CBSE Questions for Class 12 Commerce Maths Three Dimensional Geometry Quiz 8 - MCQExams.com

ABC is a triangle where $$A = ( 2,3,5 ) , B = ( - 1,2,2 )$$ and $$C (\lambda,5 , \mu )$$ if the median through A is equally inclined to the positive axis then $$\lambda + \mu$$ is 
  • 7
  • 6
  • 15
  • 9
Projection of a vector on $$3$$ coordinate axes are $$6, - 3, 2$$ respectively. Then DC's of vector are-
  • $$6, - 3, 2$$
  • $$\dfrac{6}{5},\dfrac{{ - 3}}{5},\dfrac{2}{5}$$
  • $$\dfrac{-6}{7},\dfrac{{ - 3}}{2},\dfrac{2}{7}$$
  • $$\dfrac{  6}{7},\dfrac{{ - 3}}{7},\dfrac{2}{7}$$
If the foot of the perpendicular from $$(0,0,0)$$ to a plane is $$P(1,2,2)$$. Then, the equation of the plane is
  • $$-x+2y+8z-9=0$$
  • $$x+2y+2z-9=0$$
  • $$x+y+z-5=0$$
  • $$x+2y-3z+3=0$$
$$P\left(1,1,1\right)$$ and $$Q\left(\lambda,\lambda,\lambda\right)$$ are two points in the space such that $$PQ=\sqrt{27},$$ the value of $$\lambda$$ can be 
  • $$-4$$
  • $$-2$$
  • $$2$$
  • $$0$$
$$A(2,3,7),B(-1,3,2)$$ and $$C(q,5,r)$$ are the vertices of $$\Delta ABC$$. If the median through A is equally inclined to the coordinate axes then the coordinates of the vertex C is
  • $$(7,5,14)$$
  • $$(7,5,12)$$
  • $$(7,5,10)$$
  • $$(7,5,16)$$
If the points $$\bar a + \bar b,\bar a - \bar b,\bar a + k\bar b$$ are collinear, then  
  • $$k$$ has only one real value
  • $$k$$ has two real value
  • $$k$$ has no real values
  • $$k$$ has infinite number of real values
A line $$AB$$ in three-dimensional space males angles $${45}^{o}$$ and $${120}^{o}$$ with the positive x-axis and the positive y-axis respectively. If $$AB$$ makes an acute angle $$\theta$$ with the positive z-axis, then $$\theta$$ equal
  • $${45}^{o}$$
  • $${60}^{o}$$
  • $${75}^{o}$$
  • $${30}^{o}$$
If the points $$(\alpha, - 1), (2, 1)$$ and $$(4, 5)$$ are collinear, then find $$\alpha $$ by vector method.
  • $$4$$
  • $$1$$
  • $$8$$
  • None of these
The Cartesian equation of line $$6x - 2 = 3y + 1 = 2z - 2$$ is given by

  • $$\dfrac{{3x - 1}}{3} = \dfrac{{3y + 1}}{6} = \dfrac{{z - 1}}{3}$$
  • $$\dfrac{{3x + 1}}{3} = \dfrac{{3y - 1}}{6} = \dfrac{{z - 1}}{3}$$
  • $$\dfrac{{3x - 1}}{3} = \dfrac{{3y - 1}}{6} = \dfrac{{z - 1}}{3}$$
  • $$\dfrac{{3x - 1}}{6} = \dfrac{{3y - 1}}{3} = \dfrac{{z - 1}}{3}$$
The direction ratios of the joining $$A(1,\,2,\ 1)$$ and $$(2,\ 1,\ 2)$$ are
  • $$3,\ 3,\ 3$$
  • $$-1,\ 1,\ -1$$
  • $$3,\ 1,\ 3$$
  • $$\dfrac{1}{\sqrt{3}},\ \dfrac{1}{\sqrt{3}},\ \dfrac{1}{\sqrt{3}}$$
The direction ratios of $$AB$$ are $$- 2, 2, 1$$ . If coordinates of A are $$( 4,1,5 )$$ and $$l( A B ) = 6$$ , then coordinates of $$ B $$ ?
