Question 1. The cosines of the angle between any two diagonals of a cube is

 (a) 13
(b) 12
(c) 23
(d) 13√
 (a) 13
Answer: (a) 13
Question 2. Which of the following is false?

 (a) 30°, 45°, 60° can be the direction angles of a line is space.
(b) 90°, 135°, 45° can be the direction angles of a line is space.
(c) 120°, 60°, 45° can be the direction angles of a line in space.
(d) 60°, 45°, 60° can be the direction angles of a line in space.
 (a) 30°, 45°, 60° can be the direction angles of a line is space.
Answer: (a) 30°, 45°, 60° can be the direction angles of a line is space.
Question 3. A line makes angles α, β and γ with the coordinate axes. If α + β = 90°, then γ is equal to
(a) 0°
(b) 90°
(c) 180°
(d) None of these
Answer: (b) 90°
Question 4. If a line makes an angle θ_{1}, θ_{2}, θ_{3} with the axis respectively, then cos 2θ_{1} + cos 2θ_{2} + cos 2θ_{3} =

 (a) 4
(b) 2
(c) 3
(d) 1
 (a) 4
Answer: (d) 1
Question 5. The direction cosines of a line passing through two points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) are
Answer: (c) x2−x1PQ,y2−y1PQ,z2−z1PQ
Question 6. The equation of a line which passes through the point (1, 2, 3) and is parallel to the vector 3i^+2j^−2k^, is
Answer: (b) r=(i^+2j^+3k^)+λ(3i^+2j^−2k^)
Question 7. The equation of line passing through the point (3, 2, 4) and equally inclined to the axes are

 (a) x – 3 = y + 2 = z – 4
(b) x + 3 = y – 2 = z + 4
(c) x+31=y−22=z+43
(d) None of these
 (a) x – 3 = y + 2 = z – 4
Answer: (b) x + 3 = y – 2 = z + 4
Question 8. If l, m and n are the direction cosines of line l, then the equation of the line (l) passing through (x_{1}, y_{1}, z_{1}) is
Answer: (a) x−x1l=y−y1m=z−z1n
Question 9. The certesian equation of the line l when it passes through the point (x_{1}, y_{1}, z_{1}) and parallel to the vector
b = ai^+bj^+ck^, is

 (a) x – x_{1} = y – y_{1} = z – z_{1}
(b) x + x_{1} = y + y_{1} = z + z_{1}
(c) x+x1a=y+y1b=z+z1c
(d) x−x1a=y−y1b=z−z1c
 (a) x – x_{1} = y – y_{1} = z – z_{1}
Answer: (d) x−x1a=y−y1b=z−z1c
Bihar Board 12th math chapter 11 mcq solution
Question 10. The equation of the straight line passing through the point (a, b, c) and parallel to Zaxis is
Answer: (d) x−a0=y−b0=z−c1
Question 11. The coordinates of a point on the line x+23=y+12=z−32 at a distance of 612√ from the point (1, 2, 3) is

 (a) (56, 43, 111)
(b) (5617,4317,11117)
(c) (2, 1, 3)
(d) (2, 1, 3)
 (a) (56, 43, 111)
Answer: (b) (5617,4317,11117)
Question 12. Find the coordinatets of the point where the line through the points (5, 1, 6) and (3, 4, 1) crosses the yzplane.

 (a) (0,−172,132)
(b) (0,172,−132)
(c) (10,192,132)
(d) (0, 17, 13)
 (a) (0,−172,132)
Answer: (b) (0,172,−132)
Question 13. The point A(1, 2, 3), B(1, 2, 1) and C(2, 3, 2) are three vertices of a parallelogram ABCD. Find the equation of CD.
Answer: (d) x−21=y−32=z−22
Question 14. The equation of the line joining the points (3, 4, 11) and (1, 2, 7) is
Answer: (b) x+3−2=y−43=z−112
Question 15. The vector equation of the line through the points A(3, 4, 7) and B(1, 1, 6) is
Answer: (c) r=(3i^+4j^−7k^)+λ(−2i^−5j^+13k^)
Question 16. The vactor equation of the symmetrical form of equation of straight line x+53=y+47=z−62 is
Answer: (d) r=(5i^−4j^+6k^)+μ(3i^+7j^+2k^)
Question 17. The angle between the straight lines

