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CBSE Questions for Class 12 Commerce Maths Three Dimensional Geometry Quiz 9 - MCQExams.com

A line with direction cosines proportional to 2 , 1, 2 meets each of the lines x = y + a = z and x + a = 2y = 2z. The co-ordinates of each of the point of intersection are given by ______________.
  • (3a, 3a, 3a), (a,a,a)
  • (3a,2a, 3a), (a,a,a)
  • (3a, 2a, 3a), (a, a, 2a)
  • (2a, 3a, 3a), (2a, a, a)
the points (α,β)(γ,δ),(α,δ)and(γ,β)   where α,β,γ,δ  are different real numbers, are 
  • Collinear
  • vertices of square
  • vertices of rhomubs
  • concyclic
The angle between the lines 2x=3y=z and 6x=y=4z is?
  • 0o
  • 45o
  • 90o
  • 30o
A plane passes through the point (0,1,0) and (0,0,1) and makes an angle of π4 with the plane yz=0 then the point which satisfies the desired plane is?
  • (2,1,4)
  • (2,1,2)
  • (2,1,4)
  • (2,2,4)
The angle between two adjacent sides a and b of parallelogram is π6. If a=(2,2,1) and |b|=2|a|, then area of this parallelogram is ______
  • 9
  • 18
  • 92
  • 34
The equation of the plane passing through the point (1,2,1) and perpendicular to the line joining the points (3,1,2) and (2,3,4) is ________.
  • ˉr(5ˆi+2ˆj+2ˆk)=1
  • ˉr(5ˆi+2ˆj+2ˆk)=1
  • ˉr(5ˆi2ˆj+2ˆk)=5
  • ˉr(5ˆi2ˆj2ˆk)=1
The direction ratios of the line perpendicular to the lines

