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CBSE Questions for Class 12 Commerce Maths Three Dimensional Geometry Quiz 1 - MCQExams.com

If a line makes an angle of π/4 with the positive direction of each of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is
  • π6
  • π3
  • π4
  • π2
A line AB in three-dimensional space makes angles 45o and 120o with the positive x-axis and the positive y-axis respectively. lf AB makes an acute angle e with the positive z-axis, then e equals
  • 45o
  • 60o
  • 75o
  • 30o
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: 
  • 23
  • 15
  • 35
  • 25
A line in the 3-dimensional space makes an angle θ(0<θπ2) with both the x and y axes, then the set of all values of θ is the interval :
  • [0,π4]
  • [π6,π3]
  • [π4,π2]
  • [π3,π2]
The points with position vectors 60ˆi+3ˆj, 40ˆi8ˆj, aˆi52ˆj are collinear if
  • a=40
  • a=40
  • a=20
  • None of these
The vector equation r=i2jk+t(6jk) represents a straight line passing through the points:
  • (0,6,1) and (1,2,1)
  • (0,6,1) and (1,4,2)
  • (1,2,1) and (1,4,2)
  • (1,2,1) and (0,6,1)
A line makes the same angle θ with each of the X and Z-axes. If the angle β, which it makes with Y-axis, is such that sin2β=3sin2θ, then cos2θ equals
  • 25
  • 15
  • 35
  • 23
The points with position vectors 60i+3j,40i8j and ai52j are collinear if
  • a=40
  • a=40
  • a=20
  • None of these
Equation to a line parallel to the vector 2ˆiˆj+ˆk and passing through the point ˆi+ˆj+ˆk
  • (x1)ˆi+(y1)ˆj+(z1)ˆk=λ(2ˆiˆj+ˆk)
  • (x2)ˆi+(y1)ˆj+(z+1)ˆk=λ(ˆi+ˆj+ˆk)
  • 2xˆiyˆj+zˆk=λ(ˆi+ˆj+ˆk)
  • (x1)ˆi+(y+1)ˆj+(z+1)ˆk=λ(ˆi+ˆj+ˆk)
The direction ratios of the diagonal of the cube joining the origin to the opposite corner are (when the 3 concurrent edges of the cube are coordinate axes)
  • 23,23,23
  • 1,1,1
  • 2,2,1
  • 1,2,3

If the direction cosines of a line are (1c,1c,1c) then c=______

  • 3
  • 1/3
  • ±3
  • ±13
l=m=n=1 represents the direction cosines of 

  • xaxis
  • yaxis
  • zaxis
  • none of these
 List IList II 
1) d.c's of x axisa) (1,1,1) 
2) d.c's of y axisb)(]3]3,]3)
3) d.c's of z axisc) (1,0,0)
4) d.c's of a line makes equal angles with axesd) (0,1,0)
 e) (0,0,1)
The correct order for 1, 2, 3, 4 is
  • c,d,e,b
  • a,b,c,e
  • c,d,a,b
  • b,c,a,e
If P(x,y,z) moves such that x=0,z=0, then the locus of P is the line whose d.cs are
  • y-axis
  • 1,0,0
  • 0,1,0
  • 0,0,0
If P is a point on the line passing through the point A with position vector 2¯i+¯j3¯k and parallel to ¯i+2¯j+¯k such that AP=26 then the position vector of P is
  • 4¯i+5¯j+¯k
  • 3¯j+5¯k
  • 4¯i+5¯j¯k
  • 3¯j4¯k
If the points A(¯a),B(¯b),C(¯c) satisfy the relation 3a8b+5c=0 then the points are
  • Vertices of an equilateral triangle
  • Collinear
  • Vertices of a right angled triangle
  • Vertices of a isosceles triangle
Cosine of the angle between two diagonals of a cube is equal to
  • 26
  • 13
  • 12
  • None of these
Area of ΔABC is
  • 45 squares units
  • 55 squares units
  • 65 squares units
  • none of these
A vector is equally inclined to the x-axis, y-axis and z-axis respectively, its direction cosines are
  • <13,13,13>
  • <13,13,13>
  • <13,13,13> or <13,13,13>
  • None of these
What are the DR's of vector parallel to (2,1,1) and (3,4,1)?
  • (1,5,2)
  • (2,5,2)
  • (1,5,2)
  • (1,5,2)
The direction cosines of the vectors 2i+j2k is equal to
  • 23,13,23
  • 23,13,23
  • 13,23,23
  • 23,23,13
The Cartesian equation of the plane passing through the point (3,2,1) and parallel to the vectors ¯b=¯i2¯j+4¯k and ¯c=3¯i+2¯j5¯k is
  • 2x17y8z+63=0
  • 3x+17y+8z36=0
  • 2x+17y+8z+36=0
  • 3x16y+8z63=0
A straight line is equally inclined to all the three coordinate axes. Then an angle made by the line with the y-axis is
  • cos1(13)
  • cos1(13)
  • cos1(23)
  • π4
If the foot of the perpendicular from (0,0,0) to a plane is (1,2,3), then the equation of the plane is
  • 2x+y+3z=14
  • x+2y+3z=14
  • x+2y+3z+14=0
  • x+2y3z=14
A line makes angles α,β,γ,δ with four diagonals of a cube. then  rϵ{α,β,γ,δ}cos2(r)=
  • 43
  • 43
  • 34
  • 35
Find vector equation for the line passing through the points 3¯i+4¯j7¯k,¯i¯j+6¯k.

