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Chapter 12: Intro to 3D Geometry

NCERT Solutions for CBSE Class 10 Mathematics — 100 solved questions with detailed explanations.

100

Questions

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Solved Questions

Q1. The distance between points (1,2,3) and (4,6,3) is:

5
7
25
√34

Difficulty: Easy · Topic: Distance Formula

Solution

d = √((4-1)²+(6-2)²+(3-3)²) = √(9+16+0) = √25 = 5.

Q2. The coordinates of the midpoint of the line segment joining (2,4,6) and (8,10,12) are:

(5,7,9)
(10,14,18)
(3,3,3)
(4,6,8)

Difficulty: Easy · Topic: Midpoint

Solution

Midpoint = ((2+8)/2, (4+10)/2, (6+12)/2) = (5, 7, 9).

Q3. Distance between (0,0,0) and (1,3,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(3-0)²+(2-0)²)=3.74.

Q4. Distance between (0,0,0) and (2,3,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(3-0)²+(2-0)²)=4.12.

Q5. Distance between (0,0,0) and (3,3,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(3-0)²+(2-0)²)=4.69.

Q6. Distance between (0,0,0) and (4,3,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(3-0)²+(2-0)²)=5.39.

Q7. Distance between (0,0,0) and (5,3,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(3-0)²+(2-0)²)=6.16.

Q8. Distance between (0,0,0) and (6,3,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(3-0)²+(2-0)²)=7.0.

Q9. Distance between (0,0,1) and (1,3,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(3-0)²+(3-1)²)=3.74.

Q10. Distance between (0,0,1) and (2,3,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(3-0)²+(3-1)²)=4.12.

Q11. Distance between (0,0,1) and (3,3,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(3-0)²+(3-1)²)=4.69.

Q12. Distance between (0,0,1) and (4,3,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(3-0)²+(3-1)²)=5.39.

Q13. Distance between (0,0,1) and (5,3,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(3-0)²+(3-1)²)=6.16.

Q14. Distance between (0,0,1) and (6,3,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(3-0)²+(3-1)²)=7.0.

Q15. Distance between (0,0,2) and (1,3,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(3-0)²+(4-2)²)=3.74.

Q16. Distance between (0,0,2) and (2,3,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(3-0)²+(4-2)²)=4.12.

Q17. Distance between (0,0,2) and (3,3,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(3-0)²+(4-2)²)=4.69.

Q18. Distance between (0,0,2) and (4,3,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(3-0)²+(4-2)²)=5.39.

Q19. Distance between (0,0,2) and (5,3,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(3-0)²+(4-2)²)=6.16.

Q20. Distance between (0,0,2) and (6,3,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(3-0)²+(4-2)²)=7.0.

Q21. Distance between (0,0,3) and (1,3,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(3-0)²+(0-3)²)=4.36.

Q22. Distance between (0,0,3) and (2,3,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(3-0)²+(0-3)²)=4.69.

Q23. Distance between (0,0,3) and (3,3,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(3-0)²+(0-3)²)=5.2.

Q24. Distance between (0,0,3) and (4,3,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(3-0)²+(0-3)²)=5.83.

Q25. Distance between (0,0,3) and (5,3,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(3-0)²+(0-3)²)=6.56.

Q26. Distance between (0,0,3) and (6,3,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(3-0)²+(0-3)²)=7.35.

Q27. Distance between (0,1,0) and (1,4,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(4-1)²+(2-0)²)=3.74.

Q28. Distance between (0,1,0) and (2,4,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(4-1)²+(2-0)²)=4.12.

Q29. Distance between (0,1,0) and (3,4,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(4-1)²+(2-0)²)=4.69.

Q30. Distance between (0,1,0) and (4,4,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(4-1)²+(2-0)²)=5.39.

Q31. Distance between (0,1,0) and (5,4,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(4-1)²+(2-0)²)=6.16.

Q32. Distance between (0,1,0) and (6,4,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(4-1)²+(2-0)²)=7.0.

Q33. Distance between (0,1,1) and (1,4,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(4-1)²+(3-1)²)=3.74.

