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CBSE Questions for Class 12 Commerce Maths Three Dimensional Geometry Quiz 6 - MCQExams.com

If (2,1,2) and (K,3,5) are the triads of direction ratios of two lines and the angle between them is 45, then a value of K is
  • 2
  • 3
  • 4
  • 6
The angle between two diagonals of a cube is.
  • 30o
  • 45o
  • cos1(13)
  • cos1(13)
If a line segment OP makes angles of π4 and π3 with X-axis and Y-axis, respectively. Then, the direction cosines are
  • 12,32,12
  • 12,12,12
  • 1,3,1
  • 1,13,1
If a line makes { 45 }^{ o }, { 60 }^{ o } with positive direction of axes x and y then the angles it makes with the z-axis is
  • { 30 }^{ o }
  • { 90 }^{ o }
  • { 45 }^{ o }
  • { 60 }^{ o }
The vector equation of a plane passing through a point whose. P.V. is \overrightarrow a and perpendicular to a vector \overrightarrow n, is
  • \overrightarrow r . \overrightarrow n = \overrightarrow a . \overrightarrow n
  • \overrightarrow r \times \overrightarrow n = \overrightarrow a \times \overrightarrow n
  • \overrightarrow r + \overrightarrow n = \overrightarrow a + \overrightarrow n
  • \overrightarrow r - \overrightarrow n = \overrightarrow a - \overrightarrow n
If direction cosines of a vector of magnitude 3 are \dfrac {2}{3}, -\dfrac {1}{3}, \dfrac {2}{3} and a > 0, then vector is ____
  • 2i + j + 2k
  • 2i - j + 2k
  • i - 2j + 2k
  • i + 2j + 2k
If a plane passing through the point (2, 2, 1) and is perpendicular to the planes 3x + 2y + 4z + 1 = 0 and 2x + y + 3z + 2 = 0. Then, the equation of the plane is
  • 2x - y - z - 1 = 0
  • 2x + 3y + z - 1 = 0
  • 2x + y + z + 3 = 0
  • x - y + z - 1 = 0
What is the sum of the squares of direction cosines of x-axis?
  • 0
  • \dfrac{1}{3}
  • 1
  • 3
What is the sum of the squares of direction cosines of the line joining the points (1, 2, -3) and (-2, 3, 1) ?
  • 0
  • 1
  • 3
  • \frac{2}{\sqrt{26}}
Direction ratios of the line which is perpendicular to the lines with direction ratios -1,2,2 and 0,2,1 are
  • 1,1,2
  • 2,-1,2
  • -2,1,2
  • 2,1,-2
What are the direction cosines of a line which is equally inclined to the positive directions of the axes?
  • \left( \cfrac { 1 }{ \sqrt { 3 } } ,\cfrac { 1 }{ \sqrt { 3 } } ,\cfrac { 1 }{ \sqrt { 3 } } \right)
  • \left( -\cfrac { 1 }{ \sqrt { 3 } } ,\cfrac { 1 }{ \sqrt { 3 } } ,\cfrac { 1 }{ \sqrt { 3 } } \right)
  • \left( -\cfrac { 1 }{ \sqrt { 3 } } ,-\cfrac { 1 }{ \sqrt { 3 } } ,\cfrac { 1 }{ \sqrt { 3 } } \right)
  • \left( \cfrac { 1 }{ 3 } ,\cfrac { 1 }{ 3 } ,\cfrac { 1 }{ 3 } \right)
If (1, -2, -2) and (0, 2, 1) are direction ratios of two lines, then the direction cosines of a perpendicular to both the lines are
  • \left( \dfrac{1}{3}, - \dfrac{1}{3} , \dfrac{2}{3}\right )
  • \left( \dfrac{2}{3}, - \dfrac{1}{3} , \dfrac{2}{3}\right )
  • \left( -\dfrac{2}{3}, - \dfrac{1}{3} , \dfrac{2}{3}\right )
  • \left( \dfrac{2}{\sqrt{14}}, - \dfrac{1}{\sqrt{14}} , \dfrac{3}{\sqrt{14}}\right )
The direction ratios of the line perpendicular to the lines with direction ratios 1, -2, -2 and 0, 2, 1 are
  • 2, -1, 2
  • -2, 1, 2
  • 2, 1, -2
  • -2, -1, -2
What are the direction ratios of normal to the plane 2x - y + 2z + 1 = 0 ?
