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CBSE Questions for Class 12 Commerce Maths Three Dimensional Geometry Quiz 5 - MCQExams.com

Equation of the plane passing through a point with position vector 3ˆi3ˆj+ˆk & normal to the line joining the points with position vectors 3ˆi+4ˆjˆk & 2ˆiˆj=5ˆk is
  • ¯r.(ˆi5ˆj+6ˆk)+18=0
  • ¯r.(ˆi5ˆj+6ˆk)=22
  • ¯r.(ˆi+5ˆj6ˆk)+18=0
  • ¯r.(ˆi+5ˆj+6ˆk)+12=0
Vector equation of the plane passing through a point having position vector 2ˆi+3ˆj4ˆk and perpendicular to the vector 2ˆiˆj+2ˆk is
  • r(2ˆiˆj+2ˆk)+7=0
  • r(2ˆiˆj+2ˆk)=7
  • r(3ˆi2ˆj3ˆk)=0
  • r(2ˆiˆj+2ˆk)=9
A line makes the same angle θ with each of the x and z axis. If the angle β , which it makes with yaxis is such that sin2β=3sin2θ  then cos2θ equals 
  • 35
  • 15
  • 23
  • 25
Direction cosines of the vector ˉv=a1ˆi+a2ˆj+a3ˆk are
  • <a1,a2,a3>
  • <a1,a2,a3>
  • <a1|ˉv|,a2|ˉv|,a3|ˉv|,>
  • none of these
The Equation of the plane through a point 2ˆiˆj+4ˆk & parallel to the plane ¯r.(2ˆi+4ˆj7ˆk)=6 is
  • ¯r.(2ˆi+4ˆj7ˆk)=21
  • ¯r.(2ˆi+4ˆj7ˆk)=14
  • ¯r.(2ˆi+4ˆj7ˆk)=42
  • ¯r.(2ˆi+4ˆj7ˆk)=28
The line passes through the points \left ( 5,1,a \right ) & \left ( 3,b,1 \right ) crosses the yz plane at the point \displaystyle \left ( 0,\frac{17}{2},-\frac{13}{2} \right ) ,then
  • a= 4, b= 6
  • a= 6, b= 4
  • a= 8, b= 2
  • a= 2, b= 8
The scalar product form of equation of plane \displaystyle \overline{r}=\left ( s-2t \right )\hat{i}+\left ( 3-t \right )\hat{j}+\left ( 2s-t \right )\hat{k} is
  • \displaystyle \overrightarrow{r}\cdot \left ( 2\hat{i}-5\hat{j}-\hat{k} \right )+15=0
  • \displaystyle \overrightarrow{r}\cdot \left ( 2\hat{i}-5\hat{j}-\hat{k} \right )=15
  • \displaystyle \overrightarrow{r}\cdot \left ( 2\hat{i}-5\hat{j}-\hat{k} \right )=3
  • \displaystyle \overrightarrow{r}\cdot \left ( 2\hat{i}-5\hat{j}-\hat{k} \right )= -3
If the three points with position vectors \displaystyle \bar{a}-2\bar{b}+3\bar{c}, \ 2\bar{a}+\lambda \bar{b}-4\bar{c}, \ -7\bar{b}+10\bar{c} are collinear, then \displaystyle \lambda=
  • 1
  • 2
  • 3
  • none of these
Find the equation of the plane containing the vectors \displaystyle \bar{\alpha} and \displaystyle\bar{ \beta} and passing through the point \displaystyle \bar{a}
  • \displaystyle (\bar{r}-\bar{a})\cdot (\bar{\alpha} \times \bar{\beta}) =0
  • \displaystyle (\bar{r}+\bar{a})(\bar{\alpha }\times \bar{\beta}) =0
  • \displaystyle (\bar{r}-\bar{a}) (\bar {a}\cdot \bar{b}) =0
  • none of these