x72=y+173=z61 and, x+51=y+32=z42 are proportional to
  • 4,5,7
  • 4,5,7
  • 4,5,7
  • 4,5,7
The angle between the pair of lines x22=y15=z+33 and x+21=y48=z54 is
  • cos1(21938)
  • cos1(23938)
  • cos1(24938)
  • cos1(26938)
The projections of a line segment on X,Y and Z axes are 12,4 and 3 respectively. The length and direction cosines of the line segment are
  • 13;1213,413,313
  • 19;1219,419,319
  • 11;1211,411,311
  • None of these
The Cartesian equation of a line are x22=y+13=z32. What is its vector equation?
  • r=(2ˆi+3ˆj2ˆk)+λ(2ˆiˆj+3ˆk)
  • r=(2ˆiˆj+3ˆk)+λ(2ˆi+3ˆj2ˆk)
  • r=(2ˆi+3ˆj2ˆk)
  • none of these
A line passes through the point A(2,4,5) and is parallel to the line x+33=y45=z+86. The vector equation of the line is
  • r=(3ˆi+4ˆj8ˆk)+λ(2ˆi+4ˆj5ˆk)
  • r=(2ˆi+4ˆj5ˆk)+λ(3ˆi+5ˆj+6ˆk)
  • r=(3ˆi+5ˆj+6ˆk)+λ(2ˆi+4ˆj5ˆk)
  • r=(2ˆi+4ˆj5ˆk)+λ(3ˆi+5ˆj+6ˆk)
A line passes through the point A(5,2,4) and it is parallel to the vector (2ˆiˆj+3ˆk). The vector equation of the line is
  • r=(2ˆiˆj+3ˆk)+λ(5ˆi2ˆj+4ˆk)
  • r=(5ˆi2ˆj+4ˆk)+λ(2ˆiˆj+3ˆk)
  • r.(5ˆi2ˆj+4ˆk)=14
  • none of these
The Cartesian equations of a line are x12=y+23=z51. Its vector equation is
  • r=(ˆi+2ˆj5ˆk)+λ(2ˆi+3ˆjˆk)
  • r=(2ˆi+3ˆjˆk)+λ(ˆi2ˆj+5ˆk)
  • r=(ˆi2ˆj+5ˆk)+λ(2ˆi+3ˆjˆk)
  • none of these
If the points A(1,3,2),B(4,2,2) and C(5,5,λ) are collinear then the value of λ is
  • 5
  • 7
  • 8
  • 10
The direction cosines of the perpendicular from the origin to the plane r(6ˆi3ˆj+2ˆk)+1=0 are?
  • 67,37,27
  • 67,37,27
  • 67,37,27
  • None of these
The direction consines of the line drawn from P(5,3,1)toQ(1,5,2) is
  • (6,2,3)
  • (2,4,1)
  • (4,8,1)
  • (67,27,37)
If a straight line makes an angle of 60 with each of the X and Y axes, the angle which it makes with the Z axis is
  • π3
  • π4
  • π2
  • 3π4
The points (p+1,1),(2p+1,3) and (2p+2,2p) are collinear if 
  • p=1
  • p=1/2
  • p=2
  • p=12
The equation of the plane which passes through the x-axis and perpendicular to the line (x1)cosθ=(y+2)sinθ=(z3)0 is
  • xtanθ+ysecθ=0
  • xsecθ+ytanθ=0
  • xcosθ+ysinθ=0
  • xsinθycosθ=0
The direction cosines of the normal to the plane 5y+4=0 are?
  • 0,45,0
  • 0,1,0
  • 0,1,0
  • None of these
If O is the origin and P(1,2,3) is a given point, then the equation of the plane through P and perpendicular to OP is?
  • x+2y3z=14
  • x2y+3z=12
  • x2y3z=14
  • None of these
 If the directions cosines of a line are k,k,k, then
  • k>0
  • 0<k<1
  • k=1
  • k=13 or 13
What is the equation of the plane which passes through the z-axis and is perpendicular to the line  
xacosθ=y+2sinθ=z30?
  • x+ytanθ=0
  • y+xtanθ=0
  • xcosθysinθ=0
  • xsinθycosθ=0
If P1:rn1d1=0,P2:rn2d2=0 and P3:rn3d3=0 are three planes and n1,n2 and n3 are three non-copllanar vectors, then three lines P1=0,P2=0;P2=0,P3=0 and P3=0,P1=0  are
  • parallel lines
  • coplanar lines
  • coincident lines
  • concurrent lines
If L1=0 is the reflected ray, then its equation is
  • x+104=y54=z+23
  • x+103=y+155=z+145
  • x+104=y+155=z+143
  • none of these
State true or false.
The equation of a line, which is parallel to 2ˆi+ˆj+3ˆk and which passes through the point (5,-2,4) is x52=y+21=z43
  • True
  • False
State true or false.
The vector equation of the line x53=y47=z62 is r=5ˆi4ˆj+6ˆk+λ(3ˆi+7ˆj+2ˆk)
  • True
  • False
State true or false.
The unit vector normal to the plane x+2y+3z6=0 is 114ˆi+214ˆj+314ˆk
  • True
  • False
If α, β, γ are direction angles of a line and α=60o, β=45o, γ= ____.
  • 30o or 90o
  • 45o or 60o
  • 90o or 30o
  • 60o or 120o
If cosα, cosβ, cosγ are direction 
cosines of line, then the value of sin2α+sin2β+sin2γ is
  • 1
  • 2
  • 3
  • 4
If cosα, cosβ, cosγ are direction 
cosines of line, then the value of sin2α+sin2β+sin2γ is
  • 1
  • 2
  • 3
  • 4
The direction ratios of the line which is perpendicular to the two lines x72=y+173=z61andx+51=y+32=z62 are
  • 4,5,7
  • 4,5,7
  • 4,5,7
  • 4,5,8
The angle between the lines 2x = 3 y = - z  and 6 x = -y = -4 z is 
  • 45
  • 30
  • 0
  • 90
Te direction ratios of the line 3x+1=6y2=1z are 
  • 2,1,6
  • 2,1,6
  • 2,1,6
  • 2,1,6
Position vectors of two points are 
P(2ˆi+ˆj+3ˆk) and Q(4ˆi2ˆj+ˆk)
Equation of plane passing through Q and perpendicular of PQ is 
  • r.(6ˆi+3ˆj+2ˆk)=28
  • r.(6ˆi+3ˆj+2ˆk)=32
  • r.(6ˆi+3ˆj+2ˆk)+28=0
  • r.(6ˆi+3ˆj+2ˆk)+32=0
which of the following group is not direction cosines of a line:
  • 1,1,1
  • 0,0,1
  • 1,0,0
  • 0,1,0
Direction cosines of 3i be
  • 3,0,0
  • 1,0,0
  • 1,0,0
  • 3,0,0
A(1,0,0),B(0,2,0),C(0,0,3) form the triangle ABC. Then the direction ratios of the line joining orthocenter and circumcentre of ABC are
  • 58,43,36
  • 59,44,37
  • 59,44,111
  • None of these
Assertion (A): 
Three points with position vectors a,b, c are collinear if a×b+b×c+c×a=0