  • (2, -1, 2)
  • (1, \frac{1}{2}, 1)
  • (1, -2, 1)
  • None of the above
The equation of the plane perpendicular to the yz- plane and passing through the point (1,-2,4) and (3,-4,5) is 
  • y+2z=5
  • 2y+z=5
  • y+2z=6
  • 2y+z=6
Let a vector \vec{r} make angles 60^o, 30^o with it and y-axes respectively.Find the angle \vec{r} make with z-axis.
  • 30^o
  • 60^o
  • 90^o
  • 120^o
What are the direction cosines of \vec{r} ?
  • \left (\frac{1}{2}, -\frac{\sqrt{3}}{2}, 0 \right )
  • \left (\frac{1}{2}, \frac{\sqrt{3}}{2}, 0 \right )
  • \left (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0 \right )
  • \left (-\frac{1}{2}, \frac{\sqrt{3}}{2}, 0 \right )
A straight line passes through (1, -2, 3) and perpendicular to the plane 2x + 3y - z = 7.Find the direction ratios of normal to plane.  
  • < 2, 3, -1 >
  • < 2, 3, 1 >
  • < -1, 2, 3 >
  • None of the above
What are the direction ratios of the line if it passes through the intersection of the planes x=3z+4 and y=2z-3?
  • (1,2,3)
  • (2,1,3)
  • (3,2,1)
  • (1,3,2)
The direction cosines of the straight line given by the planes x=0 and z=0 are
  • 1,0,0
  • 0,0,1
  • 1,1,0
  • 0,1,0
  • 0,1,1
The points, whose position vectors are 60i + 3j, 40i - 8j and ai - 52j collinear, if
  • a = 40
  • a = -40
  • a = 20
  • a = -20
The equation of the plane perpendicular to the line \cfrac { x-1 }{ 1 } =\cfrac { y-2 }{ -1 } =\cfrac { z+1 }{ 2 } and passing through the point (2,3,1) is
  • r.\left( \hat { i } +\hat { j } +2\hat { k } \right) =1
  • r.\left( \hat { i } -\hat { j } +2\hat { k } \right) =1
  • r.\left( \hat { i } +\hat { j } +2\hat { k } \right) =7
  • None of these
The point P(x, y, z) lies in the first octant and its distance from the origin is 12 units. If the position vector of P make 45^{\circ} and 60^{\circ} with the x-axis and y-axis respectively, then the coordinates of P are
  • (3\sqrt {3}, 6, 3\sqrt {2})
  • (4\sqrt {3}, 8, 4\sqrt {2})
  • (6\sqrt {2}, 6, 6,)
  • (6, 6, 6\sqrt {2})
  • (4\sqrt {2}, 8, 4\sqrt {3})
The direction cosines l,m,n of two lines are connected by the relations l+m+n=0, lm=0, then the angle between them is
  • \cfrac { \pi }{ 3 }
  • \cfrac { \pi }{ 4 }
  • \cfrac { \pi }{ 2 }
  • 0
The angle between the straight lines x - 1 = \dfrac{2y + 3}{3} = \dfrac{z + 5}{2} and x = 3r + 2;\, y = -2r - 1; \,z = 2, where r is a parameter, is
  • \dfrac{\pi}{4}
  • \cos^{-1} \left( \dfrac{-3}{\sqrt{182}} \right )
  • \sin^{-1} \left( \dfrac{-3}{\sqrt{182}} \right )
  • \dfrac{\pi}{2}
  • 0
The angle between the lines \dfrac {x-7}{1} = \dfrac {y+3}{-5} = \dfrac {z}{3} and \dfrac {2-x}{-7} = \dfrac {y}{2} = \dfrac {z+1}{5} is equal to :
  • \dfrac {\pi}{4}
  • \dfrac {\pi}{3}
  • \dfrac {\pi}{2}
  • \dfrac {\pi}{6}
  • 0
If \alpha ,\beta ,\gamma are the angles which a directed line makes with the positive directions of the coordinate axes, then \sin ^{ 2 }{ \alpha  } +\sin ^{ 2 }{ \beta  } +\sin ^{ 2 }{ \gamma  } is equal to
  • 1
  • 4
  • 3
  • 2
If the two lines \cfrac { x-1 }{ 2 } =\cfrac { 1-y }{ -a } =\cfrac { z }{ 4 } and \cfrac { x-3 }{ 1 } =\cfrac { 2y-3 }{ 4 } =\cfrac { z-2 }{ 2 } are perpendicular, then the value of a is equal to
  • -4
  • 5
  • -5
  • 4
  • -2