 (a) 45°
(b) 30°
(c) 60°
(d) 90°
 (a) 45°
Answer: (d) 90°
Question 18.
Answer: (d) π6
Question 19. The angle between the line 2x = 3y = z and 6x = y = 4z is

 (a) 30°
(b) 45°
(c) 90°
(d) 0°
 (a) 30°
Answer: (c) 90°
12th math chapter 11 most mcq notes and solution
Question 20. The angle between the lines 3x = 6y = 2z and x−2−5=y−17=z−31 is

 (a) π6
(b) π4
(c) π3
(d) π2
 (a) π6
Answer: (d) π2
Question 21. The angle between the lines x = 1, y = 2 and y = 1, z = 0 is

 (a) 90°
(b) 30°
(c) 60°
(d) 0°
 (a) 90°
Answer: (a) 90°
Question 22. The angle between the lines passing through the points (4, 7, 8), (2, 3, 4) and (1, 2, 1), (1, 2, 5) is

 (a) 0
(b) π2
(c) π4
(d) π6
 (a) 0
Answer: (a) 0
Question 23. The shortest distance between the lines x−33=y−8−1=z−31 and x+3−3=y+72=z−64 is equal

 (a) 3√30
(b) √30
(c) 2√30
(d) None of these
 (a) 3√30
Answer: (a) 3√30
Question 24. The shortest distance between the lines x = y = z and x + 1 – y = z0 is

 (a) 12
(b) 12√
(c) 13√
(d) 16√
 (a) 12
Answer: (d) 16√
Question 25. The shortest distance between the lines x = y + 2 = 6z – 6 and x + 1 = 2y = 12z is

 (a) 12
(b) 2
(c) 1
(d) 32
 (a) 12
Answer: (b) 2
Question 26. The direction cosines of the unit vector perpendicular to the plane r⋅(6i^−3j^−2k^)+1=0 passing through the origin are

 (a) 67,37,27
(b) 6, 3, 2
(c) −67,37,27
(d) 6, 3, 2
 (a) 67,37,27
Answer: (c) −67,37,27
Question 27. Find the vector equation of the plane which is at a distance of 8 units from the origin and which is normal to the vector 2i^+j^+2k^.
Answer: (c) r⋅(2i^+j^+2k^)=24
Question 28. Find the length of perpendicular from the origin to the plane r(3i^−4j^+12k^).

 (a) 513
(b) 513√
(c) 523
(d) 5√13
 (a) 513
Answer: (a) 513
Question 29. The equation of the plane passing through three non collinear points with position vectors a, b, c is

 (a) r.(b × c + c × a + a × b) = 0
(b) r.(b × c + c × a + a × b) = [abc] (c) r.(a × (b + c)) = [abc] (d) r.(a + b + c) = 0
 (a) r.(b × c + c × a + a × b) = 0
Answer: (b) r.(b × c + c × a + a × b) = [abc]
Question 30. Four points (0, 1, 1) (4, 4, 4) (4, 5, 1) and (3, 9, 4) are coplanar. Find the equation of the plane containing them.

 (a) 5x + 7y + 11z – 4 =0
(b) 5x – 7y + 11z + 4 = 0
(c) 5x – 7y – 11z – 4 = 0
(d) 5x + 7y – 11z + 4 = 0
 (a) 5x + 7y + 11z – 4 =0
Answer: (b) 5x – 7y + 11z + 4 = 0
12th math chapter 11 bihar board objective solution
Question 31. Find the equation of plane passing through the points P(1, 1, 1), Q(3, 1, 2), R(3, 5, 4).

