JEE (Main) Mathematics - Three-Dimensional Geometry MCQs

1. Q46. The coordinates of a point in space are (3, 4, 5). The position vector of this point is:
   • a) \(3i + 4j + 5k\)
   • b) \(4i + 5j + 3k\)
   • c) \(5i + 3j + 4k\)
   • d) \(3i + 5j + 4k\)
   Answer: a) \(3i + 4j + 5k\)
2. Q47. The distance between the points P(1, 1, 1) and Q(4, 5, 6) is:
   • a) \( \sqrt{27} \)
   • b) \( \sqrt{30} \)
   • c) \( \sqrt{34} \)
   • d) \( \sqrt{50} \)
   Answer: c) \( \sqrt{34} \)
3. Q48. The coordinates of a point dividing the line segment joining A(1, 2, 3) and B(4, 5, 6) in the ratio 4:5 are:
   • a) \( (3.4, 4.4, 5.4) \)
   • b) \( (3.5, 4.5, 5.5) \)
   • c) \( (3.8, 4.8, 5.8) \)
   • d) \( (4.2, 5.2, 6.2) \)
   Answer: a) \( (3.4, 4.4, 5.4) \)
4. Q49. The direction cosines of a line with direction ratios (1, 2, 3) are:
   • a) \( \left( \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}} \right) \)
   • b) \( \left( \frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}}, \frac{3}{\sqrt{6}} \right) \)
   • c) \( \left( \frac{1}{\sqrt{10}}, \frac{2}{\sqrt{10}}, \frac{3}{\sqrt{10}} \right) \)
   • d) \( \left( \frac{1}{\sqrt{12}}, \frac{2}{\sqrt{12}}, \frac{3}{\sqrt{12}} \right) \)
   Answer: a) \( \left( \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}} \right) \)
5. Q50. The angle between the lines with direction ratios (1, 1, 1) and (2, 2, 2) is:
   • a) \( 0^\circ \)
   • b) \( 45^\circ \)
   • c) \( 60^\circ \)
   • d) \( 90^\circ \)
   Answer: a) \( 0^\circ \)
6. Q51. The equation of the line passing through the points (1, 2, 3) and (4, 5, 6) is:
   • a) \( \frac{x - 1}{3} = \frac{y - 2}{3} = \frac{z - 3}{3} \)
   • b) \( \frac{x - 1}{2} = \frac{y - 2}{2} = \frac{z - 3}{2} \)
   • c) \( \frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 3}{3} \)
   • d) \( \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} \)
   Answer: a) \( \frac{x - 1}{3} = \frac{y - 2}{3} = \frac{z - 3}{3} \)
7. Q52. The lines \( L_1: x = 1 + t, y = 2 + 2t, z = 3 + t \) and \( L_2: x = 3 + s, y = 4 + s, z = 5 + 2s \) are:
   • a) Parallel
   • b) Skew
   • c) Coincident
   • d) Intersecting
   Answer: b) Skew
8. Q53. The shortest distance between two skew lines is:
   • a) 1
   • b) 2
   • c) 3
   • d) \( \sqrt{3} \)
   Answer: b) 2
9. Q54. If the direction ratios of a line are \( (4, 3, 5) \), the direction cosines are:
   • a) \( \left( \frac{4}{\sqrt{50}}, \frac{3}{\sqrt{50}}, \frac{5}{\sqrt{50}} \right) \)
   • b) \( \left( \frac{4}{\sqrt{60}}, \frac{3}{\sqrt{60}}, \frac{5}{\sqrt{60}} \right) \)
   • c) \( \left( \frac{4}{\sqrt{70}}, \frac{3}{\sqrt{70}}, \frac{5}{\sqrt{70}} \right) \)
   • d) \( \left( \frac{4}{\sqrt{80}}, \frac{3}{\sqrt{80}}, \frac{5}{\sqrt{80}} \right) \)
   Answer: a) \( \left( \frac{4}{\sqrt{50}}, \frac{3}{\sqrt{50}}, \frac{5}{\sqrt{50}} \right) \)
10. Q55. The angle between the lines with direction ratios (2, 3, 1) and (1, 2, 3) is:
    • a) \( 45^\circ \)
    • b) \( 60^\circ \)
    • c) \( 90^\circ \)
    • d) \( 120^\circ \)
    Answer: b) \( 60^\circ \)
11. Q56. The equation of a line in vector form passing through the point (1, 2, 3) with direction ratios (2, 3, 1) is:
    • a) \( \mathbf{r} = (1, 2, 3) + t(2, 3, 1) \)
    • b) \( \mathbf{r} = (1, 2, 3) + t(1, 2, 3) \)
    • c) \( \mathbf{r} = (2, 3, 4) + t(2, 3, 1) \)
    • d) \( \mathbf{r} = (1, 2, 3) + t(1, 1, 1) \)
    Answer: a) \( \mathbf{r} = (1, 2, 3) + t(2, 3, 1) \)
12. Q57. The direction ratios of the line joining the points (2, 3, 4) and (5, 6, 7) are:
    • a) (3, 3, 3)
    • b) (4, 4, 4)
    • c) (1, 1, 1)
    • d) (2, 2, 2)
    Answer: a) (3, 3, 3)
13. Q58. The angle between two lines \( L_1: x = 1 + t, y = 2 + 2t, z = 3 + t \) and \( L_2: x = 3 + s, y = 4 + s, z = 5 + 2s \) is:
    • a) \( 60^\circ \)
    • b) \( 90^\circ \)
    • c) \( 45^\circ \)
    • d) \( 120^\circ \)
    Answer: a) \( 60^\circ \)
14. Q59. The shortest distance between the lines \( L_1: x = 1 + t, y = 2 + 2t, z = 3 + t \) and \( L_2: x = 3 + s, y = 4 + s, z = 5 + 2s \) is:
    • a) 1
    • b) 2
    • c) 3
    • d) \( \sqrt{5} \)
    Answer: b) 2
15. Q60. The direction ratios of a line passing through the points (1, 2, 3) and (4, 5, 6) are:
    • a) (3, 3, 3)
    • b) (2, 2, 2)
    • c) (1, 1, 1)
    • d) (4, 4, 4)
    Answer: a) (3, 3, 3)

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