = \frac{y - 1}{2}\), so

    \(\displaystyle l_1:\ \frac{x - 0}{2} = \frac{y - 1}{2} = \frac{z - 0}{1}\)

For \(l_2\), \(\frac{2x - 5}{16} = \frac{x - \frac{5}{2}}{8}\), so

    \(\displaystyle l_2:\ \frac{x - \frac{5}{2}}{8} = \frac{y - 2}{-1} = \frac{z - 5}{4}\)

So the direction ratios of \(l_1\) are 2, 2, 1 and those of \(l_2\) are 8, −1, 4.

Checking the statements

(A) Direction ratio of \(l_1\) are 2, 2, 1 — Correct. They match the standard form of \(l_1\).

(B) Direction cosines of \(l_1\) are \(\displaystyle\frac{2}{3},\frac{-2}{3},\frac{1}{3}\) — Incorrect. Dividing 2, 2, 1 by \(\sqrt{4 + 4 + 1} = 3\) gives \(\frac{2}{3}, \frac{2}{3}, \frac{1}{3}\); the printed set reads −2 straight from \(\frac{1 - y}{-2}\).

(C) Direction ratio of \(l_2\) are 16, -1, 4 — Incorrect. They are 8, −1, 4, and 16, −1, 4 is not proportional to them.

(D) Angle between \(l_1\) and \(l_2\) is \(\cos^{-1}\left(\displaystyle\frac{38}{3\sqrt{273}}\right)\) — Incorrect. With direction ratios 2, 2, 1 and 8, −1, 4:

    \(\displaystyle \cos\theta = \frac{|(2)(8) + (2)(-1) + (1)(4)|}{\sqrt{4 + 4 + 1}\,\sqrt{64 + 1 + 16}}\)

    \(\displaystyle = \frac{18}{3 \times 9} = \frac{2}{3}\)

So the angle is \(\cos^{-1}\left(\frac{2}{3}\right)\). The printed value comes from the wrong direction ratios 2, −2, 1 and 16, −1, 4:

    \(\displaystyle \frac{(2)(16) + (-2)(-1) + (1)(4)}{\sqrt{4 + 4 + 1}\,\sqrt{256 + 1 + 16}} = \frac{38}{3\sqrt{273}}\)