12th Math chapter 11: full solution, notes, MCQ, & free PDF | Bihar board

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12th Math chapter 11 Bihar board with NCERT-Details & Highlights

name of content	12th Math chapter 11 full solution
content	Math objective type MCQ question with answer
name of subjects	class 12 Math
important for exam	Bihar board & other state board
pattern & Level	NCERT & State level question

12th Math chapter 11 all objective solutions – download PDF

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

• • (a) 13
    (b) 12
    (c) 23
    (d) 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.

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 co-ordinate 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 θ₁, θ₂, θ₃ with the axis respectively, then cos 2θ₁ + cos 2θ₂ + cos 2θ₃ =

• • (a) -4
    (b) -2
    (c) -3
    (d) -1

Answer: (d) -1

Question 5. The direction cosines of a line passing through two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) 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

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 (x₁, y₁, z₁) is

Answer: (a) x−x1l=y−y1m=z−z1n

Question 9. The certesian equation of the line l when it passes through the point (x₁, y₁, z₁) and parallel to the vector
b = ai^+bj^+ck^, is

• • (a) x – x₁ = y – y₁ = z – z₁
    (b) x + x₁ = y + y₁ = z + z₁
    (c) x+x1a=y+y1b=z+z1c
    (d) x−x1a=y−y1b=z−z1c

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 Z-axis 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)

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 yz-plane.

• • (a) (0,−172,132)
    (b) (0,172,−132)
    (c) (10,192,132)
    (d) (0, 17, 13)

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°

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°

Answer: (c) 90°

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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

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°

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

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

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√

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

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

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

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

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

Answer: (b) 5x – 7y + 11z + 4 = 0

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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

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

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) non-coplanar
    (c) perpendicular
    (d) None of the above

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

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

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

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

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√

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

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√

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°

Answer: (b) 90°

Question 43. Distance of the point (α, β, γ) from y-axis is

• • (a) β
    (b) |β|
    (c) |β| + |γ|
    (d) α2+γ2−−−−−−√

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

Answer: (a) 1

Question 45. The reflection of the point (α, β, γ) in the xy-plane is

• • (a) (α, β, 0)
    (b) (0, 0, γ)
    (c) (-α, -β, -γ)
    (d) (α, β, -y)

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

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

Answer: (d) A pair of perpendicular planes

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