Three Dimensional Geometry VBQs Class 12 Mathematics

VBQs Three Dimensional Geometry  Class 12 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 12 Mathematics with solutions. The following Three Dimensional Geometry  Class 12 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 12 examinations.

Three Dimensional Geometry  VBQs Class 12 Mathematics

Question. The radius of the sphere x² + y² + z² = 12x + 4y + 3z is
(a) 13/2
(b) 13
(c) 26
(d) 52 

Question. If a plane meets the coordinate axes at A, B and C such that the centroid of the triangle is (1, 2, 4), then the equation of the plane is
(a) x + 2y + 4z = 12
(b) 4x + 2y + z = 12
(c) x + 2y + 4z = 3
(d) 4x + 2y + z = 3 

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

Question. The plane through the point (– 1, – 1, – 1) and containing the line of intersection of the planes r̅ ⋅(î +3ĵ – k̂) = 0 and r̅ ⋅(ĵ + 2k̂) = 0
(a) r̅ ⋅(î + 2ĵ – 3k̂) = 0
(b) r̅ ⋅(î + 4ĵ + k̂) = 0
(c) r̅ ⋅(î+ 5ĵ – 5k̂) =  0
(d) r̅ ⋅(î + ĵ + 3k̂) = 0 

Question. The equation r̅² – 2r̅, ⋅ c̅ + h = 0 ⋅|c̅|> √h, represents 
(a) circle
(b) ellipse
(c) cone
(d) sphere 

Question. If a line in the space makes angles α,β and γ with the coordinate axes, then cos ²α + cos ²β + cos  ²γ + sin ²α + sin ²β + sin ²γ equals
(a) –1
(b) 0
(c) 1
(d) 2 

Question. If the points (1, 2, 3) and (2, –1, 0) lie on the opposite sides of the plane 2x + 3y – 2z = k, then
(a) k < 1
(b) k > 2
(c) k < 1 or k > 2
(d) 1 < k < 2

Question. The equation of sphere concentric with the sphere x² + y² + z² – 4x – 6y – 8z – 5 = 0 and which passes through the origin, is
(a) x² + y² + z² – 4x – 6y – 8z = 0
(b) x² + y² + z² – 6y – 8z = 0
(c) x² + y² + z² = 0
(d) x² + y² + z² – 4x – 6y – 8z – 6 = 0 

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 

Question. The center and radius of the sphere
x² + y² + z² + 3x – 4z + 1 = 0 are

Question. The pro ections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are :
(a) 6/5, -3/5, 2/5
(b) 6/7, -3/7, 2/7
(c) -6/7, -3/7, 2/7
(d) 6, -3, 2

Question. A line in the 3-dimensional space makes an angle θ (0 < θ < π/2) with both the x and y axes. Then the set of all values of θ is the interval:
(a) (0, π/4]
(b) [π/6, π/3]
(c) [π/4, π/2]
(d) (π/3, π/2]

Question. The angle between the lines whose direction cosines satisfy the equations l + m+ n = 0 and l² + m² + n² is
(a) π/6
(b) π/2
(c) π/3
(d) π/4

Question. A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(–1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and PQR is:
(a) cos^–1(17/31)
(b) cos^–1(19/35)
(c) cos^–1(9/35)
(d) cos–1(7/31)

Question. An angle between the lines whose direction cosines are given by the equations, l + 3m + 5n = 0 and 5lm – 2mn + 6nl = 0, is
(a) cos^–1(1/8)
(b) cos^–1(1/6)
(c) cos^–1(1/3)
(d) cos^–1(1/4)

Question. If the length of the perpendicular from the point (β, 0, β) (β ≠ 0) to the line, x/1 = (y − 1)/0 = (z+1)/−1 is √(3/2), then β is equal to:
(a) 1
(b) 2
(c) –1
(d) –2

Question. Let A (2, 3, 5), B (– 1, 3, 2) and C (λ, 5, μ) be the vertices of a ΔABC. If the median through A is equally inclined to the coordinate axes, then:
(a) 5λ – 8μ = 0
(b) 8λ – 5μ = 0
(c) 10λ – 7μ = 0
(d) 7λ – 10μ = 0

