NCERT Class 11 Maths Introduction To Three Dimensional Geometry Questions

Question. The distance of the point (4, 3, 5) from the y-axis is: 
a. √34
b. 5
c. √41
d. √15
Answer : C

Question. If centroid of tetrahedron OABC, where A, B, C are given by (a, 2, 3), (1, b, 2) and (2, 1, c) respectively be (1, 2, – 1), then distance of P(a, b, c) from origin is equal to:
a. √107
b. √14
c. √107 /14
d. None of these
Answer : A

Question. A line which makes angle 60° with y-axis and z-axis, then the angle which it makes with x-axis is:
a. 45°
b. 60°
c. 75°
d. 30°
Answer : A

Question. A line passes through the points (6, –7, –1) and (2,–3, 1). The direction cosines of line, so directed that the angle made by it with the positive direction of x-axis is acute, are:

Answer : A

Question. If the x-co-ordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, –2) is 4, then its z-co-ordinate is:
a. 2
b. 1
c. –1
d. –2
Answer : C

Question. If the direction cosines of a line are (1/c, 1/c, 1/c), then:
a. c > 0
b. c = ± √3
c. 0 < c < 1
d. c > 2
Answer : B

Question. If the direction ratio of two lines are given by 3lm− 4ln +mn = 0 and l + 2m+ 3n = 0 , then the angle between the lines is:
a. π/2
b. π/3
c. π/4 
d. π/6
Answer : A

Question. If a line makes angles α, β, γ, δ with four diagonals of a cube, then the value of sin² α + sin² β + sin² γ + sin² δ is:
a. 4/3
b. 1
c. 8/3
d. 7/3
Answer : C

Question. The vector equation of line through the point A(3, 4, –7) and B(1, –1, 6) is
a. r = (3i + 4j− 7k) +λ (i − j+ 6k)
b. r = (i − j+ 6k) +λ (3i + 4j− 7k)
c. r = (3i + 4j− 7k) +λ (−2i −5j+13k)
d. r = (i − j+ 6k) +λ (4i + 3j−k)
Answer : C

Question. If r is a vector of magnitude 21 and has d.r.’s 2, –3, 6. Then r is equal to:
a. 6i − 9j+18k
b. 6i + 9j+18k
c. 6i − 9j−18k
d. 6i + 9j−18k
Answer : A

Question. The projection of a line on co-ordinate axes are 2, 3, 6. Then the length of the line is:
a. 7
b. 5
c. 1
d.11
Answer : B

Question. The angle between two lines x + 1 / 2 = y + 1 / 2 = z - 4 / -1 and x - 4 / 1 = x + 4 / 2 = z + 1 / 2 is
a. cos⁻¹(1/9)
b. cos⁻¹(1/9)
c. cos⁻¹(1/9)
d. cos⁻¹(1/9)
Answer : D

Question. The point of intersection of the lines, x - 5 / 3 = y - 7 / -1 = z + 2 / 1 = x + 3 / -36 = y - 3 / 2 = z - 6 / 4 is:
a. 21, 5/3, 10/3
b. ( 2,10, 4)
c. (−3, 3, 6)
d. (5, 7, − 2)
Answer : A

Question. The cartesian equations of a line are 6x − 2 . = 3y +1 = 2z − 2 The vector equation of the line is:
a. r = (1/3  1 - 1/3 j+k ) + λ (I + 2j + 3k)
b. r = (3i −3j+ k) +λ (i + 2j+ 3k)
c. r = (i + j+ k) +λ (i + 2j+ 3k)
d. None of these
Answer : A

Question. The angle between the lines whose direction cosines are proportional to (1, 2, 1) and (2, –3, 6) is:

Answer : A

Question. The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 , 0 l² + m² − n² = is given by:
a. 2π/3
b. π/6
c. 5π/6
d. π/3
Answer : D

Question. The xy-plane divides the line joining the points (–1, 3, 4) and (2, –5, 6)
a. Internally in the ratio 2:3
b. Internally in the ratio 3:2
c. Externally in the ratio 2:3
d. Externally in the ratio 3:2
Answer : C