If { l }_{ 1 },{ m }_{ 1 },{ n }_{ 1 } and { l }_{ 2 },{ m }_{ 2 },{ n }_{ 2 } are DCs of the two lines inclined to each other at an angle \theta, then the DCs of the internal bisector of the angle between these lines are
  • \displaystyle \frac { { l }_{ 1 }+{ l }_{ 2 } }{ 2\sin { \frac { \theta  }{ 2 }  }  } ,\frac { { m }_{ 1 }+{ m }_{ 2 } }{ 2\sin { \frac { \theta  }{ 2 }  }  } ,\frac { { n }_{ 1 }+{ n }_{ 2 } }{ 2\sin { \frac { \theta  }{ 2 }  }  }
  • \displaystyle \frac { { l }_{ 1 }+{ l }_{ 2 } }{ 2\cos { \frac { \theta  }{ 2 }  }  } ,\frac { { m }_{ 1 }+{ m }_{ 2 } }{ 2\cos { \frac { \theta  }{ 2 }  }  } ,\frac { { n }_{ 1 }+{ n }_{ 2 } }{ 2\cos{ \frac { \theta  }{ 2 }  }  }
  • \displaystyle \frac { { l }_{ 1 }-{ l }_{ 2 } }{ 2\sin { \frac { \theta  }{ 2 }  }  } ,\frac { { m }_{ 1 }-{ m }_{ 2 } }{ 2\sin { \frac { \theta  }{ 2 }  }  } ,\frac { { n }_{ 1 }-{ n }_{ 2 } }{ 2\sin { \frac { \theta  }{ 2 }  }  }
  • \displaystyle \frac { { l }_{ 1 }-{ l }_{ 2 } }{ 2\cos { \frac { \theta  }{ 2 }  }  } ,\frac { { m }_{ 1 }-{ m }_{ 2 } }{ 2\cos { \frac { \theta  }{ 2 }  }  } ,\frac { { n }_{ 1 }-{ n }_{ 2 } }{ 2\cos{ \frac { \theta  }{ 2 }  }  }
Determine if the points (1,5) (2,3) and (-2,-11) are collinear.
  • True
  • False
In each of the following find the value of k, for which the points are collinear.
(i) (7,-2), (5,1), (3,k)
(ii) (8,1), (k,-4), (2,-5)
  • (i) k = 4
  • (i) k = 5
  • (ii) k = 3
  • (ii) k = 2
Equation of a plane containing L_{1} and L_{2} is
  • x + y + z = 0
  • 3x -2y -z =0
  • x -3y + 2z = 0
  • x + y + z = 42
The acute angle between the lines x =-2 + 2t, y = 3 -4t, z = -4 + t and x= 2 -t, y= 3 + 2t, z= -4 + 3t is
  • \displaystyle \sin^{-1}\frac{1}{\sqrt{3}}
  • \displaystyle \cos^{-1}\frac{1}{\sqrt{6}}
  • \displaystyle \cos^{-1}\frac{1}{\sqrt{5}}
  • \displaystyle \cos^{-1}2/3
Find the value of p for which the points (-5,1), (1,p) and (4, -2) are collinear.
  • 1
  • 0
  • -1
  • 2
If the foot of the perpendicular from the origin to a plane is \displaystyle \left ( a,b,c \right ), the equation of the plane is
  • \displaystyle \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=3
  • \displaystyle ax+by+cz=3
  • \displaystyle ax+by+cz=a^2+b^2+c^2
  • \displaystyle ax+by+cz=a+b+c
Let N be the foot of the perpendicular of length p, from the origin to a plane and l, m, n be the direction cosines of ON, the equation of the plane is
  • px\, +\, my\, +\, nz= l
  • lx\, +\, py\, +\, nz= m
  • lx\, +\, my\, +\, pz= n
  • lx\, +\, my\, +\, nz= p
For what value of m, the points (3,5), (m,6) and \begin{pmatrix} \dfrac { 1 }{ 2 },\dfrac {15 }{ 2 } \end{pmatrix} are collinear?
