JEE Main - Equation of a line in space and skew lines Online Test (MCQ Quiz)

1. What is the vector form of the equation of a line passing through a point with position vector 'a' and parallel to a vector 'b'?

A) r . b = a . b B) r = a + λb C) r . a = b . a D) r = λa + b

2. What is the Cartesian form of the equation of a line passing through a point (x1, y1, z1) and direction ratios (l, m, n)?

A) x/l + y/m + z/n = 1 B) (x - x1)/l = (y - y1)/m = (z - z1)/n C) lx + my + nz = 1 D) (x + x1)/l = (y + y1)/m = (z + z1)/n

3. What is the vector form of the equation of a line passing through two points with position vectors 'a' and 'b'?

A) r = a + λ(a - b) B) r = (1 - λ)a + λb C) r = λa + (1 - λ)b D) r = a + λb

4. What is the Cartesian form of the equation of a line passing through two points (x1, y1, z1) and (x2, y2, z2)?

A) (x - x1)/(x2 - x1) = (y - y1)/(y2 - y1) = (z - z1)/(z2 - z1) B) (x - x1)/(x1 - x2) = (y - y1)/(y1 - y2) = (z - z1)/(z1 - z2) C) x - x1 = y - y1 = z - z1 D) (x - x2)/(x1 - x2) = (y - y2)/(y1 - y2) = (z - z2)/(z1 - z2)

5. Two lines are parallel if their direction vectors are proportional. What is the condition for direction vectors b1 and b2?

A) b1 . b2 = 0 B) b1 x b2 = 0 C) b1 = λb2 for some scalar λ D) b1 + b2 = 0

6. Two lines are perpendicular if the dot product of their direction vectors is zero. If direction vectors are b1 and b2, what is the condition?

A) b1 = λb2 B) b1 x b2 = 0 C) b1 . b2 = 0 D) b1 + b2 = 0

7. What is the condition for two lines in space to be coplanar?

A) They must intersect. B) They must be parallel. C) The shortest distance between them is zero. D) They must be skew.

8. What are skew lines?

A) Lines that are parallel. B) Lines that intersect. C) Lines that are not parallel and do not intersect. D) Lines that are perpendicular.

9. If two lines are given by r = a1 + λb1 and r = a2 + μb2, what is the condition for them to be parallel?

A) a1 . a2 = 0 B) b1 . b2 = 0 C) b1 x b2 = 0 D) a1 x a2 = 0

10. If two lines are given by r = a1 + λb1 and r = a2 + μb2, what is the condition for them to be perpendicular?

A) b1 x b2 = 0 B) a1 . a2 = 0 C) b1 . b2 = 0 D) a1 x a2 = 0

11. What is the shortest distance between two parallel lines r = a1 + λb and r = a2 + μb?

A) |(a2 - a1) . b| / |b| B) |(a2 - a1) x b| / |b| C) |(a1 - a2) . b| / |b| D) |(a1 - a2) x b| / |b|

12. What is the shortest distance between two skew lines r = a1 + λb1 and r = a2 + μb2?

A) |(a2 - a1) . (b1 x b2)| / |b1 x b2| B) |(a1 - a2) . (b1 x b2)| / |b1 x b2| C) |(a2 - a1) x (b1 . b2)| / |b1 x b2| D) |(a2 - a1) . (b1 + b2)| / |b1 x b2|

13. If the shortest distance between two skew lines is zero, what does it imply?

A) The lines are parallel. B) The lines are perpendicular. C) The lines are coplanar (intersecting). D) The lines are identical.

14. The equation of a line passing through (2, 3, 4) and parallel to vector 3i + 2j - 8k in vector form is:

A) r = (2i + 3j + 4k) + λ(3i + 2j - 8k) B) r = (3i + 2j - 8k) + λ(2i + 3j + 4k) C) r = (2i + 3j + 4k) - λ(3i + 2j - 8k) D) r = (2i + 3j + 4k) + λ(2i + 3j + 4k)

15. The equation of a line passing through (2, 3, 4) and parallel to vector 3i + 2j - 8k in Cartesian form is:

A) (x - 2)/3 = (y - 3)/2 = (z - 4)/-8 B) (x - 3)/2 = (y - 2)/3 = (z + 8)/4 C) (x + 2)/3 = (y + 3)/2 = (z + 4)/-8 D) x/2 = y/3 = z/4

16. Find the direction ratios of the line passing through points P(2, 4, 6) and Q(3, 5, 7).

A) 2, 4, 6 B) 3, 5, 7 C) 1, 1, 1 D) -1, -1, -1

17. Find the equation of the line passing through (1, 2, 3) and having direction ratios 4, 5, 6.

A) (x - 1)/4 = (y - 2)/5 = (z - 3)/6 B) (x + 1)/4 = (y + 2)/5 = (z + 3)/6 C) x/1 = y/2 = z/3 D) (x - 4)/1 = (y - 5)/2 = (z - 6)/3