Question. A line makes the same angle θ, with each of the x and z axis. If the angle β, which it makes with y-axis, is such that sin²β = 3sin²θ then cos²θ equals
(a) 2/5
(b) 1/5
(c) 3/5
(d) 2/3

Question. Let ABC be a triangle with vertices at points A (2, 3, 5), B (–1, 3, 2) and C (λ, 5, μ) in three dimensional space. If the median through A is equally inclined with the axes, then (λ, μ) is equal to :
(a) (10, 7)
(b) (7, 5)
(c) (7, 10)
(d) (5, 7)

Question. The vertices B and C of a “ABC lie on the line, (x + 2)/3 = (y − 1)/0 = z/4 such that BC = 5 units. Then the area (in sq. units) of this triangle, given that the point A (1, –1, 2), is:
(a) 5√17
(b) 2√34
(c) 6
(d) √34

Question. If the pro ections of a line segment on the x, y and z-axes in 3-dimensional space are 2, 3 and 6 respectively, then the length of the line segment is :
(a) 12
(b) 7
(c) 9
(d) 6

Question. If a point R(4, y, z) lies on the line segment oining the points P(2, –3, 4) and Q(8, 0, 10), then distance of R from the origin is :
(a) 2√14
(b) 2√21
(c) 6
(d) √53

Question. The acute angle between two lines such that the direction cosines l, m, n, of each of them satisfy the equations l + m + n = 0 and l² + m² – n² = 0 is :
(a) 15°
(b) 30°
(c) 60°
(d) 45°

Question. A plane P meets the coordinate axes at A, B and C respectively. The centroid of ΔABC is given to be (1,1,2). Then the equation of the line through this centroid and perpendicular to the plane P is:
(a) (x − 1)/2 = (y − 1)/1 = (z − 1)/1
(b) (x − 1)/1 = (y − 1)/1 = (z − 1)/2
(c) (x − 1)/2 = (y − 1)/2 = (z − 1)/1
(d) (x − 1)/1 = (y − 1)/2 = (z − 1)/2

Question. A line AB in three-dimensional space makes angles 45° and 120° with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle θ with the positive z – axis, then θ equals
(a) 45°
(b) 60°
(c) 75°
(d) 30°

Question. The length of the pro ection of the line segment joining the points (5, –1, 4) and (4, –1, 3) on the plane, x + y + = 7 is:
(a) 2/3
(b) 1/3
(c) √(2/3)
(d) 2/√3

Question. If a line makes an angle of π/4 with the positive directions of each of x- axis and y- axis, then the angle that the line makes with the positive direction of the z-axis is
(a) π/4
(b) π/2
(c) π/6
(d) π/3

Question. ABC is triangle in a plane with vertices A (2, 3, 5), B (–1, 3, 2) and C (λ, 5, μ). If the median through A is equally inclined to the coordinate axes, then the value of (λ³ + μ³ + 5) is :
(a) 1130
(b) 1348
(c) 1077
(d) 676

Question. If two lines L₁ and L₂ in space, are defined by (image 32) then L₁ is perpendicular to L₂, for all non-negative reals λ and μ, such that :
(a) λ + μ = 1
(b) λ ≠ μ
(c) λ + μ = 0
(d) λ = μ

Question. A plane containing the point (3, 2, 0) and the line (x – 1)/1 = (y – 2)/5 = (z – 3)/4 also contains the point :
(a) (0, 3, 1)
(b) (0, 7, –10)
(c) (0, –3, 1)
(d) 0, 7, 10

Question. The distance of the point (1, 3, –7) from the plane passing through the point (1, –1, –1), having normal perpendicular to both the lines (image 89)
(a) 10/√74
(b) 20/√74
(c) 10/√83
(d) 5/√83

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

Question. If non ero numbers a, b, c are in H.P., then the straight line x/a + y/b + 1/c = 0 always passes through a fixed point. That point is
(a) (– 1, 2)
(b) (– 1, – 2)
(c) (1, – 2)
(d) (1, – 1/2)

Question. The distance of the point (1, –5, 9) from the plane x – y + z = 5 measured along the line x = y = z is :
(a) 10/√3
(b) 20/3
(c) 3√10
(d) 10√3