  • 9
  • 5
  • 3
  • 2
Find the direction cosines l,m,n of a line which are connected by the relation l+m-n=0 and 2ml-2mn+nl=0
  • \displaystyle \frac { -2 }{ \sqrt { 6 } } ,\frac { 1 }{ \sqrt { 6 } } ,\frac { -1 }{ \sqrt { 6 } }
  • \displaystyle \frac { 2 }{ \sqrt { 6 } } ,\frac { -1 }{ \sqrt { 6 } } ,\frac { 1 }{ \sqrt { 6 } }
  • \displaystyle \frac { -2 }{ \sqrt { 6 } } ,\frac { -1 }{ \sqrt { 6 } } ,\frac { -1 }{ \sqrt { 6 } }
  • \displaystyle \frac { 2 }{ \sqrt { 6 } } ,\frac { 1 }{ \sqrt { 6 } } ,\frac { 1 }{ \sqrt { 6 } }
The direction ratios of two lines are 1,-2,-2 and 0,2,1. The direction cosines of the line perpendicular to the above lines are 
  • \displaystyle \frac { 2 }{ 3 } ,\frac { -1 }{ 3 } ,\frac { 2 }{ 3 }
  • \displaystyle \frac { -1 }{ 3 } ,\frac { 2 }{ 3 } ,\frac { 2 }{ 3 }
  • \displaystyle \frac { 1 }{ 4 } ,\frac { 3 }{ 4 } ,\frac { 1 }{ 2 }
  • None of these
If the projection of a line segment on x,y and z axes are respectively 3,4 and 5, then the length of the line segment is
  • 3\sqrt { 2 }
  • 5\sqrt { 2 }
  • 6\sqrt { 2 }
  • None of these
Lines OA,OB are drawn from O with direction cosines proportional to (1,-2,-1),(3,-2,3). Find the direction cosines of the normal to the plane AOB
  • \displaystyle \left< \pm \frac { 4 }{ \sqrt { 29 } } \pm \frac { 3 }{ \sqrt { 29 } } \pm \frac { -2 }{ \sqrt { 29 } } \right>
  • \displaystyle \left< \pm \frac { 2 }{ \sqrt { 29 } } \pm \frac { 3 }{ \sqrt { 29 } } \pm \frac { -2 }{ \sqrt { 29 } } \right>
  • \displaystyle \left< \pm \frac { 8 }{ \sqrt { 29 } } \pm \frac { 6 }{ \sqrt { 29 } } \pm \frac { -2 }{ \sqrt { 29 } } \right>
  • \displaystyle \left< \pm \frac { 8 }{ \sqrt { 29 } } \pm \frac { 3 }{ \sqrt { 29 } } \pm \frac { -2 }{ \sqrt { 29 } } \right>
If the points (p,0), (0,q) and (1,1) are collinear, then \dfrac { 1 }{ p }+\dfrac { 1 }{ q } is equal to:
  • -1
  • 1
  • 2
  • 0
The angle between the straight lines whose direction cosines are given by 2l+2m-n=0,mn+nl+lm=0, is
  • \displaystyle \frac { \pi }{ 2 }
  • \displaystyle \frac { \pi }{ 3 }
  • \displaystyle \frac { \pi }{ 4 }
  • None of these
Equation of the line L is -
  • \vec{r} = 2\hat{k} + \lambda(\hat{i} + \hat{k})
  • \vec{r} = 2\hat{k} + \lambda(2\hat{j} + \hat{k})
  • \vec{r} = 2\hat{k} + \lambda(\hat{j} + \hat{k})
  • none of these
If direction ratios of the normal of the plane which contains the lines \displaystyle\frac{x-2}{3}=\displaystyle\frac{y-4}{2}=\displaystyle\frac{z-1}{1}\;\&\;\displaystyle\frac{x-6}{3}=\displaystyle\frac{y+2}{2}=\displaystyle\frac{z-2}{1} are (a,\,1,\,-26), then a is equal to
  • 5
  • 6
  • 7
  • 8
The equation of the plane which contains the lines \vec{r}\, =\, \hat{i}\, +\, 2 \hat{j}\, -\, \hat{k}\, +\, \lambda\, (\hat{i}\, +\, 2\, \hat{j}\, -\, \hat{k}) and \vec{r}\, =\, \hat{i}\, +\, 2 \hat{j}\, -\, \hat{k}\, +\, \mu\, (\hat{i}\, +\, \hat{j}\, +\, 3 \hat{k}) must be
  • \vec{r}.\, (7 \hat{i}\, -\, 4\, \hat{j}\, -\, \hat{k})\, =\, 0
  • 7(x\, -\, 1)\, -\, 4(y\, -\, 2)\, -\, (z\, +\, 1)\, =\, 0
  • \vec{r}.\, (\hat{i}\, +\, 2\, \hat{j}\, -\, \hat{k})\, =\, 0
  • \vec{r}.\, (\hat{i}\, +\, \hat{j}\, +\, 3\, \hat{k})\, =\, 0
If the foot of the perpendicular from the origin to a plane is \left( a,b,c \right) , the equation of the plane is 
  • \displaystyle \frac { x }{ a } +\frac { y }{ b } +\frac { z }{ c } =3
  • ax+by+cz=3
  • ax+by+cz={a}^{2}+{b}^{2}+{c}^{2}
  • ax+by+cz=a+b+c
Are the points (1, 1), (2, 3) and (8, 11) collinear ?
