JEE (Main) Mathematics - Three Dimensional Geometry: MCQs (31-45)

31. The equation of a line passing through the point P(2, 3, 4) and having direction ratios (1, -2, 3) is:
- A) \( x - 2 = \frac{y - 3}{-2} = \frac{z - 4}{3} \)
- B) \( x + 2 = \frac{y - 3}{2} = \frac{z + 4}{3} \)
- C) \( x - 2 = \frac{y + 3}{2} = \frac{z - 4}{-3} \)
- D) \( x - 2 = \frac{y - 3}{3} = \frac{z - 4}{2} \)
Answer: A) \( x - 2 = \frac{y - 3}{-2} = \frac{z - 4}{3} \)
32. The general equation of a line in space is:
- A) \( ax + by + cz + d = 0 \)
- B) \( x = x_1 + a t, y = y_1 + b t, z = z_1 + c t \)
- C) \( a x + b y + c z + d = 0 \)
- D) \( x = a + b t, y = c + d t \)
Answer: B) \( x = x_1 + a t, y = y_1 + b t, z = z_1 + c t \)
33. The direction ratios of a line parallel to the x-axis are:
- A) (1, 0, 0)
- B) (0, 1, 0)
- C) (0, 0, 1)
- D) (1, 1, 1)
Answer: A) (1, 0, 0)
34. If two lines \( L_1: x = 2 + t, y = 3 + 2t, z = 1 + 3t \) and \( L_2: x = 4 + s, y = 5 + 2s, z = 2 + 3s \) are parallel, then:
- A) \( t = s \)
- B) \( t = -s \)
- C) Direction ratios of \( L_1 \) and \( L_2 \) are proportional
- D) Direction ratios of \( L_1 \) and \( L_2 \) are not proportional
Answer: C) Direction ratios of \( L_1 \) and \( L_2 \) are proportional
35. The shortest distance between two skew lines can be calculated using:
- A) Cross product of their direction ratios
- B) Dot product of their direction ratios
- C) Distance formula between two points
- D) Vector cross product of their direction vectors
Answer: A) Cross product of their direction ratios
36. The shortest distance between the lines \( L_1: \frac{x - 2}{3} = \frac{y - 1}{-1} = \frac{z - 4}{5} \) and \( L_2: \frac{x - 1}{2} = \frac{y - 3}{4} = \frac{z - 5}{-1} \) is given by:
- A) \( \frac{| 5 - 4 |}{\sqrt{3^2 + (-1)^2 + 5^2}} \)
- B) \( \frac{| 1 + 3 |}{\sqrt{2^2 + 4^2 + (-1)^2}} \)
- C) \( \frac{| 4 + 5 |}{\sqrt{1^2 + 4^2 + (-3)^2}} \)
- D) \( \frac{| 2 + 5 |}{\sqrt{3^2 + (-2)^2 + 4^2}} \)
Answer: B) \( \frac{| 1 + 3 |}{\sqrt{2^2 + 4^2 + (-1)^2}} \)
37. The equation of the line passing through the points (1, 2, 3) and (4, 5, 6) in symmetric form is:
- A) \( \frac{x - 1}{3} = \frac{y - 2}{3} = \frac{z - 3}{3} \)
- B) \( \frac{x - 1}{3} = \frac{y - 2}{3} = \frac{z - 3}{4} \)
- C) \( \frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z - 3}{3} \)
- D) \( \frac{x - 1}{3} = \frac{y - 2}{4} = \frac{z - 3}{5} \)
Answer: A) \( \frac{x - 1}{3} = \frac{y - 2}{3} = \frac{z - 3}{3} \)
38. Skew lines are:
- A) Lines that intersect at a point
- B) Lines that are parallel
- C) Lines that are not parallel and do not intersect
- D) Lines that lie in the same plane
Answer: C) Lines that are not parallel and do not intersect
39. The shortest distance between two skew lines is given by the magnitude of the vector:
- A) Cross product of their direction ratios
- B) Dot product of their direction ratios
- C) Normal to both lines
- D) Angle between the two lines
Answer: A) Cross product of their direction ratios
40. The angle between two skew lines is:
- A) Zero
- B) 90°
- C) The angle between the shortest distance vector and the direction vector of each line
- D) The angle between their direction ratios
Answer: C) The angle between the shortest distance vector and the direction vector of each line
41. The equation of a line in parametric form is given by:
- A) \( x = a + t, y = b + t, z = c + t \)
- B) \( x = a + bt, y = c + dt, z = e + ft \)
- C) \( x = a + bt, y = b + dt, z = c + ft \)
- D) None of the above
Answer: B) \( x = a + bt, y = c + dt, z = e + ft \)
42. The shortest distance between two skew lines is:
- A) Always zero
- B) The perpendicular distance from one line to another
- C) The distance between the parallel projections of the lines
- D) The distance between the intersection points of the lines
Answer: B) The perpendicular distance from one line to another
43. If two lines \( L_1: r = (2, 3, 4) + t(1, 1, 1) \) and \( L_2: r = (1, 0, -1) + s(2, 1, 3) \), the shortest distance between them is:
- A) 1
- B) 2
- C) \( \sqrt{3} \)
- D) \( \sqrt{5} \)
Answer: C) \( \sqrt{3} \)
44. The direction ratios of two skew lines are \( (1, 2, 3) \) and \( (2, 3, 4) \). The shortest distance between them is:
- A) 1
- B) \( \sqrt{2} \)
- C) 2
- D) \( \sqrt{3} \)
Answer: B) \( \sqrt{2} \)
45. The equation of the plane passing through the point (1, 2, 3) and perpendicular to the vector (4, -1, 2) is:
- A) \( 4x - y + 2z = 4 \)
- B) \( 4x - y + 2z = 7 \)
- C) \( 4x + y - 2z = 7 \)
- D) \( 4x + y + 2z = 7 \)
Answer: B) \( 4x - y + 2z = 7 \)

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JEE (Main) Mathematics - Three Dimensional Geometry: MCQs (31-45)

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