Question. Equation of the plane which passes through the point of intersection of lines (x – 1)/3 = (y – 2)/1 = (z – 3)/2 and (x – 3)/1 = (y – 1)/2 = (z – 2)/3 and has the largest distance from the origin is:
(a) 7x + 2y + 4 = 54
(b) 3x + 4y + 5 = 49
(c) 4x + 3y + 5 = 50
(d) 5x + 4y + 3 = 57

Question. The angle between the lines 2x = 3y = – z and 6x = – y = – 4 is
(a) 0°
(b) 90°
(c) 45°
(d) 30°

Question. If the straight lines x = 1+ s, y = -3 – λs, z = 1+ λs and x = t/2, y = 1 + t, z = 2 – t, with parameters s and t respectively, are co-planar, then λ equals.
(a) 0
(b) –1
(c) – 1/2
(d) –2

Question. The two lines x = ay + b , z = cy + d and x = a’y + b’, z = c’y + d’ will be perpendicular, if and only if
(a) aa’ + cc’ + 1 = 0
(b) aa’ + bb’ + cc’ + 1 = 0
(c) aa’ + bb’ +cc’ = 0
(d) (a + a’) (b + b’) +(c + c’) = 0.

Question. The shortest distance between the lines (x – 1)/0 = (y+1)/-1 = z/1 and x + y + z + 1 = 0, 2x – y + z + 3 = 0 is :
(a) 1
(b) 1/√3
(c) 1/√2
(d) 1/2

Question. The foot of the perpendicular drawn from the point (4, 2, 3) to the line oining the points (1, –2, 3) and (1,1, 0) lies on the plane :
(a) 2x + y – z = 1
(b) x – y – 2z = 1
(c) x – 2y + z = 1
(d) x + 2y – z = 1

Question. If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4 + 22 = 0 measured parallel to line, x/1 = y/4 = z/5 is Q, then PQ is equal to :
(a) 6√5
(b) 3√5
(c) 2√42
(d) √42

Question. A equation of a plane parallel to the plane x – 2y + 2z –5 = 0 and at a unit distance from the origin is :
(a) x – 2y + 2z – 3 = 0
(b) x – 2y + 2z + 1 = 0
(c) x – 2y + 2z – 1 = 0
(d) x – 2y + 2z + 5 = 0

Question. If the angle between the line x = (y – 1)/2 = (z – 3)/λ and the plane x + 2y + 3z = 4 is cos^–1 (√(5/14)), then λ equals
(a) 3/2
(b) 2/5
(c) 5/3
(d) 2/3

Question. If the point (2, α, β) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and is perpendicular to the plane 2x – 5y = 15, then 2α – 3β is equal to :
(a) 12
(b) 7
(c) 5
(d) 17

Question. If the angle between the line 2(x + 1) = y = + 4 and the plane 2x – y + √λ + 4 = 0 is π/6, then the value of λ is:
(a) 135/7
(b) 45/11
(c) 45/7
(d) 135/11

Question. The equation of the plane containing the straight line x/2 = y/3 = z/4 and perpendicular to the plane containing the straight lines x/3 = y/4 = z/2 and x/4 = y/2 = z/3 is :
(a) x – 2y + z = 0
(b) 3x + 2y – 3z = 0
(c) x + 2y – 2z = 0
(d) 5x + 2y – 4z = 0

Question. Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is
(a) 3/2
(b) 5/2
(c) 7/2
(d) 9/2

Question. The length of the perpendicular from the point (2, –1, 4) on the straight line, (x+3)/10 = (y – 2)/-7 = z/1 is :
(a) greater than 3 but less than 4
(b) less than 2
(c) greater than 2 but less than 3
(d) greater than 4

Question. The equation of the plane containing the line 2x – 5y + z = 3; x + y + 4z = 5, and parallel to the plane, x + 3y + 6z = 1, is:
(a) x + 3y + 6z = 7
(b) 2x + 6y + 12z = – 13
(c) 2x + 6y + 12z = 13
(d) x + 3y + 6z = –7

Question. A plane passing through the point (3, 1, 1) contains two lines whose direction ratios are 1, –2, 2 and 2, 3, –1 respectively. If this plane also passes through the point (a, – 3, 5), then a is equal to :
(a) 5
(b) –10
(c) 10
(d) –5

Question. If for some α and β in R, the intersection of the following three planes
x + 4y – 2z = 1
x + 7y – 5z = β
x + 5y + αz = 5
is a line in R³, then α + β is equal to:
(a) 0
(b) 10
(c) 2
(d) –10

Question. If the distance between the plane, 23x – 10y – 2z + 48 = 0 and the plane containing the lines (x + 1)/2 = (y – 3)/4 = (z + 1)/3 and (x + 3)/2 = (y + 2)/6 = (z – 1)/λ (λ ∈ R) is equal to k/√633 , then k is equal to ______.