  • collinear
  • Non collinear
  • coplaner
  • None of above
The planes 3x-y+z+1=0,5x+y+3z=0 intersect in the line PQ. The equation of the plane through the point (2,1,4) and perpendicular to PQ is
  • x+y-2z=5
  • x+y-2z=-5
  • x+y+2z=5
  • x+y+2z=-5
The direction ratios of a line followed by the insect during its journey from A to G along the shortest path are
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  • (3, 4, 5)
  • (0, 4, 5)
  • (7, 0, 5)
  • (0, 0, 1)
The direction cosines of a vector \displaystyle \hat {i} + \hat {j} + \sqrt {2} \hat {k} are 
  • \displaystyle \frac {1}{2} , \frac {1} {2} , 1
  • \displaystyle \frac {1}{\sqrt 2} , \frac {1} {\sqrt 2} , \frac {1} {2}
  • \displaystyle \frac {1}{2} , \frac {1} {2} , \frac {1} {\sqrt 2}
  • \displaystyle \frac {1}{\sqrt 2} , \frac {1} {\sqrt 2} , \frac {1} {\sqrt 2}
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
3y+4z-6=0
  • \left( 0,\cfrac { 24 }{ 25 } ,\cfrac { 18 }{ 25 }  \right)
  • \left( 0,\cfrac { 24 }{ 25 } ,\cfrac { 24 }{ 25 }  \right)
  • \left( 0,\cfrac { 18 }{ 25 } ,\cfrac { 24 }{ 25 }  \right)
  • None of these
If a line makes angles \alpha, \beta, \gamma and \delta with the diagonals of a cube, Then, \cos^2\alpha +\cos^2\beta +\cos^2\gamma +\cos^2\delta =\dfrac {a}{b}, where a and b are in lowest form, find a+b
  • 7
  • 6
  • 8
  • None of these
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
2x+3y+4z-12=0
  • \left( \cfrac { 24 }{ 29 } ,\cfrac { 36 }{ 49 } ,\cfrac { 48 }{ 29 }  \right)
  • \left( \cfrac { 24 }{ 49 } ,\cfrac { 36 }{ 49 } ,\cfrac { 48 }{ 49 }  \right)
  • \left( \cfrac { 24 }{ 29 } ,\cfrac { 36 }{ 29 } ,\cfrac { 48 }{ 29 }  \right)
  • \left( \cfrac { 24 }{ 49 } ,\cfrac { 36 }{29 } ,\cfrac { 48 }{ 49 }  \right)
If \alpha, \beta and \gamma are the angles which a half ray makes with the positive direction of the axes, then \sin^2\alpha+\sin^2\beta+\sin^2\gamma is equal to  
  • 1
  • 2
  • 0
  • -1
The equation of the plane containing the line \dfrac { x+1 }{ -3 } =\dfrac { y-3 }{ 2 } =\dfrac { z+2 }{ 1 } and the point \left( 0,7,-7 \right) is
  • x+y+z=1
  • x+y+z=2
  • x+y+z=0
  • None of these
If the three points A(1,6), B(3,-4) and C(x,y) are collinear, then the equation satisfying by x and y is
  • 5x+y-11=0
  • 5x+13y+5=0
  • 5x-13y+5=0
  • 13x-5y+5=0
If the line \vec{OR} makes angles \theta_1, \theta_2, \theta_3 with the planes XOY, YOZ, ZOX respectively, then \cos^2\theta_1+\cos^2\theta_2+\cos^2\theta_3 is equal to
  • 1
  • 2
  • 3
  • 4
The direction cosines of the line segment joining points (-3, 1, 2) and (1, 4, -10) is.
