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 $$45^{\circ}$$, then a value of $$K$$ is
  • $$2$$
  • $$3$$
  • $$4$$
  • $$6$$
The angle between two diagonals of a cube is.
  • $$30^o$$
  • $$45^o$$
  • $$cos^{-1}(\frac{1}{3})$$
  • $$cos^{-1}(\frac{1}{\sqrt{3}})$$
If a line segment $$OP$$ makes angles of $$\dfrac {\pi}{4}$$ and $$\dfrac {\pi}{3}$$ with X-axis and Y-axis, respectively. Then, the direction cosines are
  • $$\dfrac {1}{\sqrt {2}}, \dfrac {\sqrt {3}}{2}, \dfrac {1}{\sqrt {2}}$$
  • $$\dfrac {1}{\sqrt {2}}, \dfrac {1}{2}, \dfrac {1}{2}$$
  • $$1, \sqrt {3}, 1$$
  • $$1, \dfrac {1}{\sqrt {3}}, 1$$
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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