 (a) x + 2y = 0
(b) x – y = 2
(c) x + 2y = 2
(d) x + y = 2
 (a) x + 2y = 0
Answer: (d) x + y = 2
Question 32. The vector equation of the plane passing through the origin and the line of intersection of the plane r.a = λ and r.b = µ is

 (a) r.(λa – µb) = 0
(b) r.(λb – µa) = 0
(c) r.(λa + µb)= 0
(d) r.(λb + µa) = 0
 (a) r.(λa – µb) = 0
Answer: (b) r.(λb – µa) = 0
Question 33. The vector equation of a plane passing through the intersection of the planes r⋅(i^+j^+k^)=6 and r⋅(2i^+3j^+4k^)=−5 and the point (1, 1, 1) is
Answer: (c) r⋅(20i^+23j^+26k^)=69
Question 34.

 (a) coplanar
(b) noncoplanar
(c) perpendicular
(d) None of the above
 (a) coplanar
Answer: (a) coplanar
Question 35. The equation of the plane through the point (0, 4, 6) and (2, 9, 3) and perpendicular to the plane x – 4y – 2z = 8 is

 (a) 3x + 3y – 2z = 0
(b) x – 2y + z = 2
(c) 2x + y – z = 2
(d) 5x – 3y + 2z = 0
 (a) 3x + 3y – 2z = 0
Answer: (c) 2x + y – z = 2
Question 36. The angle between the planes r⋅(i^+2j^+k^)=4 and r(−i^+j^+2k^)=9 is

 (a) 30°
(b) 60°
(c) 45°
(d) None of these
 (a) 30°
Answer: (b) 60°
Question 37. The angle between the panes x + y = 0 and y – z = 1 is

 (a) π6
(b) π4
(c) π3
(d) π2
 (a) π6
Answer: (c) π3
Question 38. If the angle between the planes 2x – y + 2z = 3 and 3x + 6y + cz = 4 is cos−1(421), then c2 =

 (a) 1
(b) 4
(c) 9
(d) 5
 (a) 1
Answer: (b) 4
Question 39. The distance of the plane 2x – 3y + 4z – 6 = 0 from the origin is A. Here, A refers to

 (a) 6
(b) 6
(c) −629√
(d) 629√
 (a) 6
Answer: (b) 6
BSEB 12th math chapter 11 most mcq solution
Question 40. Find the length of perpendicular from origin to the plane r⋅(3i^−4j^−12k^)+39=0

 (a) 1
(b) 3
(c) 17
(d) None of these
 (a) 1
Answer: (b) 3
Question 41. The distance of the origin from the plane through the points (1, 1, 0), (1, 2, 1) and (2, 2, 1) is

 (a) 311√
(b) 522√
(c) 3
(d) 422√
 (a) 311√
Answer: (b) 522√
Question 42. The angle between the straight line x−12=y+3−1=z−52 and the plane 4x – 2y + 4z = 9 is

 (a) 60°
(b) 90°
(c) 45°
(d) 30°
 (a) 60°
Answer: (b) 90°
Question 43. Distance of the point (α, β, γ) from yaxis is

 (a) β
(b) β
(c) β + γ
(d) α2+γ2−−−−−−√
 (a) β
Answer: (d) α2+γ2−−−−−−√
Question 44. The distance of the plane r⋅(27i^+37j^−67k^)=1 from the origin is

 (a) 1
(b) 7
(c) 17
(d) None of these
 (a) 1
Answer: (a) 1
Question 45. The reflection of the point (α, β, γ) in the xyplane is

 (a) (α, β, 0)
(b) (0, 0, γ)
(c) (α, β, γ)
(d) (α, β, y)
 (a) (α, β, 0)
Answer: (d) (α, β, y)
Question 46. The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, 1), C(4, 5, 0) and D(2, 6, 2), is equal to

 (a) 9 sq. units
(b) 18 sq. units
(c) 27 sq. units
(d) 81 sq. units
 (a) 9 sq. units
Answer: (a) 9 sq. units
Question 47. The locus represented by xy + yz = 0 is

 (a) A pair of perpendicular lines
(b) A pair of parallel lines
(c) A pair of parallel planes
(d) A pair of perpendicular planes
 (a) A pair of perpendicular lines
Answer: (d) A pair of perpendicular planes
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