Question. The shortest distance between the –axis and the line x + y + 2z – 3 = 0 = 2x + 3y + 4z – 4, is
(a) 1
(b) 2
(c) 4
(d) 3

Question. Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the image of R in the plane P is:
(a) (6, 5, 2)
(b) (6, 5, –2)
(c) (4, 3, 2)
(d) (3, 4, –2)

Question. The coordinates of the foot of perpendicular from the point (1, 0, 0) to the line (x – 1)/2 = (y+1)/-3 = (z+10)/8 are
(a) (2, – 3, 8)
(b) (1, – 1, – 10)
(c) (5, – 8, – 4)
(d) (3, – 4, – 2)

Question. If (a, b, c) is the image of the point (1, 2, –3) in the line, (x+1)/2 = (y – 3)/-2 = z/-1, then a + b + c is equals to:
(a) 2
(b) – 1
(c) 3
(d) 1

Question. The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines r̅ = (î + ĵ) + λ(î + 2ĵ – k̂) and r̅ = (î + ĵ) + μ(-î + ĵ – 2k̂) is :
(a) 3
(b) 1/3
(c) √3
(d) 1/√3

Question. If the line, (x – 1)/2 = (z + 1)/3 = (z – 2)/4 meets the plane, x + 2y + 3 = 15 at a point P, then the distance of P from the origin is:
(a) √5/2
(b) 2√5
(c) 9/2
(d) 7/2

Question. A plane passing through the points (0, –1, 0) and (0, 0, 1) and making an angle π/4 with the plane y – + 5 = 0, also passes through the point:
(a) (– √2 , 1, –4)
(b) ( √2 , –1, 4)
(c) (– √2 , –1, –4)
(d) ( √2 , 1, 4)

Question. Let P be the plane, which contains the line of intersection of the planes, x + y + – 6 = 0 and 2x + 3y + + 5 = 0 and it is perpendicular to the xy-plane. Then the distance of the point (0, 0, 256) from P is equal to:
(a) 17/√5
(b) 63√5
(c) 205√5
(d) 11/√5

Question. The equation of a plane containing the line of intersection of the planes 2x – y – 4 = 0 and y + 2z – 4 = 0 and passing through the point (1, 1, 0) is :
(a) x – 3y – 2z = –2
(b) 2x – z = 2
(c) x – y – z = 0
(d) x + 3y + z = 4

Question. The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0 is :
(a) r̅ × (î – k̂) + 2 = 0
(b) r̅ × (î – k̂) – 2 = 0
(c) r̅ × (î + k̂) + 2 = 0
(d) r̅ × (î – k̂) + 2 = 0

Question. The equation of a plane containing the line (x+1)/-3 = (y – 3)/2 = (z+2)/1 and the point (0, 7, – 7) is
(a) x + y + z = 0
(b) x + 2y + z = 21
(c) 3x – 2y + 5z + 35 = 0
(d) 3x + 2y + 5z + 21 = 0

Question. Statement 1: The shortest distance between the lines x/2 = y/-1 = z/2 and (x – 1)/4 = (y – 1)/-2 = (z – 1)/4 is √2.
Statement 2: The shortest distance between two parallel lines is the perpendicular distance from any point on one of the lines to the other line.
(a) Statement 1 is true, Statement 2 is false.
(b) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(c) Statement 1 is false, Statement 2 is true.
(d) Statement 1 is true, Statement 2 is true, , Statement 2 is not a correct explanation for Statement 1.