The number of straight lines that are equally inclined to the three-dimensional coordinate axes, is
  • 2
  • 4
  • 6
  • 8
If the direction ratios of two lines are given by 3lm-4ln+mn=0 and l+2m+2n=0, then, the angle between the lines is
  • \cfrac { \pi }{ 2 }
  • \cfrac { \pi }{ 3 }
  • \cfrac { \pi }{ 4 }
  • \cfrac { \pi }{ 6 }
The angle between two diagonals of a cube will be
  • \sin ^{ -1 }{ \left( \dfrac { 1 }{ 3 } \right) }
  • \cos ^{ -1 }{ \left( \dfrac { 1 }{ 3 } \right) }
  • Variable
  • None of these
A = \begin{bmatrix} l_{1}& m_{1} & n_{1}\\ l_{2} & m_{2} & n_{2}\\ l_{3} & m_{3} & n_{3}\end{bmatrix} and B = \begin{bmatrix}p_{1} & q_{1} & r_{1}\\ p_{2} & q_{2} & r_{2}\\ p_{3} & q_{3} & r_{3}\end{bmatrix} where p_{1}, q_{1}, r_{1} are the co-factors of the elements l_{i},m_{i},n_{i} for i = 1, 2, 3. If (l_{1}, m_{1}, n_{1}), (l_{2}, m_{2}, n_{2}) and (l_{3},m_{3}, n_{3}) are the direction cosines of three mutually perpendicular lines then (p_{1}, q_{1}, r_{1}), (p_{2}, q_{2}, r_{2}) and (p_{3}, q_{3}, r_{3}) are
  • The direction cosines of three mutually perpendicular lines
  • The direction ratios of three mutually perpendicular lines which are not direction cosines
  • The direction cosines of three lines which need not be perpendicular
  • The direction ratios but not the direction cosines of three lines which need not be perpendicular
If a line makes the angle \alpha ,\beta ,\gamma with three dimensional coordinate axes respectively, then \cos { 2\alpha  } +\cos { 2\beta  } +\cos { 2\gamma  } is equal to
  • -2
  • -1
  • 1
  • 2
If the direction cosines of a line are \left (\dfrac {1}{c}, \dfrac {1}{c}, \dfrac {1}{c}\right ), then
  • 0 < c < 1
  • c > 2
  • c = \pm \sqrt {2}
  • None of these
A line makes the same angle \theta with each of the x and z-axes. If the angle \beta, which it makes with y-axis, is such that \sin ^{ 2 }{ \beta  } =3\sin ^{ 2 }{ \theta  } , then \cos ^{ 2 }{ \theta  } is equal to
  • \dfrac { 2 }{ 3 }
  • \dfrac { 1 }{ 5 }
  • \dfrac { 3 }{ 5 }
  • \dfrac { 2 }{ 5 }
If a lines makes angles \alpha , \beta , \gamma , \delta with four digonals of a cube. Then \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma + \cos ^2 \delta will be :
  • 4/3
  • 3/4
  • 1/4
  • None of these
The acute angle between the lines whose direction ratios are given by l+m-n=0 and { l }^{ 2 }+{ m }^{ 2 }-{ n }^{ 2 }=0, is
  • 0
  • \pi /6
  • \pi /4
  • \pi /3
ABCD is a trapezium in which AB and CD are parallel sides. If l(AB) =  3 l (CD) and \bar {DC} = 2 \hat {i} - 5 \hat {k}. Then vector  \bar {AB} is
  • \frac{3}{\sqrt{29} } (2 \hat{i} - 5 \hat {k})
  • \frac {\sqrt{29}} {3} (5 \hat{i} - 2 \hat {k})
  • -6 \hat {i} + 15 \hat {k}
  • A or B
If a line makes { 45 }^{ o },{ 60 }^{ o } with positive direction of axes x and y then the angle it makes with the z-axis is:
  • { 30 }^{ o }\quad
  • { 90 }^{ o }\quad
  • { 45 }^{ o }\quad
  • { 60 }^{ o }\quad
The dc's (l,m,n) of two lines are connected between the relation l + m + n = 0, \ lm = 0, then the angle between the lines is 
  • \displaystyle\frac{\pi}{3}
  • \dfrac{\pi}{4}
  • \dfrac{\pi}{2}
  • \dfrac{\pi}{6}
The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes 2x+y-2z=5 and 3x-6y-2z=7 is?