18. What is the vector equation of the line passing through the origin and the point (2, 3, 4)?

A) r = λ(2i + 3j + 4k) B) r = (2i + 3j + 4k) + λ(2i + 3j + 4k) C) r = (2i + 3j + 4k) D) r = λ(i + j + k)

19. Find the angle between the lines with direction ratios (1, 2, 3) and (2, -1, 1).

A) cos⁻¹(3 / sqrt(14)sqrt(6)) B) cos⁻¹(1 / sqrt(14)sqrt(6)) C) cos⁻¹(0) D) cos⁻¹(1)

20. If the direction cosines of a line are l, m, n, then which of the following is true?

A) l + m + n = 1 B) l² + m² + n² = 1 C) l² - m² - n² = 0 D) l² + m² - n² = 1

21. Find the angle between the line (x-2)/3 = (y+1)/2 = (z-3)/1 and the line (x+1)/2 = (y-2)/1 = (z+2)/3.

A) cos⁻¹(13 / sqrt(14)sqrt(14)) B) cos⁻¹(10 / sqrt(14)sqrt(14)) C) cos⁻¹(0) D) cos⁻¹(1)

22. The lines (x - 1)/(-1) = (y - 2)/2 = (z - 3)/3 and (x - 1)/3 = (y - 2)/2 = (z - 3)/(-1) are:

A) Parallel B) Perpendicular C) Intersecting but not perpendicular D) Skew

23. Find the shortest distance between the lines r = i + 2j + 3k + λ(2i + 3j + 4k) and r = 2i + 3j + 4k + μ(3i + 4j + 5k).

A) 0 B) 1 / sqrt(6) C) sqrt(6) D) 1

24. The lines x = x1 + at, y = y1 + bt, z = z1 + ct and x = x2 + du, y = y2 + eu, z = z2 + fu are parallel if:

A) a/d = b/e = c/f B) ae - bd = 0 C) af - cd = 0 D) bf - ce = 0

25. For the lines to be coplanar, the scalar triple product of the vector joining any point on the first line to any point on the second line and the direction vectors of the two lines must be zero. What is this condition?

A) (a2 - a1) . (b1 x b2) = 0 B) (a1 - a2) . (b1 x b2) = 0 C) (a2 - a1) x (b1 . b2) = 0 D) (a2 - a1) . (b1 + b2) = 0

26. Find the point where the line (x-2)/3 = (y+1)/2 = (z-3)/1 intersects the xy-plane.

A) (2, -1, 3) B) (11, 5, 7) C) (5, 1, 5) D) (2, -1, 0)

27. What is the distance of the point (1, 2, 3) from the line (x-2)/3 = (y+1)/2 = (z-3)/1?

A) sqrt(38 / 14) B) sqrt(14 / 38) C) sqrt(38) D) sqrt(14)

28. The lines r = (i + j) + λ(i - j + k) and r = (2i + k) + μ(i - 2j + k) are:

A) Parallel B) Perpendicular C) Skew D) Intersecting

29. Find the shortest distance between the lines x = 1 + 2t, y = 2 + 3t, z = 3 + 4t and x = 2 + 3u, y = 4 + 4u, z = 5 + 5u.

A) sqrt(2) / sqrt(14) B) sqrt(14) / sqrt(2) C) 0 D) 1 / sqrt(14)

30. The equation of the line passing through the origin and perpendicular to the lines with direction ratios (1, 2, 3) and (2, -1, 1) is:

A) r = λ(i + 5j - 5k) B) r = λ(5i - j + 5k) C) r = λ(5i + 5j + k) D) r = λ(i - j + k)

31. Find the value of λ for which the lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-2)/λ = (y-4)/6 = (z-5)/8 are parallel.

A) 2 B) 3 C) 4 D) 1

32. If the lines (x-1)/a = (y-2)/b = (z-3)/c and (x-2)/2 = (y-3)/3 = (z-4)/4 are perpendicular, then:

A) 2a + 3b + 4c = 0 B) a + b + c = 0 C) a/2 = b/3 = c/4 D) 4a + 6b + 8c = 0

33. What is the condition for the lines x = a1 + λb1 and x = a2 + μb2 to be coplanar?

A) (a2 - a1) . (b1 x b2) = 0 B) (a1 - a2) . (b1 x b2) = 0 C) (a2 - a1) x (b1 . b2) = 0 D) b1 . b2 = 0

34. Find the shortest distance between the origin and the line (x-2)/3 = (y+1)/2 = (z-3)/1.

A) sqrt(14) B) sqrt(38) C) sqrt(14 / 38) D) sqrt(38 / 14)

35. The lines (x - x1)/l1 = (y - y1)/m1 = (z - z1)/n1 and (x - x2)/l2 = (y - y2)/m2 = (z - z2)/n2 intersect if:

A) |(x2-x1)(m1n2 - m2n1) + (y2-y1)(n1l2 - n2l1) + (z2-z1)(l1m2 - l2m1)| = 0 B) |(x2-x1)(m1n2 - m2n1) + (y2-y1)(n1l2 - n2l1) + (z2-z1)(l1m2 - l2m1)| > 0 C) l1l2 + m1m2 + n1n2 = 0 D) l1/l2 = m1/m2 = n1/n2

