CBSE Class 12 Mathematics Three Dimensional Geometry MCQs Set 04

Question. If the straight lines x-1/k = y-2/2 = z-3/2 and x-2/3 = y-3/k = z-1/2 intersect at a point, then the integer is equal to
(a) –5
(b) 5
(c) 2
(d) –2
Answer: a

Question. The intersection point of the lines x-1/2 = y-2/3 = z-3/4 and x- y/5 = y-1/2=z is
(a) (1, 1, 1)
(b) (-1,-1,-1)
(c) (1,2,3)
(d) (2,2,2)
Answer: b

Question. The distance of the point( -15,-10) from the point of intersection of the line
(a) 11
(b) 12
(c) 11
(d) 13
Answer: d

Question. The image of the point (1 ,6 3 ) 1 in the line x/1= y-1/2 = z-2/3, is −
(a) (1, 6, 9)
(b) (1, 0, 7)
(c) (0, 2, 7)
(d) (7, 9, 0)
Answer: b

Question. Two lines x-1/2= y+1/3 = z-1/4 and x-3/1=y-k/2=z intersect at a point, if k is equal to
(a) 2/9
(b) 1/2
(c) 9/2
(d) 1/6
Answer: c

Question. A line with positive direction cosines passes through the point P(2,-1,2) and makes equal angles with the coordinate axes. The line meets the plane 2x+y+z=9 z + at point Q. The length of the line segment PQ equals
(a) 1
(b) √2
(c) √3
(d) 2
Answer: c

Question. The lines x/1=y/2= z/3 and x-1/-2=y-2/-4=z-3/-6 are
(a) interecting
(b) skew
(c) parallel
(d) coincident
Answer: d

Question. The length of the perpendicular drawn from ( 1,2,3) to the line x-6/3 = y-7/2 = z-7/-2 is 
(a) 4
(b) 5
(c) 6
(d) 7
Answer: d

Question. The shortest distance between the lines x-3/3 = y-8/-1 = z-3/1 and x+3/-3 = y+7/2 = z-6/4 is
(a) √30
(b) 2 √30
(c) 5 √30
(d) 3 √30
Answer: d

Question. The coordinates of the point where the line through (3,-4,-5) and (2,-3,1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, 1, 0) −is, 
(a) (1, 2, 7)
(b) (-1, 2, -7)
(c) (1, 2, 7) 
(d) None of these
Answer: c

Question. If A(0,0,0), B (a,0,0),C(0,b,0) are the vertices of a tetrahedron, then the volume of tetrahedron is
(a) abc cu units
(b) abc/2cu units
(c) abc/3 cu units
(d) abc/6 cu units
Answer: d

Question. The angle between the lines whose direction cosines are given by 2l-m+2n=0,lm + mn + nl =0, is
(a) π/6
(b) π/4
(c) π/3
(d) π/2
Answer: d

Question. The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5,1,-2 ) then its z-coordinate is 
(a) 1
(b) −1
(c) 1/2
(d) 0
Answer: b

Question. The angle between the lines whose direction are given by l m+ n = 0 and l² +m² -n²=0 is 
(a) π/6
(b) π/4
(c) π/3
(d) π/2
Answer: c

Question. A(3, 2, 0),B(5, 3, 2) and C( 9, 6, 3) − − are the vertices of a ∆ABC. If the bisector of ∠BAC meets BC at D, coordinates of D are
(a) (19/8.57/16,17/16)
(b) (-19/8, 57/16, 17/16)
(c) (19/8, -57/16, 17/16)
(d) (19/8, 57/16, -17/16)
Answer: a

Question. The angle between any two diagonals of a cube is
(a) sin⁻¹ 2/3
(b) cos⁻¹ 1/2
(c) cos⁻¹ (1/√3)
(d) cos⁻¹(1/3)
Answer: d

Question. The volume of the tetrahedron included between the plane 3x+ 4y- 5z- 60 = 0 and the coordinate planes is
(a) 60
(b) 600
(c) 720
(d) 400
Answer: b

Question. A line segment has length 63 and direction ratios are 3,- 2, 6. If the line makes an obtuse angle with x-axis,the components of the line vector are
(a) 27,-18,54 
(b) −27 ,18, 54 
(c) − 27, 18, -54 
(d) 27,-18, -54
Answer: c

Question. Which one of the following planes contains the z-axis?
(a) x – z = 0
(b) z + y = 0
(c) 3x + 2y = 0
(d) 3x + 2z = 0
Answer: c

Question. The angle between a line whose direction ratios are in the ratio 2 : 2 : 1 and a line joining (3, 1, 4) to (7, 2, 12) is
(a) cos^–1(2/3)
(b) cos^–1(–2/3)
(c) tan^–1(2/3)
(d) None of these
Answer: a

Question. Any three numbers which are proportional to the direction cosines of a line, are called.
(a) direction angles
(b) direction ratios
(c) another set of direction cosines
(d) None of the above
Answer: b

Question. The angle between the line x-2/a = y-2/b = z-2/c and the plane ax + by + cz + 6 = 0 is
(a) sin^-1(1/√a² + b² + c²)
(b) 45°
(c) 60°
(d) 90°
Answer: d

Question. Under what condition do ⟨1/√2 , 1/2 , k⟩ represent direction cosines of a line?
(a) k = 1/2
(b) k = -1/2
(c) k = ± 1/2
(d) k can take any value
Answer: c

Question. The projection of the line segment joining the points (–1, 0, 3) and (2, 5, 1) on the line whose direction ratios are (6, 2, 3) is
(a) 6
(b) 7
(c) 22/7
(d) 3
Answer: c