Q34. Distance between (0,1,1) and (2,4,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(4-1)²+(3-1)²)=4.12.

Q35. Distance between (0,1,1) and (3,4,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(4-1)²+(3-1)²)=4.69.

Q36. Distance between (0,1,1) and (4,4,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(4-1)²+(3-1)²)=5.39.

Q37. Distance between (0,1,1) and (5,4,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(4-1)²+(3-1)²)=6.16.

Q38. Distance between (0,1,1) and (6,4,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(4-1)²+(3-1)²)=7.0.

Q39. Distance between (0,1,2) and (1,4,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(4-1)²+(4-2)²)=3.74.

Q40. Distance between (0,1,2) and (2,4,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(4-1)²+(4-2)²)=4.12.

Q41. Distance between (0,1,2) and (3,4,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(4-1)²+(4-2)²)=4.69.

Q42. Distance between (0,1,2) and (4,4,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(4-1)²+(4-2)²)=5.39.

Q43. Distance between (0,1,2) and (5,4,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(4-1)²+(4-2)²)=6.16.

Q44. Distance between (0,1,2) and (6,4,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(4-1)²+(4-2)²)=7.0.

Q45. Distance between (0,1,3) and (1,4,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(4-1)²+(0-3)²)=4.36.

Q46. Distance between (0,1,3) and (2,4,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(4-1)²+(0-3)²)=4.69.

Q47. Distance between (0,1,3) and (3,4,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(4-1)²+(0-3)²)=5.2.

Q48. Distance between (0,1,3) and (4,4,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(4-1)²+(0-3)²)=5.83.

Q49. Distance between (0,1,3) and (5,4,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(4-1)²+(0-3)²)=6.56.

Q50. Distance between (0,1,3) and (6,4,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(4-1)²+(0-3)²)=7.35.

Q51. Distance between (0,2,0) and (1,5,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(5-2)²+(2-0)²)=3.74.

Q52. Distance between (0,2,0) and (2,5,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(5-2)²+(2-0)²)=4.12.

Q53. Distance between (0,2,0) and (3,5,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(5-2)²+(2-0)²)=4.69.

Q54. Distance between (0,2,0) and (4,5,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(5-2)²+(2-0)²)=5.39.

Q55. Distance between (0,2,0) and (5,5,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(5-2)²+(2-0)²)=6.16.

Q56. Distance between (0,2,0) and (6,5,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(5-2)²+(2-0)²)=7.0.

Q57. Distance between (0,2,1) and (1,5,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(5-2)²+(3-1)²)=3.74.

Q58. Distance between (0,2,1) and (2,5,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(5-2)²+(3-1)²)=4.12.

Q59. Distance between (0,2,1) and (3,5,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(5-2)²+(3-1)²)=4.69.

Q60. Distance between (0,2,1) and (4,5,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(5-2)²+(3-1)²)=5.39.

Q61. Distance between (0,2,1) and (5,5,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(5-2)²+(3-1)²)=6.16.

Q62. Distance between (0,2,1) and (6,5,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(5-2)²+(3-1)²)=7.0.

Q63. Distance between (0,2,2) and (1,5,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(5-2)²+(4-2)²)=3.74.

Q64. Distance between (0,2,2) and (2,5,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(5-2)²+(4-2)²)=4.12.

Q65. Distance between (0,2,2) and (3,5,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(5-2)²+(4-2)²)=4.69.

Q66. Distance between (0,2,2) and (4,5,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(5-2)²+(4-2)²)=5.39.

Q67. Distance between (0,2,2) and (5,5,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(5-2)²+(4-2)²)=6.16.

Q68. Distance between (0,2,2) and (6,5,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(5-2)²+(4-2)²)=7.0.

Q69. Distance between (0,2,3) and (1,5,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(5-2)²+(0-3)²)=4.36.

Q70. Distance between (0,2,3) and (2,5,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(5-2)²+(0-3)²)=4.69.

Q71. Distance between (0,2,3) and (3,5,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(5-2)²+(0-3)²)=5.2.

Q72. Distance between (0,2,3) and (4,5,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(5-2)²+(0-3)²)=5.83.