  • $$( 0,5 , - 7 )$$
  • $$( 8 , - 3,3 )$$
  • $$( 0,7,5 )$$
  • $$( 8,3,3 )$$
If the lines $${L}_{1}\ and {L}_{2}$$ are given by $$\bar { r } =\left( \bar { i } +2\bar { j } -\bar { k }  \right) +t\left( \bar { 2i } -3\bar { j } +\bar { k }  \right) \ and\bar { r } =\left( \bar { i } +\bar { j } +\bar { k }  \right) +s\left( 2\bar { i } +\bar { j } -\bar { k }  \right)$$, then 
  • $${L}_{1}\ and {L}_{2}$$ are perpendicular
  • $${L}_{1}\ and {L}_{2}$$ are parallel
  • $$\left ({L}_{1},{L}_{2}\right)={45}^{o}$$
  • $$\left ({L}_{1},{L}_{2}\right)={60}^{o}$$
The vector equation of line passing through the point $$(-1,-1,2)$$ and parallel to the line $$2x-2=3y+1=6z-2$$
  • $$(-\hat { i } -\hat { j } +2\hat { k } )+\lambda (3\hat { i } +2\hat { j } +\hat { k } )$$
  • $$(-\hat { i } -\hat { j } +2\hat { k } )+\lambda (2\hat { i } +3\hat { j } +6\hat { k } )$$
  • $$(-\hat { i } -\hat { j } +2\hat { k } )+\lambda (\hat { i } +2\hat { j } +3\hat { k } )$$
  • $$(-\hat { i } -\hat { j } +2\hat { k } )+\lambda (2\hat { i } +3\hat { j } +\hat { k } )$$
If $$\bar {a}, \bar {b}$$ and $$\bar {c}$$ are non-zero non collinear vectors and $$\theta(\neq 0 , \pi)$$ is the angle between $$\bar {b}$$ and $$\bar {c}$$ if $$(\bar {a}\times \bar {b}) \times \bar {c}=\dfrac {1}{2} |\bar {b}|\bar {c}|\bar {a}$$. then $$\sin \theta =$$
  • $$\sqrt{\dfrac{2}{3}}$$
  • $$\dfrac{\sqrt{3}}{2}$$
  • $$\dfrac{4\sqrt{2}}{3}$$
  • $$\dfrac{2\sqrt{2}}{3}$$
If $$A=(1,2,-1), B=(2,0,3), C=(3,-1,2)$$ then the angle between $$\overline { AB } $$ and $$\overline { AC } $$ is
  • $${0}^{o}$$
  • $${90}^{o}$$
  • $$\cos ^{ -1 }{ \left( \cfrac { 20 }{ \sqrt { 21 } \sqrt { 22 } } \right) } $$
  • $$\cos ^{ -1 }{ \left( \cfrac { 15 }{ \sqrt { 21 } \sqrt { 11 } } \right) } $$
A line d.c's proportional to $$(2,1,2)$$ meets each of the lines $$x=y+a=z$$ and $$x+a=2y=2z$$. Then the coordinates of each of the points of intersection are given by
  • $$(3a,2a,3a); (a,a,2a)$$
  • $$(3a,2a,3a); (a,a,a)$$
  • $$(3a,3a,3a); (a,a,a)$$
  • $$(2a,3a,3a); (2a,a,a)$$
If $$\frac{x-14}{l}=\frac{y-2}{m}=\frac{z+1}{n}$$ is the equation of the line through (1,2,-1) and (-1,0,1), then (l,m,n) is 
  • (-1,0,1)
  • (1,1,-1)
  • (1,2,-1)
  • (0,1,0)
If $$A = (1,2,3) , B  = (2,10,1), Q$$ are collinear points and $$Q_{x}=-1$$ then $$Q_{z}$$ is
  • $$-3$$
  • $$7$$
  • $$-14$$
  • $$-7$$
The direction cosines to two lines at right angles are (1,2,3) and (-2,$$\frac{1}{2}$$,$$\frac{1}{3}$$), then the direction cosine perpendicular to both given lines are:
  • $$\sqrt{\frac{25}{2198}}$$,$$\sqrt{\frac{19}{2198}}$$,$$\sqrt{\frac{729}{2198}}$$
  • $$\sqrt{\frac{24}{2198}}$$,$$\sqrt{\frac{38}{2198}}$$,$$\sqrt{\frac{730}{2198}}$$
  • $$\frac{1}{3}$$,-2,$$\frac{-7}{2}$$
  • None of the above
The direction cosines of a vector $$ \overrightarrow { A }  $$ are $$ \cos \alpha = \frac {4} { 5 \sqrt {2}} , \cos\beta =\frac { 1 }{ \sqrt { 2 }  } , \cos\gamma =\frac { 3 }{ 5\sqrt { 2 }  }  $$ then, the vector $$ \overrightarrow {A} $$ is 
  • $$ 4\hat { i } +\hat { j } +3\hat { k } $$
  • $$ 4\hat { i } +5\hat { j } +3\hat { k } $$
  • $$ 4\hat { i } -5\hat { j } +3\hat { k } $$
  • $$ 4\hat { i } -\hat { j } -3\hat { k } $$
If  $$\bar { a }, \bar { b }, \bar { c }$$ are non-coplaner vector , then the vectors $$2\bar { a }- 4\bar { b }+ 4\bar { c }, \bar { a }- 2\bar { b }+ 4\bar { c }$$ and $$-\bar { a }+ 2\bar { b }+ 4\bar { c }$$ are parellel.