  • ¯r=(32λ)¯i+(45λ)¯j+(7+13λ)¯k
  • ¯r=(2λ)¯i+(4+5λ)¯j+(713λ)¯k
  • ¯r=(32λ)¯i(45λ)¯j+(7+13λ)¯k
  • ¯r=(32λ)¯i+(45λ)¯j(7+13λ)¯k
Find vector equation of line passing through  (1,4,1),(1,2,2).
  • ¯r=¯i+(42λ)¯j+(1+λ)¯k
  • ¯r=¯i+(4+2λ)¯j+(1+λ)¯k
  • ¯r=¯i(42λ)¯j+(1+λ)¯k
  • ¯r=¯i+(42λ)¯j+(1λ)¯k
The projections of a directed line segment on the coordinate axes are 12,4,3 respectively.
What are the direction cosines of the line segment?
  • (12/13,4/13,3/13)
  • (12/13,4/13,3/13)
  • (12/13,4/13,3/13)
  • (12/13,4/13,3/13)
From the point P(3,1,11), a perpendicular is drawn on the line L given by the equation x2=y23=z34. Let Q be the foot of the perpendicular.
What are the direction ratios of the line segment PQ?
  • (1,6,4)
  • (1,6,4)
  • (1,6,4)
  • (2,6,4)
Find the vector equation of line joining the points (2,1,3) and (4,3,1)
  • ¯r=2(13λ)¯i(1+2λ)¯j(34λ)¯k
  • ¯r=2(13λ)¯i(1+2λ)¯j+(34λ)¯k
  • ¯r=2(13λ)¯i+(1+2λ)¯j+(34λ)¯k
  • ¯r=2(1+3λ)¯i+(1+2λ)¯j+(3+4λ)¯k
The vector equation of line passing through two points A(x1,y1,z1),B(x2,y2,z2) is
  • r=ab
  • r=a+λ(ba)
  • r=a+λb
  • r=a+b
Which of the following represents direction cosines of the line:
  • 0,12,12
  • 0,32,12
  • 0,32,12
  • 12,12,12
A line passes through the points (6, -7, -1) and (2, -3, 1). What are the direction ratios of the line ? 
  • (4,4,2)
  • (4,4,2)
  • (4,4,2)
  • (2,1,1)
The length of perpendicular from the origin to the plane which makes intercepts \dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5} respectively on the coordinate axes is 
  • \dfrac{1}{\sqrt[5]{2}}
  • \dfrac{1}{10}
  • \displaystyle\sqrt[5]{2}
  • 5
ABCD is a trapezium in which AB and CD are parallel.that the mid points of the sides AB ,CD and the intersection of the diagonals are collinear.
  • True
  • False
If points (1,2), (3 , 5) and (0 , b ) are collinear the value of b is  
  • \dfrac{1}{2}
  • \dfrac{7}{2}
  • 2
  • -1
The direction angles of the line x = 4z + 3, y = 2 - 3z are \alpha, \beta and \gamma, then \cos \alpha + \cos \beta + \cos \gamma = ________.
  • \dfrac {2}{\sqrt {26}}
  • \dfrac {8}{\sqrt {26}}
  • 1
  • 2
The angle between the lines whose direction cosines are \left( \dfrac {\sqrt{3}}{4}, \dfrac {1}{4}, \dfrac {\sqrt{3}}{2} \right) and \left( \dfrac {\sqrt{3}}{4}, \dfrac {1}{4}, -\dfrac {\sqrt{3}}{2} \right) is :
  • \pi
  • \dfrac {\pi}{2}
  • \dfrac {\pi}{3}
  • \dfrac {\pi}{4}
If a line makes the angles \alpha , \beta and \gamma with the axes, then what is the value of 1+\cos 2\alpha +\cos 2\beta+\cos 2\gamma equal to ?
  • -1
  • 0
  • 1
  • 2
Direction cosines of the line \cfrac { x+2 }{ 2 } =\cfrac { 2y-5 }{ 3 } ,z=-1 are ____
  • \cfrac { 4 }{ 5 } ,\cfrac { 3 }{ 5 } ,0