Question. The plane which bisects the line oining the points (4, – 2, 3) and (2, 4, – 1) at right angles also passes through the point:
(a) (4, 0, 1)
(b) (0, –1, 1)
(c) (4, 0, –1)
(d) (0, 1, –1)

Question. If an angle between the line, (x+1)/2 = (y – 2)/1 = (z – 3)/-2 and the plane, x – 2y – kx = 3 is cos^–1(2√2/3), then a value of k is
(a) √(5/3)
(b) √(3/5)
(c) – 3/5
(d) – 5/3

Question. The lines r̅ = (î – ĵ) + l(2î + k̂) and r̅ = (2î – ĵ) + m(î + ĵ – k̂)
(a) do not intersect for any values of l and m
(b) intersect for all values of l and m
(c) intersect when l = 2 and m = 1/2
(d) intersect when l = 1 and m = 2

Question. Equation of the line of the shortest distance between the lines x/1 = y/-1 = z/1 and (x – 1)/0 = (y+1)/-2 = z/1 is :
(a) x/1 = y/-1 = z/-2
(b) (x – 1)/1 = (y+1)/-1 = z/-2
(c) (x – 1)/1 = (y+1)/-1 = z/1
(d) x/-2 = y/1 = z/2

Question. The length of the perpendicular drawn from the point (3,-1,11) to the line x/2 = (y – 2)/3 = (z -3)/4 is :
(a) √29
(b) √33
(c) √53
(d) √66

Question. Let S be the set of all real values of l such that a plane passing through the points (–λ², 1, 1), (1, –λ², 1) and (1, 1, –λ²) also passes through the point- (–1, –1, 1). Then S is equal to :
(a) {√3}
(b) {√3, –√3}
(c) {1, –1}
(d) {3, –3}

Question. If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through (a, 7, 1) is , then a is equal to _________.

Question. On which of the following lines lies the point of inter – section of the line, (x – 4)/2 = (y – 5)/2 = (z – 3)/1 and the plane, x + y + z = 2 ?
(a) (x+3)/3 = (4 – y)/3 = (z + 1)/-2
(b) (x – 4)/1 = (y – 5)/1 = (z – 5 )/-1
(c) (x – 1)/1 = (y – 3)/2 = (z + 4)/-5
(d) (x – 2)/2 = (y – 3)/2 = (z + 3)/3

Question. If for some a ∈ R, the lines L₁ : (x + 1)/2 = (y – 2)/-1 = (z – 1)/1 and L₂ : (x + 2)/a = (y+1)/(5 – a) = (z+1)/1 are coplanar, then the line L₂ passes through the point :
(a) (10, 2, 2)
(b) (2, – 10, – 2)
(c) (10, – 2, – 2)
(d) (– 2, 10, 2)

Question. The system of linear equations
x + y + z = 2
2x + 3y + 2z = 5
2x + 3y + (a² – 1)z = a + 1
(a) is inconsistent when a = 4
(b) has a unique solution for |a| = √3
(c) has infinitely many solutions for a = 4
(d) is inconsistent when |a| = √3

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

Question. The plane containing the line (x – 3)/2 = (y+2)/–1 = (z – 1)/3 and also containing its pro ection on the plane 2x + 3y – z = 5, contains which one of the following points?
(a) (2, 2, 0)
(b) (–2, 2, 2)
(c) (0, – 2, 2)
(d) (2, 0, –2)

Question. The lines (x – 2)/1 = (y – 3)/1 = (z – 4)/-k and (x – 1)/k = (y – 4)/2 = (z – 5)/1 are coplanar if
(a) k = 3 or –2
(b) k = 0 or –1
(c) k = 1 or –1
(d) k = 0 or –3

Question. If Q(0, –1, –3) is the image of the point P in the plane 3x – y + 4 = 2 and R is the point (3, –1, –2), then the area (in sq. units) of ΔPQR is :
(a) 2√13
(b) √91/4
(c) √91/2
(d) √65/2

Question. The image of the line (x – 1)/3 = (y – 3)/1 = (z – 4)/-5 in the plane 2x – y + z + 3 = 0 is the line:
(image 104)