  • True
  • False
The angle between the lines $$\frac{{x - 2}}{3} = \frac{{y + 1}}{{ - 2}},z = 2$$ and $$\frac{{x - 1}}{1} = \frac{{2y + 3}}{3} = \frac{{z + 5}}{2}$$ is equal to 
  • $$\pi /2$$
  • $$\pi /3$$
  • $$\pi /6$$
  • none of these
The direction ratios of the line joining the points $$(4, 3, -5)$$ and $$(-2, 1, -8)$$ are
  • $$\dfrac{6}{7}, \dfrac{2}{7}, \dfrac{3}{7}$$
  • $$6, 2, 3$$
  • $$5,8,0$$
  • $$3,7,9$$
If $$\left(\dfrac {1}{2},\dfrac {1}{3},n\right)$$ are the direction cosines of a line then the value of $$n$$ is
  • $$\dfrac {\sqrt {23}}{6}$$
  • $$\dfrac {23}{6}$$
  • $$\dfrac {2}{3}$$
  • $$\dfrac {3}{2}$$
The angle between the pair of lines with direction ratios (1, 1, 2) and $$(\sqrt{3} - 1, -\sqrt{3} - 1, 4)$$ is 
  • $$30^o$$
  • $$45^o$$
  • $$60^o$$
  • $$90^o$$
A line makes angles $$\alpha,\beta,\gamma,\delta$$ with the four diagonals of a cube then $$\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma+\cos^{2}\delta$$ is equal to
  • $$1$$
  • $$4/3$$
  • $$3/4$$
  • $$4/5$$
The direction ratios of the line, given by the planes x - y + z - 5 = 0, x - 3y - 6 = 0 are 
  • (3, 1, -2)
  • (2, -4, 1)
  • (1,-1, 1)
  • (0,2,1)
If $$\overline { O A } = 3 \overline { i } + \overline { j } - \overline { k }$$, $$| \overline { A B } | = 2 \sqrt { 6 }$$ and AB has the direction ratios 1, -1 , 2 then $$| O B | =$$ 
  • $$\sqrt { 35 }$$
  • $$\sqrt { 41 }$$
  • $$\sqrt { 26 }$$
  • $$\sqrt { 55 }$$
The direction cosines of a vector A are $$\cos { \alpha  } =\frac { 4 }{ 5\sqrt { 2 }  } ,$$ $$cos \beta =\frac { 1 }{ \sqrt { 2 }  } ,$$ and $$cos \gamma = \frac{ 3 }{ 5\sqrt { 2 }  } ,$$ then vector A is
  • $$4i+j+3k$$
  • $$4i+5j+3k$$
  • 4i-5j-3k
  • none
The vector $$a = \alpha 1 + 2 j + \beta k$$ lies in the plane of the vectors $$b = i + jt$$ and $$c = j + k$$ and bisects the angle between $$b$$ and $$c$$. Then which one of the following gives possible values $$\alpha$$ and $$\beta$$.