  • \displaystyle\frac{4}{13}, \frac{3}{13},- \frac{12}{13}
  • \displaystyle\frac{-4}{13}, \frac{3}{13}, -\frac{12}{13}
  • \displaystyle\frac{-4}{13}, \frac{-3}{13}, \frac{12}{13}
  • None of these
The direction cosine of a line which is perpendicular to both the lines whose direction ratios are 1, 2, 2 and 0, 2, 1 are
  • \displaystyle \frac { -2 }{ 3 } ,\frac { 1 }{ 3 } ,\frac { 2 }{ 3 }
  • \displaystyle \frac { 2 }{ 3 } ,\frac { -1 }{ 3 } ,\frac { 2 }{ 3 }
  • \displaystyle \frac { 2 }{ 3 } ,\frac { 1 }{ 3 } ,\frac { -2 }{ 3 }
  • \displaystyle \frac { 2 }{ 3 } ,\frac { -1 }{ 3 } ,\frac { -2 }{ 3 }
Three district points A, B and C with p.v.s. and \displaystyle \vec { a } ,\vec { b }  and \displaystyle \vec { c }  respectively are collinear if there exist non-zero scalars x, y, z such that
  • \displaystyle x\vec { a } +y\vec { b } +z\vec { c } =0 and \displaystyle x+y+z=0
  • \displaystyle x\vec { a } +y\vec { b } +z\vec { c } and \displaystyle x+y+z\neq 0
  • \displaystyle x\vec { a } +y\vec { b } +z\vec { c } \neq 0 and \displaystyle x+y+z=0
  • \displaystyle x\vec { a } +y\vec { b } +z\vec { c } =3 and \displaystyle x+y+z\neq 0
If a line makes angles \alpha, \beta, \gamma with the coordinate axes, then the value of \cos 2\alpha + \cos 2\beta + \cos 2\gamma is
  • 3
  • -2
  • 2
  • -1
The points (k -1, \ k +2), (k, \ k +1), (k +1, \ k) are collinear for 
  • any value of k
  • k=-\dfrac{1}{2} only
  • no value of k
  • integral values of k only
A plane passing through (-1, 2, 3) and whose normal makes equal angle with the coordinate axes is
  • x + y + z + 4 = 0
  • x - y + z + 4 = 0
  • x + y + z - 4 = 0
  • x + y + z = 0
If \cos { \alpha  } ,\cos { \beta  } ,\cos { \gamma  } are the direction cosines of a vector \vec { a } , then \cos { 2\alpha  } +\cos { 2\beta  } +\cos { 2\gamma  } is equal to
  • 2
  • 3
  • -1
  • 0
The vector equation of the plane which is at a distance of \cfrac { 3 }{ \sqrt { 14 }  } from the origin and the normal from the origin is 2\hat { i } -3\hat { j } +\hat { k } is
  • \vec { r } .(2\hat { i } -3\hat { j } +\hat { k } )=3
  • \vec { r } .(\hat { i } +\hat { j } +\hat { k } )=9
  • \vec { r } .(\hat { i } +2\hat { j } )=3
  • \vec { r } .(2\hat { i } +\hat { k } )=3
The direction ratios of two lines AB, AC are 1, -1, -1 and 2, -1,The direction ratios of the normal to the plane ABC are
  • 2, 3, -1
  • 2, 2, 1
  • 3, 2, -1
  • -1, 2, 3
If A(3, 4, 5), B(4, 6, 3), C(-1, 2, 4) and D(1, 0, 5) are such that the angle between the lines \overline{DC} and \overline{AB} is \theta then cos\,\theta =
  • \dfrac{7}{9}
  • \dfrac{2}{9}
  • \dfrac{4}{9}
  • \dfrac{5}{9}
If the angles made by a straight line with the coordinate axes are \alpha, \dfrac{\pi}{2}-\alpha, \beta then \beta=
  • 0
  • \dfrac{\pi}{6}
  • \dfrac{\pi}{2}
  • \pi
0:0:1


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