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Mathematics - Three Dimensional Geometry

Practise 176 NDA previous year questions on Three Dimensional Geometry, compiled from papers spanning 2010 to 2026. Each question is pulled directly from the actual exam paper it appeared in, so you're practising in the exact style and difficulty NDA tests.

Three Dimensional Geometry has been asked in 17 of the 21 years of NDA papers analysed (2006–2026), with 176 questions in all. 31% of them were easy, 65% moderate and 4% tough. Most of its questions came from 2014, with 18 questions that year. In the last three years (2024–2026) it had 28 questions across 5 papers, about 5.6 per paper.

The chapter's 6 topics are not weighted equally: Direction Cosines and Direction Ratios (58 questions, 33%), Plane in Space (41 questions, 23%), and Basics of point in 3 dimension (33 questions, 19%) have been asked the most. Direction Cosines and Direction Ratios has also led over the last five years, with 17 questions. Direction Cosines and Direction Ratios has produced the most tough questions (2).

  1. 1Passage: , , and are four points. Question: Consider the following statements : I. is parallel to . II. is perpendicular to . Which of the s2026 (Phase 1)Direction Cosines and Direction Ratios
  2. 2Passage: , , and are four points. Question: What is equal to ?2026 (Phase 1)Basics of point in 3 dimension
  3. 3Passage: A line passing through the point is perpendicular to the plane given by . Question: What is the equation of the line ?2026 (Phase 1)Line in Space
  4. 4Passage: A line passing through the point is perpendicular to the plane given by . Question: What are the direction ratios of a line paralle2026 (Phase 1)Plane in Space
  5. 5Passage: The equation of the sphere is . Question: What is the radius of the sphere passing through origin and concentric with the sphere ?2026 (Phase 1)Sphere
  6. 6Passage: The equation of the sphere is . Question: If the radius of the sphere is units, what is the value of ?2026 (Phase 1)Sphere
  7. 7If a line in 3 dimensions makes angles , and with the positive directions of the coordinate axes, then what is equal to?2025 (Phase 1)Direction Cosines and Direction Ratios
  8. 8, and are three vertices of a rectangle. What is the area of the rectangle?2025 (Phase 1)Basics of point in 3 dimension
  9. 9If a line where , makes an angle with the positive direction of -axis, then what is equal to?2025 (Phase 1)Direction Cosines and Direction Ratios
  10. 10What is the equation of the plane passing through the point and perpendicular to the line whose direction ratios are ?2025 (Phase 1)Plane in Space
  11. 11A line makes angles , and with the positive directions of the coordinate axes. If and , then what is equal to?2025 (Phase 1)Direction Cosines and Direction Ratios
  12. 12For the following two (02) items: A plane is parallel to the line having direction ratios and contains the line of intersection of the plane2025 (Phase 2)Plane in Space
  13. 13For the following two (02) items: A plane is parallel to the line having direction ratios and contains the line of intersection of the plane2025 (Phase 2)Plane in Space
  14. 14For the following two (02) items: Suppose is the sphere with the smallest radius that passes through the points , and . What is the radius o2025 (Phase 2)Sphere
  15. 15For the following two (02) items: Suppose is the sphere with the smallest radius that passes through the points , and . On which one of the 2025 (Phase 2)Basics of point in 3 dimension
  16. 16For the following two (02) items: Let , and are three consecutive vertices of a parallelogram . What is the fourth vertex ?2025 (Phase 2)Basics of point in 3 dimension
  17. 17If and are the end points of a diameter of a sphere , then what is equal to?2024 (Phase 1)Sphere
  18. 18If are the direction cosines of a normal to the plane , then what is the value of ?2024 (Phase 1)Direction Cosines and Direction Ratios
  19. 19A line through with direction ratios meets the plane . What is the point of intersection of line and plane?2024 (Phase 1)Line in Space
  20. 20If is the perpendicular distance from origin to the plane passing through and , then what is equal to?2024 (Phase 1)Plane in Space
  21. 21If a direction cosines of a line are connected by relation , then what is the value of ?2024 (Phase 1)Direction Cosines and Direction Ratios
  22. 22If a vector of magnitude 2 units makes an angle with with and an acute angle with , then what are the components of the vector?2024 (Phase 1)Direction Cosines and Direction Ratios
  23. 23Paragraph: Let be a sphere. Question: What is the diameter of the sphere?2024 (Phase 2)Sphere
  24. 24Paragraph: Let be a sphere. Question: The centre of the sphere lies on the plane2024 (Phase 2)Basics of point in 3 dimension
  25. 25Paragraph: Let S be the line of intersection of two planes and . Question: Which of the following are the direction ratios of S ?2024 (Phase 2)Direction Cosines and Direction Ratios
  26. 26Paragraph: Let S be the line of intersection of two planes and . Question: If are direction cosines of S, then what is the value of ?2024 (Phase 2)Direction Cosines and Direction Ratios
  27. 27Paragraph: Let be a line and P : be a plane Question: What are the direction ratios of the line?2024 (Phase 2)Line in Space
  28. 28Paragraph: Let be a line and be a plane Question: What is the point of intersection of and ?2024 (Phase 2)Line and Plane
  29. 29Paragraph: Let and be the end points of the diameter of the sphere Question: What is equal to?2023 (Phase 1)Basics of point in 3 dimension
  30. 30Paragraph: Let and be the end points of the diameter of the sphere Question: If is any point on the sphere, then what is equal to?2023 (Phase 1)Basics of point in 3 dimension
  31. 31Paragraph: Consider two lines whose direction ratios are and . They are inclined at an angle . Question: What is the value of ?2023 (Phase 1)Direction Cosines and Direction Ratios
  32. 32Paragraph: Consider two lines whose direction ratios are and . They are inclined at an angle . Question: What are the direction ratios of a 2023 (Phase 1)Direction Cosines and Direction Ratios
  33. 33Paragraph: Let a vector make angles with the positive directions of axes respectivelv, Question: What is equal to?2023 (Phase 1)Direction Cosines and Direction Ratios
  34. 34Paragraph: Let a vector make angles with the positive directions of axes respectively. Question: What is equal to?2023 (Phase 1)Direction Cosines and Direction Ratios
  35. 35Paragraph: The position vectors of two points A and B are and respectively. Question: Consider the following points: 1. 2. 3. Which of the a2023 (Phase 1)Line in Space
  36. 36If is the line with DIRECTIONS ratios and passing through , then what are the coordinates of the points on L whose distance from is 2 units?2023 (Phase 2)Basics of point in 3 dimension
  37. 37Which one of the planes is parallel to the line ?2023 (Phase 2)Line and Plane
  38. 38What is the angle between the lines and2023 (Phase 2)Direction Cosines and Direction Ratios
  39. 39What is the equation of the sphere concentric with the sphere and which passes through the origin?2023 (Phase 2)Sphere
  40. 40A point lies on the line joining and . If -coordinate of P is 7 , what is the sum of other two coordinates?2023 (Phase 2)Basics of point in 3 dimension
  41. 41The plane where meets the coordinate axes at and respectively. The equation of the sphere passing through the origin and is . What is the va2022 (Phase 1)Plane in Space
  42. 42The plane where meets the coordinate axes at and respectively. The equation of the sphere passing through the origin and is . If is the perp2022 (Phase 1)Line and Plane
  43. 43The plane where meets the coordinate axes at and respectively. The equation of the sphere passing through the origin and is . What is the eq2022 (Phase 1)Line in Space
  44. 44Let the plane pass through the point . What are the direction ratios of a normal to the plane?2022 (Phase 1)Direction Cosines and Direction Ratios
  45. 45Let the plane pass through the point . If and are the intercepts made by the plane on the coordinate axes respectively, then what is equal t2022 (Phase 1)Plane in Space
  46. 46Consider the points and . Which of the following statements is/ are correct? 1. The points are the vertices of a rectangle . 2. The mid-poin2022 (Phase 2)Basics of point in 3 dimension
  47. 47Consider the equation of a sphere . Which of the following statements is/are correct? 1. -axis is tangent to the sphere. 2. The centre of th2022 (Phase 2)Sphere
  48. 48A plane cuts intercepts 2,2,1 on the coordinate axes. What are the direction cosines of the normal to the plane?2022 (Phase 2)Direction Cosines and Direction Ratios
  49. 49Consider the following statements : 1. The direction ratios of -axis can be 2. The direction ratios of a line perpendicular to -axis can be 2022 (Phase 2)Direction Cosines and Direction Ratios
  50. 50What is the angle between the two lines having direction ratios and ?2021 (Phase 1)Direction Cosines and Direction Ratios