36. Find the foot of the perpendicular from the point (1, 2, 3) to the line (x-2)/3 = (y+1)/2 = (z-3)/1.

A) (5, 1, 5) B) (2, -1, 3) C) (11, 5, 7) D) (1, 2, 3)

37. What is the equation of the line passing through (1, 1, 1) and parallel to the line 2x - 1 = 3y - 1 = 4z - 1?

A) (x - 1)/6 = (y - 1)/4 = (z - 1)/3 B) (x - 1)/2 = (y - 1)/3 = (z - 1)/4 C) (x - 1)/12 = (y - 1)/6 = (z - 1)/4 D) (x - 1)/2 = (y - 1)/3 = (z - 1)/4

38. Two lines are given by r = a1 + λb1 and r = a2 + μb2. If a1 = i + 2j - k, b1 = 2i + j - k, a2 = 2i - j + k, b2 = i + 2j + k. Find the shortest distance.

A) sqrt(21) / 7 B) sqrt(7) / 21 C) 7 / sqrt(21) D) 21 / sqrt(7)

39. Find the equation of the plane containing the line (x-1)/2 = (y-2)/3 = (z-3)/4 and parallel to the line x/3 = y/4 = z/5.

A) x - 2y + z = 0 B) x + 2y - z = 0 C) 2x - y + z = 0 D) x - y + 2z = 0

40. The direction ratios of a line are proportional to (1, -2, 3). What are its direction cosines?

A) 1/√14, -2/√14, 3/√14 B) 1/14, -2/14, 3/14 C) √14, -2√14, 3√14 D) 1/3, -2/3, 3/3

41. If the lines (x-1)/l = (y-2)/m = (z-3)/n and (x-2)/2 = (y-3)/3 = (z-4)/4 are coplanar, then:

A) 2l + 3m + 4n = 0 B) l/2 = m/3 = n/4 C) |(2-1)(3n-4m) + (3-2)(4l-2n) + (4-3)(2m-3l)| = 0 D) l + m + n = 0

42. Find the angle between the line joining (2, 3, 4) to (3, 4, 5) and the line joining (1, 2, 3) to (2, 3, 4).

A) 0 degrees B) 30 degrees C) 60 degrees D) 90 degrees

43. The vector equation of the line passing through (1, 2, 3) and parallel to the vector 2i + 3j + 4k is:

A) r = (i + 2j + 3k) + λ(2i + 3j + 4k) B) r = (2i + 3j + 4k) + λ(i + 2j + 3k) C) r = (1 + 2λ)i + (2 + 3λ)j + (3 + 4λ)k D) r = λ(i + 2j + 3k)

44. What is the condition for the lines (x - x1)/l1 = (y - y1)/m1 = (z - z1)/n1 and (x - x2)/l2 = (y - y2)/m2 = (z - z2)/n2 to be skew?

A) l1/l2 = m1/m2 = n1/n2 B) l1l2 + m1m2 + n1n2 = 0 C) |(x2-x1)(m1n2 - m2n1) + (y2-y1)(n1l2 - n2l1) + (z2-z1)(l1m2 - l2m1)| != 0 D) |(x2-x1)(m1n2 - m2n1) + (y2-y1)(n1l2 - n2l1) + (z2-z1)(l1m2 - l2m1)| = 0

45. Find the shortest distance between the lines r = i + j + k + λ(2i - j + k) and r = 2i - j + k + μ(i + 2j - k).

A) sqrt(2) / 3 B) sqrt(3) / 2 C) 3 / sqrt(2) D) 2 / sqrt(3)

46. The lines x=t, y=2t, z=3t and x=u, y=3u, z=4u are:

A) Parallel B) Perpendicular C) Skew D) Intersecting at origin

47. Find the projection of the line segment joining (1, 2, 3) and (4, 5, 6) onto the line with direction ratios (2, 1, -2).

A) 3 B) 9 C) 6 D) 1

48. What is the equation of the line passing through (a, b, c) and parallel to the x-axis?

A) (x-a)/1 = (y-b)/0 = (z-c)/0 B) (x-a)/0 = (y-b)/1 = (z-c)/0 C) (x-a)/0 = (y-b)/0 = (z-c)/1 D) x=a, y=b, z=c

49. Find the relation between the direction cosines l, m, n of two perpendicular lines.

A) l1l2 + m1m2 + n1n2 = 0 B) l1l2 = m1m2 = n1n2 C) l1/l2 = m1/m2 = n1/n2 D) l1² + l2² = m1² + m2² = n1² + n2²

50. The lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-1)/3 = (y-2)/4 = (z-3)/5 are:

A) Parallel B) Perpendicular C) Skew D) Intersecting

Equation of a line in space and skew lines - Online Test

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