Question. Statement-1: The point A(1, 0, 7)) is the mirror image of the point B(1, 6, 3) in the line : x/1 = (y – 1)/2 = (z – 2)/3
Statement-2: The line x/1 = (y – 1)/2 = (z – 2)/3 bisects the line segment oining A(1, 0, 7) and B(1, 6, 3).
(a) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is false.
(c) Statement-1 is false, Statement-2 is true.
(d) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Question. A perpendicular is drawn from a point on the line (x – 1)/2 = (y + 1)/-1 = z/1 to the plane x + y + z = 3 such that the foot of the perpendicular Q also lies on the plane x – y + z = 3. Then the co-ordinates of Q are :
(a) (1, 0, 2)
(b) (2, 0, 1)
(c) (–1, 0, 4)
(d) (4, 0, –1)

Question. A variable plane passes through a fixed point (3, 2, 1) and meets x, y and z axes at A, B and C respectively. A plane is drawn parallel to yz – plane through A, a second plane is drawn parallel zx – plane through B and a third plane is drawn parallel to xy – plane through C. Then the locus of the point of intersection of these three planes, is
(a) x + y + z = 6
(b) x/2 + y/2 + z/1 = 1
(c) 3/x + 2/y + 1/z = 1
(d) 1/x + 1/y + 1/z = 11/6

Question. The shortest distance between the lines x/2 = y/2 = z/1 and (x+2)/-1 = (y – 4)/8 = (z – 5)/4 lies in the interval :
(a) (3, 4]
(b) (2, 3]
(c) [1, 2)
(d) [0, 1)

Question. The plane which bisects the line segment oining the points (– 3, – 3, 4) and (3, 7, 6) at right angles, passes through which one of the following points?
(a) (–2, 3, 5)
(b) (4, – 1, 7)
(c) (2, 1, 3)
(d) (4, 1, – 2)

Question. An angle between the plane, x + y + z = 5 and the line of intersection of the planes, 3x + 4y + z – 1 = 0 and 5x + 8y + 2z + 14 = 0, is
(a) cos^-1(3/√17)
(b) cos^-1(√(3/17))
(c) sin^-1(3/√17)
(d) sin^-1(√(3/17))

Question. If the lines (x + 1)/2 = (y – 1)/1 = (z + 1)/3 and (x + 2)/2 = (y – k)/3 = z/4 are coplanar, then the value of k is :
(a) 11/2
(b) – 11/2
(c) 9/2
(d) – 9/2

Question. If the line (x – 1)/2 = (y+1)/3 = (z – 1)/4 and (x – 3)/1 = (y – K)/2 = z/1 intersect, then k is equal to:
(a) –1
(b) 2/9
(c) 9/2
(d) 0

Question. If the line, (x – 3)/1 = (y + 2)/-1 = (z + λ)/-2 lies in the plane, 2x – 4y + 3 = 2, then the shortest distance between this line and the line, (x – 1)/12 = y/9 = z/4 is :
(a) 2
(b) 1
(c) 0
(d) 3

Question. If the lines x = ay + b, z = cy + d and x = a’ z + b’, y = c’ z + d’ are perpendicular, then:
(a) ab’ + bc’ + 1 = 0
(b) cc’ + a + a’ = 0
(c) bb’ + cc’ + 1 = 0
(d) aa’ + c + c’ = 0

Question. If the line, (x – 3)/2 = (y + 2)/-1 = (z + 4)/3 lies in the plane, lx + my – = 9, then l² + m² is equal to :
(a) 5
(b) 2
(c) 26
(d) 18

Question. If the three planes x = 5, 2x – 5ay + 3z – 2 = 0 and 3bx + y – 3z = 0 contain a common line, then(a, b) is equal to
(a) (8/15, – 1/5)
(b) (1/5, – 8/15)
(c) (– 8/15, 1/5)
(d) (– 1/5, 8/15)

Question. If the equation of a plane P, passing through the intersection of the planes, x + 4y – z + 7 = 0 and 3x + y + 5z = 8 is ax + by + 6z = 15 for some a, b ∈ R, then the distance of the point (3, 2, –1) from the plane P is ___________.

Question. If the plane 2x – y + 2z + 3 =0 has the distances 1/2 and 2/3 units from the planes 4x – 2y + 4z + λ = 0 and 2x – y + 2z + μ = 0, respectively, then the maximum value of λ + μ is equal to :
(a) 9
(b) 15
(c) 5
(d) 13