Frequently Asked Questions

1.How many questions are available for Three Dimensional Geometry?

This chapter includes 176 questions compiled from NDA papers spanning 2010 to 2026.

2.Are these questions only from NDA, or from other exams too?

These are NDA questions specifically for the Three Dimensional Geometry chapter. Questions from other exams like JEE Main or NTA Abhyas (JEE Main) are organised separately under their respective exam sections.

3.Is Three Dimensional Geometry an important chapter for NDA Mathematics?

Three Dimensional Geometry has appeared in 17 of the 17 years of NDA papers covered here, with 176 questions across 32 papers - about 6 per paper it appears in.

4.What is the difficulty level of Three Dimensional Geometry questions in NDA?

Of the 176 questions analysed for this chapter, 31% are easy, 65% moderate and 4% tough. 2014 had the most, with 18 questions.

5.Which topics in Three Dimensional Geometry should I focus on?

These topics have carried the most weightage in Three Dimensional Geometry:

  • Direction Cosines and Direction Ratios - 58 questions (33%)
  • Plane in Space - 41 questions (23%)
  • Basics of point in 3 dimension - 33 questions (19%)
  • Direction Cosines and Direction Ratios has produced the most tough questions (2).
6.Is Three Dimensional Geometry's weightage in NDA going up or down?

Across all 32 papers analysed, Three Dimensional Geometry has averaged about 5.5 questions per paper. In the last three years (2024–2026), that average has stayed about the same, at about 5.6 questions per paper. Use the year-wise numbers on this page for the exact trend before you decide how much time to give it.