Question. The length intercepted by a line with direction ratios 2, 7, –5 between the lines x-5/3 = y-7/-1 and x+3/-3 = y-3/2 = z-6/4 is
(a) √75
(b) √78
(c) √83
(d) None of these
Answer: b

Question. If the directions cosines of a line are k, k, k, then
(a) k > 0
(b) 0 < k < 1
(c) k = 1
(d) k = 1/√3 or -1/√3
Answer: d

Question. Two planes r . n₁ = d₁ and r . n₂ = d₂ are perpendicular to each other, if
(a) n₁ = n₂
(b) n₁ is parallel to n₂
(c) n₁ . n₂ = 0
(d) None of the above
Answer: c

Question. The projections of a line segment on the coordinate axes are 12, 4, 3. The direction cosine of the line are:
(a) • 12/13 , • 4/13 , 3/13
(b) 12/13, • 4/13 , 3/13
(c) 12/13 , 4/13 , 3/13
(d) None of these
Answer: c

Question. The angle between two planes is equal to
(a) The angle between the tangents to them from any point.
(b) The angle between the normals to them from any point.
(c) The angle between the lines parallel to the planes from any point.
(d) None of these.
Answer: b

Question. Direction ratios of two lines are a, b, c and 1/bc , 1/ca , 1/ab , The lines are
(a) Mutually perpendicular
(b) Parallel
(c) Coincident
(d) None of these
Answer: b

Question. Consider the following statements
Statement I : The vector equation of a line passing through two points whose position vectors are a and b, is r = a + λ (b – a) ∀ λ ∈ R.
Statement II : The cartesian equation of a line passing through two points (x₁, y₁, z₁) and (x₂, y₂, z₂) is x-x₁/x₂-x₁ = y-y₁/y₂-y₁ = z-z₁/z₂-z₁
Choose the correct option.
(a) Statement I is true
(b) Statement II is true
(c) Both statements are true
(d) Both statements are false
Answer: c

Question. A line makes angles of 45° and 60° with the positive axes of X and Y respectively. The angle made by the same line with the positive axis of Z, is.
(a) 30° or 60°
(b) 60° or 90°
(c) 90° or 120°
(d) 60° or 120°
Answer: d

Question. Consider the following statements
Statement I : The points (1, 2, 3), (–2, 3, 4) and (7, 0, 1) are collinear.
Statement II : If a line makes angles π/2 , 3π/4 and π/4 with X, Y and Z-axes respectively, then its direction cosines are 0, -1/√2 and 1/√2 .
Choose the correct option.
(a) Statement I is true
(b) Statement II is true
(c) Both statements are true
(d) Both statements are false
Answer: c

Question. Let the line x-2/3 = y-1/-5 = z+2/2 lie in the plane x + 3y –αz + β = 0. Then (α, β) equals
(a) (–6, 7)
(b) (5, –15)
(c) (–5, 5)
(d) (6, –17)
Answer: a

Question. The line which passes through the origin and intersect the two lines x-1/2 = y+3/4 = z-5/3 , x-4/2 = y+3/3 = z-14/4 , is
(a) x/1 = y/-3 = z/5
(b) x/-1 = y/3 = z/5
(c) x/1 = y/3 = z/-5
(d) x/1 = y/4 = z/-5
Answer: a

Question. The direction ratios of the line OP are equal and the length OP = √3 . Then the coordinates of the point P are :
(a) ( –1, – 1, –1)
(b) ( √3, √3, √3)
(c) ( √2, √2, √2)
(d) (2, 2, 2)
Answer: a

Question. Consider the following statements
Statement I : If a line in space does not pass through the origin, the direction cosines of the line does not exist.
Statement II : Two parallel lines have same set of direction cosines.
Choose the correct option.
(a) Statement I is true
(b) Statement II is true
(c) Both statements are true
(d) Both statements are false
Answer: b

Question. The ordered pair (λ, μ) such that the points (λ, μ, –6),(3, 2, –4) and (9, 8, –10) become collinear is
(a) (3, 4)
(b) (5, 4)
(c) (4, 5)
(d) (4, 3)
Answer: b

Question. The equation of the right bisector plane of the segment joining (2, 3, 4) and (6, 7, 8) is
(a) x + y + z + 15 = 0
(b) x + y + z – 15 = 0
(c) x – y + z – 15 = 0
(d) None of these
Answer: b

Question. The coordinates of a point on the line x+2/3 = y+1/2 = z-3/2 at a distance of 6/√2 from the point (1, 2, 3) is
(a) (56,,43, 111)
(b) (56/17 , 43/17 , 111/17)
(c) (2, 1, 3)
(d) (–2, –1, –3)
Answer: b

Question. If a plane passes through the point (1, 1, 1) and is perpendicular to the line x-1/3 = y-1/0 = z-1/4 , then its perpendicular distance from the origin is
(a) 3/4
(b) 4/3
(c) 7/5
(d) 1
Answer: c

Question. Which of the following are true?
I. If a, b and c are the direction ratios of a line, then ka, kb and kc is also a set of direction ratios.
II. The two sets of direction ratios of a line are in proportion.
III. There exists two sets of direction ratios of a line.
(a) I and II are true
(b) II and III are true
(c) I and III are true
(d) All are true
Answer: a

Question. If the plane x – 3y + 5z = d passes through the point (1, 2, 4), then the length of intercepts cut by it on the axes of X, Y, Z are respectively, is
(a) 15, –5, 3
(b) 1, –5, 3
(c) –15, 5, –3
(d) 1, –6, 20
Answer: a