Question. The angle between the pair of lines with direction ratios (1, 1, 2) and ( √3 − 1,− √3 − 1,4) is:
a. 30°
b. 45°
c. 60°
d. 90°
Answer : C

Question. If direction ratios of two lines are 5, −12,13 and −3, 4, 5 then the angle between them is:
a. cos⁻¹(1/ 65) 
b. cos⁻¹ (2 / 65) 
c. cos⁻¹ (3/ 65) 
d. π / 2
Answer : A

Question. The equation of the plane, which makes with co-ordinate axes a triangle with its centroid (α, β, γ), is:
a. α x +β y +γ z = 3
b. x/α + y/β + z/γ = 1
c. α x +β y +γ z = 1
d. x/α + y/β + z/γ = 3
Answer : D

Question. Angle between two planes x +2y+2z=3 and −5x + 3y + 4z = 9 is:

Answer : A

Question. The shortest distance between the lines r = (i + j− k) +λ (3i − j) and r = (4i − k) +μ (2i + 3k) is:a. 6
b. 0
c. 2
d. 4
Answer : B

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

Question. The ratio in which the plane x − 2y + 3z = 17 divides the line joining the point (–2, 4, 7) and (3, –5, 8) is:
a. 10 : 3
b. 3 : 1
c. 3 : 10
d. 10 : 1
Answer : C

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

Integer

Question. A variable plane is at a constant distance p form the origin and meets the axes in A, B and C. If the locus of the centroid of the tetrahedron OABC is x–2 + y^–2 + z^–2 =λp^–2 then the value of 160λ must be
Answer : 2560

Question. The lines x + 4 / 3 = y + 6 / 5 = z - 1 /-2 and 3x – 2y + z + 5 = 0 =
2x + 3y + 4z–k
are coplanar for k is equal to
Answer : 2

Question. The shortest distance between the z-axis and the lines x + y + 2z − 3 = 0, 2x + 3y + 4z − 4 = 0must be
Answer : 2

Question. If the volume of tetrahedron formed by planes whose equations are y + z = 0, z + x = 0, x + y = 0 and x + y + z = 1 is λ cubic unit then the value of 729λ must be
Answer : 486

Question. If the angle of intersection of the sphere x² + y² + ²z − 2x −4 y − 6z +10 = 0 with the sphere, the extremities of whose diameter are (1, 2, –3) and (5,0,1) is cos–1(λ), thenthe value of 9999|λ| must be
Answer : 6666

12.1.1 Coordinate axes and coordinate planes Let X′OX, Y′OY, Z′OZ be three mutually perpendicular lines that pass through a point O such that X′OX and Y′OY lies in the plane of the paper and line Z′OZ is perpendicular to the plane of paper. These three lines are called rectangular axes ( lines X′OX, Y′OY and Z′OZ are called x-axis, y-axis and z-axis). We call this coordinate system a three-dimensional space, or simply space.
The three axes taken together in pairs determine xy, yz, zx-plane, i.e., three coordinate planes. Each plane divide the space in two parts and the three coordinate planes together divide the space into eight regions (parts) called octant, namely (i) OXYZ (ii) OX′YZ (iii) OXY′Z (iv) OXYZ′ (v) OXY′Z′ (vi) OX′YZ′ (vii) OX′Y′Z (viii) OX′Y′Z′.
(Fig.12.1).

Let P be any point in the space, not in a coordinate plane, and through P pass planes parallel to the coordinate planes yz, zx and xy meeting the coordinate axes in the points A, B, C respectively. Three planes are
(i) ADPF || yz-plane   (ii) BDPE || xz-plane (iii) CFPE || xy-plane
These planes determine a rectangular parallelopiped which has three pairs of rectangular faces (A D P F, O B E C),(B D P E, C F A O) and (A O B D, FPEC) (Fig 12.2) 12.1.2 Coordinate of a point in space An arbitrary point P in three-dimensional space is assigned coordinates (x0, y0, z0) provided that

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