  • \cfrac { 3 }{ 5 } ,\cfrac { 4 }{ 5 } ,\cfrac { 1 }{ 5 }
  • -\cfrac { 3 }{ 5 } ,\cfrac { 4 }{ 5 } ,0
  • \cfrac { 4 }{ 5 } ,-\cfrac { 2 }{ 5 } ,\cfrac { 1 }{ 5 }
The following lines are \hat { r } =\left( \hat { i } +\hat { j }  \right) +\lambda \left( \hat { i } +2\hat { j } -\hat { k }  \right) +\mu \left( -\hat { i } +\hat { j } -\hat { 2k }  \right)
  • collinear
  • skew-lines
  • co-planar lines
  • parallel lines
L and M are two points with position vectors 2\overline { a } -\overline { b } and a+2\overline { b } respectively. The position vector of the point N which divides the line segment LM in the ratio 2:1 externally is
  • 3\overline { b }
  • 4\overline { b }
  • 5\overline { b }
  • 3\overline { a } +4\overline { b }
If the normal of the plane makes an angles \dfrac {\pi}{4}, \dfrac {\pi}{4} and \dfrac {\pi}{2} with positive X-axis, Y-axis and Z-axis respectively and the length of the perpendicular line segment from origin to the plane is \sqrt {2}, then the equation of the plane is ________.
  • x + y + z = \sqrt {2}
  • x + y + z = 1
  • x + y = 2
  • x = \sqrt {2}
Direction cosines of ray from P(1, -2, 4) to Q(-1, 1, -2) are
  • -2, 3, -6
  • 2, -3, 6
  • 2, 3, 6
  • \dfrac{-2}{7}, \dfrac{3}{7}, \dfrac{-6}{7}
If the lines x=1+a,y=-3-\lambda a,z=1+\lambda a and x=\cfrac { b }{ 2 } ,y=1+b,z=2-b are coplanar, then \lambda is equal to
  • -3
  • 2
  • 1
  • -2
Vector equation of the line 6x - 2 = 3y + 1 = 2z - 2 is 
  • \vec r = \hat i + \hat j + 3\hat k + \lambda (\hat i - 2\hat j + 3\hat k)
  • \vec r = \hat i - 2\hat j + 3\hat k + \lambda \left( {\frac{1}{3}\hat i + \frac{1}{3}\hat j + \hat k} \right)
  • \vec r = \frac{1}{3}\hat i - \frac{1}{3}\hat j\, + \hat k + \lambda (\hat i + 2\hat j + 3\hat k)
  • \vec r = - 2\hat i + \hat j\, - 2\hat k + \lambda (6\hat i + 3\hat j + 2\hat k)
The points with position vectors 60\hat{i}+3\hat{j}, 40\hat{i}-8\hat{j}, a\hat{i}-52\hat{j}  are collinear if
  • a=-40
  • a=40
  • a=20
  • None\ of\ these
Direction cosines l, m, n of two lines are connected by the equation l-5m+3n=0 and 7l^{2}+5m^{2}-3n^{2}=0. The direction cosines of one of the lines are
  • -1/\sqrt{6},1/\sqrt{6},2/\sqrt{6}
  • 2/\sqrt{6},1/\sqrt{6},-1/\sqrt{6}
  • 1/\sqrt{6},2/\sqrt{6},-1/\sqrt{6}
  • 1/\sqrt{6},1/\sqrt{6},2/\sqrt{6}
The direction ratios of the line joining the points A(4,-3,7) and B(1,3,5) are:
  • (5,0,12)
  • (3,-6,2)
  • (5,6,2)
  • \left(\dfrac{5}{2},0,6\right)
Can \dfrac{1}{\sqrt{3}}, \dfrac{2}{\sqrt{3}}, \dfrac{-2}{\sqrt{3}} be the direction cosines of any directed line?
  • Yes
  • No
  • Cannot say
  • None of these
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