  • 14x+2y-15z=1
  • -14x+2y+15z=3
  • 14x-2y+15z=27
  • 14x+2y+15z=31
ABC is a triangle where A(2,3,5), B(-1,3,2) and C(\lambda , 5, \mu). Let the median through A is equally inclined to the axes.
The value of \mu - \lambda is equal to:
  • 1
  • 2
  • 3
  • 4
The point of intersection of the line joining the points (-3, 4, -8) and (5, -6, 4) with the XY-plane is
  • \left (\dfrac {7}{3}, -\dfrac {8}{3}, 0\right )
  • \left (-\dfrac {7}{3}, -\dfrac {8}{3}, 0\right )
  • \left (-\dfrac {7}{3}, \dfrac {8}{3}, 0\right )
  • \left (\dfrac {7}{3}, \dfrac {8}{3}, 0\right )
Find the direction cosines of the line which is perpendicular to the lines with direction cosines proportional to 1, -2, -2 and 0, 2, 1.
  • \dfrac{1}{3}, \dfrac{2}{3}, -\dfrac{1}{3}
  • \dfrac{2}{3}, -\dfrac{1}{3}, \dfrac{2}{3}
  • 1, -\dfrac{1}{3}, \dfrac{2}{3}
  • None of these
The measure of acute angle between the lines whose direction ratios are 3, 2, 6 and -2, 1, 2 is __________.
  • \cos^{-1}\left (\dfrac {1}{7}\right )
  • \cos^{-1}\left (\dfrac {8}{15}\right )
  • \cos^{-1}\left (\dfrac {1}{3}\right )
  • \cos^{-1}\left (\dfrac {8}{21}\right )
The perpendicular distance from the point (3,1,1) on the plane passing through the point (1,2,3) and containing the line, \vec { r } =\hat { i } +\hat { j } +\lambda \left( 2\hat { i } +\hat { j } +4\hat { k }  \right) , is:
  • \cfrac { 1 }{ \sqrt { 11 } }
  • \cfrac { 4 }{ \sqrt { 41 } }
  • 0
  • \cfrac { 3 }{ \sqrt { 11 } }
If the angle between the lines, \displaystyle\frac{x}{2}=\frac{y}{2}=\frac{z}{1} and \displaystyle\frac{5-x}{-2}=\frac{7y-14}{P}=\frac{z-3}{4} is \cos^{-1}\displaystyle \left(\frac{2}{3}\right)
then p is equal to?
  • -\displaystyle\frac{7}{4}
  • \displaystyle\frac{2}{7}
  • -\displaystyle\frac{4}{7}
  • \displaystyle\frac{7}{2}
The acute angle between two lines such that the direction cosines l,\  m,\ n of each of them satisfy the equation l+m+n=0 and l^2+m^2-n^2=0 is 
  • 30^\circ
  • 45^\circ
  • 60^\circ
  • 15^\circ
The direction cosines of the ray P(1,-2,4) and Q(-1,1,-2) are
  • \left(-2,-3,-6\right)
  • \left(2,-3,-6\right)
  • \left(\dfrac{2}{7},\dfrac{3}{7},\dfrac{6}{7}\right)
  • \left(-\dfrac{2}{7},\dfrac{3}{7},-\dfrac{6}{7}\right)
The measure of the angle between the lines, whose direction numbers are l,m,n and m-n,n-l,l-m is ______
  • \cfrac { \pi }{ 4 }
  • \cfrac { \pi }{ 6 }
  • \cfrac { \pi }{ 2 }
  • \cfrac { \pi }{ 3 }
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