Reason (R):
Three points A,B, C are collinear if AB=t BC, where t is a scalar quantity.
  • Both A and R are individually true and R is the correct explanation of A.
  • Both A and R are individually true and R is NOT the correct explanation of A.
  • A is true but R is false.
  • A is false but R is true.
If a ray makes angles α,β,γ and δ with the four diagonals of a cube and
A:cos2α+cos2β+cos2γ+cos2δ
B:sin2α+sin2β+sin2γ+sin2δ
C:cos2α+cos2β+cos2γ+cos2δ
Arrange A,B,C in descending order
  • B,A,C
  • A,B,C
  • C,A,B
  • B,C,A
Find the angle between the pair of lines r=3i+2j4k+λ(i+2j+2k) and r=5i2k+μ(3i+2j+6k).
  • cos1(1921)
  • sin1(1921)
  • cos1(2021)
  • sin1(2021)
Statement-1  :  If a line makes acute angles α,β,γ,δ with diagonals of a cube, then cos2α+cos2β+cos2γ+cos2δ=43
Statement 2  :  If a line makes equal angle (acute) with the axes, then its direction cosine are 13,12 and 13
  • Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
  • Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
  • Statement-1 is True, Statement-2 is False
  • Statement-1 is False, Statement-2 is True
If the lines x21=y31=z4k and x1k=y42=z51 are coplanar, then k can have 
  • exactly one value
  • exactly two values
  • exactly three values
  • any value
lf α, β, γ are the angles made by a line with the coordinate axes in the positive direction, then the range of sinαsinβ+sinβsinγ+sinγsinα is
  • [12,1]
  • [1,2]
  • [12,)
  • [1,)

If the straight lines x12=y+1K=z2 and x+15=y+12=zK are coplanar, then the plane(s) containing these two lines is(are)

  • y+2z=1
  • y+z=1
  • yz=1
  • y2z=1
The intercepts made on the axes by the plane which bisects the line joining the points (1,2,3) and (3,4,5) at right angles are
  • (92,9,9)
  • (92,9,9)
  • (9,92,9)
  • (9,92,9)
lf a line makes angles 60o,45o,45o and θ with the four diagonals of a cube, then sin2θ=
  • 112
  • 1112
  • 1112
  • 3112
The direction ratios of the diagonal of a cube which joins the origin to the opposite corner are (when the three concurrent edges of the cube are coordinate axes)
  • 23,23,23
  • 1,1,1
  • 2,2,1
  • 1,2,3
The vector equation of the plane through the point i+2jk and to the line of intersection of the plane r.(3ij+k)=1 and r.(i+4j2k)=2 is
  • r.(2i+7j13k)=1
  • r.(2i7j13k)=1
  • r.(2i+7j+13k)=0
  • None of these
Equation of the line which passes through the point with p.v. (2, 1, 0) and perpendicular to the plane containing the vectors ˆi+ˆjandˆj+ˆk is
  • r=(2,1,0)+t(1,1,1)
  • r=(2,1,0)+t(1,1,1)
  • r=(2,1,0)+t(1,1,1)
  • r=(2,1,0)+t(1,1,1)
0:0:2


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