Consider the line \(\vec r = \hat{\imath} - 2\hat{\jmath} + 4\hat{k} + \lambda(-\hat{\imath} + 2\hat{\jmath} - 4\hat{k})\)
Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | A point on the given line | (I) | \(\left(\displaystyle\frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}, \frac{-4}{\sqrt{21}}\right)\) |
| (B) | direction ratios of the line | (II) | \((4, -2, -2)\) |
| (C) | direction cosines of the line | (III) | \((1, -2, 4)\) |
| (D) | direction ratios of a line perpendicular to given line | (IV) | \((-1, 2, -4)\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
Note: The given answer is as per the official NTA Answer Key. However, more than one answer is possible: Option (3) and Option (4) are both correct, as explained below.
The line passes through the point (1, −2, 4) and is parallel to \(\vec d = -\hat{\imath} + 2\hat{\jmath} - 4\hat{k}\). Its general point is
\((1 - \lambda,\ -2 + 2\lambda,\ 4 - 4\lambda)\)
(A) A point on the given line: \(\lambda = 0\) gives (III) (1, −2, 4), and \(\lambda = 2\) gives (IV) (−1, 2, −4). Both are points on the line.
(B) direction ratios of the line: any triple proportional to \((-1, 2, -4)\). Both (IV) (−1, 2, −4) and (III) (1, −2, 4) \(= -1 \times (-1, 2, -4)\) are direction ratios of the line.
(C) direction cosines of the line: dividing by \(|\vec d| = \sqrt{1 + 4 + 16} = \sqrt{21}\) gives
\(\displaystyle \left(\frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}, \frac{-4}{\sqrt{21}}\right)\)
This is (I). (II) cannot be direction cosines, since \(4^2 + (-2)^2 + (-2)^2 = 24 \ne 1\).
(D) direction ratios of a line perpendicular to given line: (II) \((4, -2, -2)\), since
\((-1)(4) + (2)(-2) + (-4)(-2) = -4 - 4 + 8 = 0\)
So (C)-(I) and (D)-(II) are fixed, while (A) and (B) can each take (III) or (IV).
Option (1) (A)-(IV), (B)-(III), (C)-(II), (D)-(I) — Incorrect. (4, −2, −2) are not direction cosines (their squares add up to 24, not 1).
Option (2) (A)-(III), (B)-(IV), (C)-(II), (D)-(I) — Incorrect. Same reason: (C) must match (I), not (II).
Option (3) (A)-(III), (B)-(IV), (C)-(I), (D)-(II) — Correct. (1, −2, 4) is the point at \(\lambda = 0\), and (−1, 2, −4) is the direction vector as printed.
Option (4) (A)-(IV), (B)-(III), (C)-(I), (D)-(II) — Should also be correct, though not the answer given by NTA. (−1, 2, −4) is the point at \(\lambda = 2\), and (1, −2, 4) is −1 times the direction vector, so it also gives direction ratios of the line.
Advice for the exam: When more than one option is correct, mark the one that reads the line most directly, here Option (3): the point at \(\lambda = 0\) and the direction ratios exactly as printed. 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.