  • $$\alpha = 1 , \beta = 2$$
  • $$\alpha = 2 , \beta = 1$$
  • $$\alpha = 1 , \beta = 1$$
  • $$\alpha = 2 , \beta = 2$$
Let $$l_{1},\ m_{1},\ n_{1};\ l_{2},\ m_{2},\ n_{2};\ l_{3},\ m_{3},\ n_{3}$$ be the direction cosines of three mutually perpendicular line then $$\begin{vmatrix} { l }_{ 1 } & m_{ 1 } & n_{ 1 } \\ { l }_{ 2 } & m_{ 2 } & n_{ 2 } \\ { l }_{ 3 } & m_{ 3 } & n_{ 3 } \end{vmatrix}$$
  • $$0$$
  • $$\pm 1$$
  • $$\pm 2$$
  • $$\pm \dfrac{1}{2}$$
The point where $$\vec{ x }$$ which is perpendicular to $$(2,-3,1)$$ and $$(1,-2,3)$$ and which satisfies the condition $$\vec { x } \cdot ( \hat { i } + 2 \hat { j } - 7 \hat{ k } ) = 10$$
  • $$\left(3,5,1\right)$$
  • $$\left(7,-5,1\right)$$
  • $$\left(3,-5,1\right)$$
  • $$\left(7,5,1\right)$$
The equation of the plane through $$\left(0,-5,1\right)$$ which is perpendicular to the planes $$2x+4y+2z+3=0$$,$$2x+5y+3z+4=0$$ is 
  • $$x+y+z=6$$
  • $$x-y+z=6$$
  • $$x-y-z=6$$
  • $$x+y+z+6=0$$
If $$A(p,q,r)$$ and $$B=(p\prime ,q\prime ,r\prime )$$ are two points on the line $$\lambda x=\mu y=yz$$ such that $$OA=3,OB=4$$ then $$pp\prime +qq\prime +rr\prime $$ is equal to 
  • $$7$$
  • $$12$$
  • $$5$$
  • $$None$$ $$of$$ $$these$$
The angle between the lines whose de's satisfy the equation $$l+m+m=0$$ and $$l^2+m^{2}-n^{2}=0$$ is 
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{\pi}{4}$$
The angle between the lines, whose direction ratios are $$1,1,2$$ and $$\sqrt { 3 } - 1 , - \sqrt { 3 } - 1,4 ,$$ is
  • $${45} ^ { \circ }$$
  • $${30} ^ { \circ }$$
  • $${60} ^ { \circ }$$
  • $${90} ^ { \circ }$$
The directions cosines of the line which is perpedicular to the lines whose direction cosines are proportional to (1, -1, 2) and (2,-1,-1) are:
  • $$\dfrac { 1 }{ \sqrt { 35 } } ,-\dfrac { 5 }{ \sqrt { 35 } } \frac { 3 }{ \sqrt { 35 } } $$
  • $$-\dfrac { 1 }{ \sqrt { 35 } } ,\frac { 5 }{ \sqrt { 35 } } \dfrac { 3 }{ \sqrt { 35 } } $$
  • $$\dfrac { 1 }{ \sqrt { 35 } } ,\dfrac { 5 }{ \sqrt { 35 } } \frac { 3 }{ \sqrt { 35 } } $$
  • None of these
If a plane passes through the point $$(1, 1, 1)$$ and is perpendicular to the line $$\dfrac{x-1}{3}=\dfrac{y-1}{0}=\dfrac{z-1}{4}$$ then its perpendicular distance from the origin is 
  • $$\dfrac{3}{4}$$
  • $$\dfrac{4}{3}$$
  • $$\dfrac{7}{5}$$
  • $$1$$
The direction cosines of a line equally inclined to three mutually perpendicular lines having direction cosines as $$l_{1},m_{1},n_{1};l_{2},m_{2},n_{2}$$ and $$l_{3},m_{3},n_{3}$$ are
  • $$l_{1}+ l_{2}+ l_{3},m_{1}+m_{2}+m_{3},n_{1}+n_{2}+n_{3}$$
  • $$\dfrac{l_{1}+l_{2}+l_{3}}{\sqrt{3}},\dfrac{m_{1}+m_{2}+m_{3}}{\sqrt{3}},\dfrac{n_{1}+n_{2}+n_{3}}{\sqrt{3}}$$
  • $$\dfrac{l_{1}+l_{2}+l_{3}}{3},\dfrac{m_{1}+m_{2}+m_{3}}{3},\dfrac{n_{1}+n_{2}+n_{3}}{3}$$
  • $$None\ of\ these$$
$$l=m=n=1$$ represent the direction cosines of the 
  • $$x-$$ axis
  • $$y-$$ axis
  • $$z-$$ axis
  • $$none\ of\ these$$
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 direction ratios of the line
$$x-y+z-5=\quad 0\quad =\quad x-3y-6\quad are$$
  • $$3,1,-2$$
  • $$2,-4,1$$
