Q1: 1. In three-dimensional geometry, which of the following represents the distance between two points in space?

• A) Magnitude of vector
• B) Scalar product
• C) Cross product
• D) Dot product

Q2: 2. What is the equation for the plane passing through a point (2, -1, 3) and perpendicular to the vector <1, 2, -3>?

• A) x + 2y - 3z = 1
• B) x - 2y + 3z = 7
• C) x + 2y - 3z = 7
• D) x - 2y + 3z = 1

Q3: 3. Which of the following represents the direction cosines of a line passing through two points in space?

• A) Sum of angles
• B) Product of angles
• C) Ratios of angles
• D) Sine of angles

Q4: 4. The equation of a line passing through the points (1, 2, 3) and (4, 5, 6) is given by

• A) x = 1 + t, y = 2 + t, z = 3 + t
• B) x = 1 + 3t, y = 2 + 3t, z = 3 + 3t
• C) x = 1 + 2t, y = 2 + 3t, z = 3 + t
• D) x = 1 + 3t, y = 2 + 2t, z = 3 + t

Q5: 5. The angle between the lines with direction vectors <1, 2, 3> and <2, -1, 4> is

• A) 45 degrees
• B) 60 degrees
• C) 90 degrees
• D) 120 degrees

Q6: 6. The shortest distance between the lines with equations x = 2 + t, y = 3 - 2t, z = 1 + 3t and x = 3 + s, y = 1 + 2s, z = 2 - s is

• A) 5 units
• B) 6 units
• C) 7 units
• D) 8 units

Q7: 7. The equation of the sphere passing through the points (1, 2, 3), (4, 5, 6), and (7, 8, 9) is

• A) x^2 + y^2 + z^2 = 14
• B) x^2 + y^2 + z^2 = 19
• C) x^2 + y^2 + z^2 = 24
• D) x^2 + y^2 + z^2 = 29

Q8: 8. The equation of the plane passing through the points (1, 2, 3), (2, 3, 4), and (3, 4, 5) is given by

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

Q9: 9. The volume of a parallelepiped formed by the vectors <1, 2, 3>, <4, 5, 6>, and <7, 8, 9> is

• A) 0
• B) 1
• C) 2
• D) 3

Q10: 10. The equation of the plane passing through the origin and perpendicular to the plane 2x - 3y + 4z = 5 is

• A) 2x - 3y + 4z = 0
• B) 2x + 3y - 4z = 5
• C) 2x + 3y - 4z = 0
• D) 2x - 3y + 4z = 5

Q11: 11. The equation of the plane passing through the points (1, 2, 3), (2, 3, 4), and parallel to the plane x - y + z = 7 is

• A) x - y + z = 1
• B) x - y + z = 2
• C) x - y + z = 3
• D) x - y + z = 4

Q12: 12. The equation of the line passing through the points (1, 2, 3) and (3, 4, 5) is given by

• A) x = 1 + t, y = 2 + t, z = 3 + t
• B) x = 1 + 2t, y = 2 + 2t, z = 3 + 2t
• C) x = 1 + 3t, y = 2 + 3t, z = 3 + 3t
• D) x = 1 + 4t, y = 2 + 4t, z = 3 + 4t

Q13: 13. The angle between the planes with equations x - y + z = 3 and 2x + y - z = 2 is

• A) 30 degrees
• B) 45 degrees
• C) 60 degrees
• D) 90 degrees

Q14: 14. The equation of the line passing through the point (1, 2, 3) and parallel to the vector <2, -1, 3> is

• A) x = 1 + 2t, y = 2 - t, z = 3 + 3t
• B) x = 1 + 2t, y = 2 + t, z = 3 + 3t
• C) x = 1 + 2t, y = 2 - t, z = 3 - 3t
• D) x = 1 + 2t, y = 2 + t, z = 3 - 3t

Q15: 15. The equation of the plane passing through the points (1, 2, 3), (2, 3, 4), and (3, 4, 5) is given by

• A) x - y + z = 1
• B) x + y - z = 1
• C) x + y + z = 1
• D) x - y - z = 1

Q16: 16. The area of the triangle formed by the points (1, 2, 3), (2, 3, 4), and (3, 4, 5) is

• A) 0
• B) 1
• C) 2
• D) 3

Q17: 17. The angle between the vectors <1, 2, 3> and <2, 4, 6> is

• A) 30 degrees
• B) 45 degrees
• C) 60 degrees
• D) 90 degrees

Q18: 18. The equation of the plane passing through the points (1, 2, 3), (2, 3, 4), and perpendicular to the vector <1, 1, 1> is

• A) x + y + z = 1
• B) x - y + z = 1
• C) x + y - z = 1
• D) x - y - z = 1

Q19: 19. The equation of the plane passing through the points (1, 2, 3), (2, 3, 4), and parallel to the plane x + y + z = 6 is

• A) x + y + z = 1
• B) x - y + z = 1
• C) x + y - z = 1
• D) x - y - z = 1

Q20: 20. The equation of the sphere passing through the points (1, 2, 3), (2, 3, 4), and (3, 4, 5) is given by

• A) x^2 + y^2 + z^2 = 14
• B) x^2 + y^2 + z^2 = 15
• C) x^2 + y^2 + z^2 = 16
• D) x^2 + y^2 + z^2 = 17

Q21: 21. The equation of the line passing through the points (1, 2, 3) and (3, 4, 5) is given by

• A) x = 1 + t, y = 2 + t, z = 3 + t
• B) x = 1 + 3t, y = 2 + 3t, z = 3 + 3t
• C) x = 1 + 2t, y = 2 + 3t, z = 3 + t
• D) x = 1 + 3t, y = 2 + 2t, z = 3 + t

Q22: 22. The angle between the lines with direction vectors <1, 2, 3> and <2, -1, 4> is

• A) 45 degrees
• B) 60 degrees
• C) 90 degrees
• D) 120 degrees

Q23: 23. The shortest distance between the lines with equations x = 2 + t, y = 3 - 2t, z = 1 + 3t and x = 3 + s, y = 1 + 2s, z = 2 - s is

• A) 5 units
• B) 6 units
• C) 7 units
• D) 8 units

Q24: 24. The equation of a line passing through the points (1, 2, 3) and (4, 5, 6) is given by

• A) x = 1 + t, y = 2 + t, z = 3 + t
• B) x = 1 + 3t, y = 2 + 3t, z = 3 + 3t
• C) x = 1 + 2t, y = 2 + 3t, z = 3 + t
• D) x = 1 + 3t, y = 2 + 2t, z = 3 + t

Q25: 25. What is the equation for the plane passing through a point (2, -1, 3) and perpendicular to the vector <1, 2, -3>?

• A) x + 2y - 3z = 1
• B) x - 2y + 3z = 7
• C) x + 2y - 3z = 7
• D) x - 2y + 3z = 1