Q73. Distance between (0,2,3) and (5,5,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(5-2)²+(0-3)²)=6.56.

Q74. Distance between (0,2,3) and (6,5,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(5-2)²+(0-3)²)=7.35.

Q75. Distance between (0,3,0) and (1,6,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(6-3)²+(2-0)²)=3.74.

Q76. Distance between (0,3,0) and (2,6,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(6-3)²+(2-0)²)=4.12.

Q77. Distance between (0,3,0) and (3,6,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(6-3)²+(2-0)²)=4.69.

Q78. Distance between (0,3,0) and (4,6,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(6-3)²+(2-0)²)=5.39.

Q79. Distance between (0,3,0) and (5,6,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(6-3)²+(2-0)²)=6.16.

Q80. Distance between (0,3,0) and (6,6,2):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(6-3)²+(2-0)²)=7.0.

Q81. Distance between (0,3,1) and (1,6,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(6-3)²+(3-1)²)=3.74.

Q82. Distance between (0,3,1) and (2,6,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(6-3)²+(3-1)²)=4.12.

Q83. Distance between (0,3,1) and (3,6,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(6-3)²+(3-1)²)=4.69.

Q84. Distance between (0,3,1) and (4,6,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(6-3)²+(3-1)²)=5.39.

Q85. Distance between (0,3,1) and (5,6,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(6-3)²+(3-1)²)=6.16.

Q86. Distance between (0,3,1) and (6,6,3):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(6-3)²+(3-1)²)=7.0.

Q87. Distance between (0,3,2) and (1,6,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(6-3)²+(4-2)²)=3.74.

Q88. Distance between (0,3,2) and (2,6,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(6-3)²+(4-2)²)=4.12.

Q89. Distance between (0,3,2) and (3,6,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(6-3)²+(4-2)²)=4.69.

Q90. Distance between (0,3,2) and (4,6,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(6-3)²+(4-2)²)=5.39.

Q91. Distance between (0,3,2) and (5,6,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(6-3)²+(4-2)²)=6.16.

Q92. Distance between (0,3,2) and (6,6,4):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(6-3)²+(4-2)²)=7.0.

Q93. Distance between (0,3,3) and (1,6,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((1-0)²+(6-3)²+(0-3)²)=4.36.

Q94. Distance between (0,3,3) and (2,6,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((2-0)²+(6-3)²+(0-3)²)=4.69.

Q95. Distance between (0,3,3) and (3,6,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((3-0)²+(6-3)²+(0-3)²)=5.2.

Q96. Distance between (0,3,3) and (4,6,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((4-0)²+(6-3)²+(0-3)²)=5.83.

Q97. Distance between (0,3,3) and (5,6,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((5-0)²+(6-3)²+(0-3)²)=6.56.

Q98. Distance between (0,3,3) and (6,6,0):

Difficulty: Easy · Topic: 3D Distance

Solution

d=√((6-0)²+(6-3)²+(0-3)²)=7.35.

Q99. Section formula in 3D: point dividing (x₁,y₁,z₁) and (x₂,y₂,z₂) in ratio m:n is

((mx₂+nx₁)/(m+n), ...)
midpoint only
((mx₁+nx₂)/(m+n), ...)
cannot be determined

Difficulty: Easy-Medium · Topic: Intro to 3D Geometry

Solution

((mx₂+nx₁)/(m+n), ...).

Q100. The distance from origin to (a,b,c) is

√(a²+b²+c²)
(a²+b²+c²)/3
a+b+c
|a|+|b|+|c|

Difficulty: Easy-Medium · Topic: Intro to 3D Geometry

Solution

√(a²+b²+c²).

Other Chapters in Mathematics

Ch 1: Sets Ch 2: Relations & Functions Ch 3: Trigonometric Functions Ch 4: Principle of Mathematical Induction Ch 5: Complex Numbers Ch 6: Linear Inequalities Ch 7: Permutations & Combinations Ch 8: Binomial Theorem Ch 9: Sequences & Series Ch 10: Straight Lines Ch 11: Conic Sections Ch 13: Limits & Derivatives Ch 14: Mathematical Reasoning Ch 15: Statistics Ch 16: Probability

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