  • $$\frac { 3 }{ \sqrt { 14 } } ,\frac { 1 }{ \sqrt { 14 } } ,\frac { -2 }{ \sqrt { 14 } } $$
  • $$\frac { 2 }{ \sqrt { 41 } } ,\frac { -4 }{ \sqrt { 41 } } ,\frac { 1 }{ \sqrt { 41 } } $$
The direction cosines of a line equally inclined to three mutually perpendicular lines having direction cosines as $$l_1$$, $$m_1$$, $$n_1$$ : $$l_2$$, $$m_2$$, $$n_2$$ and   $$l_3$$, $$m_3$$, $$n_3$$ are
  • $$l_1$$+$$l_2$$+$$l_3$$, $$m_1$$+$$m_2$$+$$m_3$$, $$n_1$$+$$n_2$$+$$n_3$$
  • $$\dfrac{l_1+l_2+l_3}{\sqrt{3}} $$, $$\dfrac{m_1+m_2+m_3}{\sqrt{3}} $$, $$\dfrac{n_1+n_2+n_3}{\sqrt{3}} $$
  • $$\dfrac{l_1+l_2+l_3}{3} $$, $$\dfrac{m_1+m_2+m_3}{3} $$, $$\dfrac{n_1+n_2+n_3}{3} $$
  • None of these
If  $$A(3\hat { i } +2\hat { j } +3\hat { k } ),B(-\hat { i } -\hat { j } +8\hat { k } ),C(-4\hat { i } +4\hat { j } +6\hat { k } )$$  are the vertices of a triangle then the equation of the line passing through the circumcentre and parallel to  $$\vec { A B }$$  is
  • $$\hat { r } =\left( -\dfrac { 4 }{ 3 } \hat { i } +\dfrac { 5 }{ 3 } \hat { j } +\dfrac { 17 }{ 3 } \hat { k } \right) +t(2\hat { i } +3\hat { j } -5\hat { k } )$$
  • $$\hat { r } =\left( \dfrac { 4 }{ 3 } \hat { i } +\dfrac { 5 }{ 3 } \hat { j } +\dfrac { 17 }{ 3 } \hat { k } \right) +t(2\hat { i } +3\hat { j } -5\hat { k } )$$
  • $$\hat { r } =\left( -\dfrac { 4 }{ 3 } \hat { i } +\dfrac { 5 }{ 3 } \hat { j } -\dfrac { 17 }{ 3 } \hat { k } \right) +t(2\hat { i } +3\hat { j } -5\hat { k } )$$
  • $$\hat { r } =\left( \dfrac { 4 }{ 3 } \hat { i } -\dfrac { 5 }{ 3 } \hat { j } +\dfrac { 17 }{ 3 } \hat { k } \right) +t(2\hat { i } +3\hat { j } -5\hat { k } )$$
The cartesian from of equation a line passing through the point position vector $$2\hat{i}-\hat{j}+2\hat{k}$$ and is in the direction of $$-2\hat{i}+\hat{j}+\hat{k}$$, is
  • $$\dfrac{x-2}{-2}=\dfrac{y+1}{1}=\dfrac{z-2}{1}$$
  • $$\dfrac{x+4}{-2}=\dfrac{y-1}{1}=\dfrac{z+2}{1}$$
  • $$\dfrac{x+2}{4}=\dfrac{y-1}{-1}=\dfrac{z-1}{2}$$
  • $$None \ of \ these$$
If $$\cos { \alpha ,\quad \cos { \beta ,\quad \cos { \gamma  }  }  }$$   are the direction cosine of a line, then find the value of $${ cos }^{ 2 }\alpha +\left( \cos { \beta +\sin { \gamma  }  }  \right)$$$$\left( \cos { \beta - \sin { \gamma  } }  \right)$$
  • $$2$$
  • $$0$$
  • $$-1$$
  • $$1$$
$$\dfrac { x - 2 } { 1 } = \dfrac { y - 3 } { 1 } = \dfrac { z - 4 } { - 1 }$$ & $$\dfrac { x - 1 } { k } = \dfrac { y - 4 } { 2 } = \dfrac { z - 5 } { 2 }$$ are coplanar then k=?
  • any value
  • exactly one value
  • exactly $$2$$ values
  • exactly $$3$$ values
The plane through (1, 1, 1) (1, -1, 1) and (-7, -3, -5) is
  • Parallel to x-axis.
  • Parallel to y-axis.
  • Perpendicular to y-axis.
  • Perpendicular to x-axis.
Direction ratio of line given by $$\dfrac { x-1 }{ 3 } =\dfrac { 6-2y }{ 10 } =\dfrac { 1-z }{ -7 } $$ are:
  • $$<3,10,-7>$$
  • $$<3,-5,7>$$
  • $$<3,5,7>$$
  • $$<3,5,-7>$$
A normal to the plane $$  x=2  $$ is...
  • $$
    (0,1,1)
    $$
  • $$
    (2,0,2)
    $$
  • $$
    (1,0,0)
    $$
  • $$
    (0,1,0)
    $$
0:0:1


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