Q1: 1. What is the equation of a plane passing through the point (1, 2, 3) and perpendicular to the vector \( \langle 2, -1, 3 \rangle \)?

• A) \( 2x - y + 3z = 7 \)
• B) \( 2x - y + 3z = 9 \)
• C) \( 2x - y + 3z = 5 \)
• D) \( 2x - y + 3z = 11 \)

Q2: 2. Which of the following is a point on the line passing through the points (2, 1, -3) and (4, 3, 1)?

• A) (1, 0, -5)
• B) (3, 2, -1)
• C) (0, 1, -4)
• D) (-1, 4, 2)

Q3: 3. The distance between the parallel planes \(2x - y + z = 5\) and \(2x - y + z = 9\) is

• A) 2 units
• B) 4 units
• C) 6 units
• D) 8 units

Q4: 4. If the line \(x = 1 + 2t, y = 3 - t, z = 2t\) is perpendicular to the plane \(2x - y + z = 4\), then t =

• A) 1
• B) 2
• C) 3
• D) 4

Q5: 5. The equation of a line passing through the point (1, 2, 3) and parallel to the vector \( \langle 3, -1, 2 \rangle \) is

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

Q6: 6. The point of intersection of the planes \(x - y + z = 2\) and \(2x + y + z = 4\) is

• A) (1, 1, 0)
• B) (2, 0, 2)
• C) (0, 2, 0)
• D) (0, 0, 2)

Q7: 7. The equation of a plane passing through the points (1, 2, 3), (4, 1, -2), and (2, 5, 1) is

• A) \( x - y + 2z = 3 \)
• B) \( 2x + 3y - z = 5 \)
• C) \( x + 2y - z = 3 \)
• D) \( 3x - 2y + z = 7 \)

Q8: 8. The shortest distance between the line passing through the points (1, -2, 3) and (3, 4, -1) and the point (2, 0, 1) is

• A) \( \sqrt{14} \)
• B) \( \sqrt{10} \)
• C) \( \sqrt{6} \)
• D) \( \sqrt{2} \)

Q9: 9. The equation of a line passing through the points (-1, 2, 4) and (3, 1, -2) is

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

Q10: 10. The angle between the planes \(2x - y + z = 3\) and \(x + 2y - z = 4\) is

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