CBSE Class 12 Mathematics 3D Geometry Worksheet Set 01

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
Answer : C

Question. The length of the pro ection of the line segment oining 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
Answer : C

Question. A line AB in three-dimensional space makes angles 45^o and 120 with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle q with the positive-axis, then q equals
(a) 45^o
(b) 60^o
(c) 75^o
(d) 30^o
Answer : B

Question. A line in the 3-dimensional space makes an angle θ (0 < 0 ≤ 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]
Answer : C

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
Answer : C

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(19/31)
(c) cos^-1(9/31)
(d) cos^-1(7/31)
Answer : B

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
Answer : B

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 - 2 / 1
(b) x - 1 /1 = y - 1 / 1 = z - 2 / 2
(c) x - 1 /2 = y - 1 / 2 = z - 2 / 1
(d) x - 1 /1 = y - 1 / 2 = z - 2 / 2
Answer : C

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
Answer : A

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
Answer : B

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)
Answer : C

Question. A line makes the same angle q, with each of the x and z axis. If the angle b, which it makes with y-axis, is such that sin² β = 3sin² θ then cos² θ equals 
(a) 5/2
(b) 5/1
(c) 5/3
(d) 3/2
Answer : C

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) 2√30
(b) 7/30√2
(c) 3√30
(d) 3
Answer : A

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
Answer : B

Question. Two lines x - 3/1 = y + 1/3 = z - 6/-1 and x + 5/7 = y - 2/-6 = z - 3/4 intersect at the point R. The reflection of R in the xy- plane has coordinates :
(a) (2, –4, –7)
(b) (2, 4, 7)
(c) (2, –4, 7)
(d) (–2, 4, 7)
Answer : A

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
Answer : C

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
Answer : A

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
Answer : D

Question. If a line makes an angle of p / 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
Answer : B

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)
Answer : B

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
Answer : C

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)
Answer : B

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
Answer : A

Question. If the angle between the lines x/2 = y/2 = z/1 and 5 - x /-2 = 7y - 14/ P = z -3 /4 is cos^–1(2/3) then P is equal to
(a) - 7/4
(b) 2/7
(c) - 4/7
(d) 7/2
Answer : D

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 _________. 
Answer : 4

Please click on below link to download CBSE Class 12 Mathematics 3D Geometry Worksheet Set A