CBSE Questions for Class 12 Commerce Maths Three Dimensional Geometry Quiz 1 - MCQExams.com

If a line makes an angle of $${\pi }/{4}$$ with the positive direction of each of $$x$$-axis and $$y$$-axis, then the angle that the line makes with the positive direction of the $$z$$-axis is
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{2}$$
A line $$AB$$ in three-dimensional space makes angles $$45^{\mathrm{o}}$$ and $$120^{\mathrm{o}}$$ with the positive $$\mathrm{x}$$-axis and the positive $$\mathrm{y}$$-axis respectively. lf $$AB$$ makes an acute angle $$e$$ with the positive $$\mathrm{z}$$-axis, then $$e$$ equals
  • $$45^{\mathrm{o}}$$
  • $$60^{\mathrm{o}}$$
  • $$75^{\mathrm{o}}$$
  • $$30^{\mathrm{o}}$$
A line makes the same angle $$\Theta$$, with each of the $$\mathrm{x}$$ and $$\mathrm{z}$$ axis. If the angle $$\beta$$, which it makes with $$\mathrm{y}$$-axis, is such that $$\sin^{2}\beta=3\sin_{\Theta}^{2}$$, then $$\cos^{2}\Theta$$ equals: 
  • $$\displaystyle \frac{2}{3}$$
  • $$\displaystyle \frac{1}{5}$$
  • $$\displaystyle \frac{3}{5}$$
  • $$\displaystyle \frac{2}{5}$$
A line in the 3-dimensional space makes an angle $$\theta \left(0 < \theta \leq \dfrac {\pi}{2}\right)$$ with both the x and y axes, then the set of all values of $$\theta$$ is the interval :
  • $$\left[0, \dfrac {\pi}{4}\right]$$
  • $$\left[\dfrac {\pi}{6}, \dfrac {\pi}{3}\right]$$
  • $$\left[\dfrac {\pi}{4}, \dfrac {\pi}{2}\right]$$
  • $$\left[\dfrac {\pi}{3}, \dfrac {\pi}{2}\right]$$
The points with position vectors $$60 \widehat{i} + 3 \widehat{j}$$, $$40 \widehat{i} - 8 \widehat{j}$$, $$a\widehat{i} -52\widehat{j}$$ are collinear if
  • $$a = -40$$
  • $$a = 40$$
  • $$a = 20$$
  • None of these
The vector equation $$r=i-2j-k+t(6j-k)$$ represents a straight line passing through the points:
  • $$(0, 6, -1)$$ and $$(1, -2, -1)$$
  • $$(0, 6, -1)$$ and $$(-1, -4, -2)$$
  • $$(1, -2, -1)$$ and $$(1, 4, -2)$$
  • $$(1, -2, -1)$$ and $$(0, -6, 1)$$
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 $$\cos ^{ 2 }{ \theta  } $$ equals
  • $$\dfrac {2}{5}$$
  • $$\dfrac {1}{5}`$$
  • $$\dfrac {3}{5}$$
  • $$\dfrac {2}{3}$$
The points with position vectors $$ 60i + 3j,  40i -8j$$ and $$ ai -52j $$ are collinear if
  • $$a = -40$$
  • $$a = 40$$
  • $$a = 20$$
  • None of these
Equation to a line parallel to the vector $$2\hat{i}-\hat{j}{+}\hat{k}$$ and passing through the point $$\hat{i}+\hat{j}{+\hat{k}}$$
  • $$(x-1)\hat{i}+(y-1)\hat{j}+(z -1)\hat{k}=\lambda(2\hat{i}-\hat{j}+\hat{k})$$
  • $$(x-2)\hat{i}+(y-1)\hat{j}+(z+1)\hat{k}=\lambda(\hat{i}+\hat{j}+\hat{k})$$
  • $$2x\hat{i}-y\hat{j}+z\hat{k}=\lambda(\hat{i}+\hat{j}+\hat{k})$$
  • $$(x-1)\hat{i}+(y+1)\hat{j}+(z+1)\hat{k}=\lambda(\hat{i}+\hat{j}+\hat{k})$$
The direction ratios of the diagonal of the cube joining the origin to the opposite corner are (when the $$3$$ concurrent edges of the cube are coordinate axes)
  • $$\displaystyle \dfrac{2}{\sqrt{3}},\dfrac{2}{3},\dfrac{2}{3}$$
  • $$1,1,1$$
  • $$2,-2,1$$
  • $$1,2,3$$

If the direction cosines of a line are $$\left(\displaystyle \dfrac{1}{c},\dfrac{1}{c},\dfrac{1}{c}\right)$$ then $$c=$$______
  • $$3$$
  • $$1/\sqrt{3}$$
  • $$\pm\sqrt{3}$$
  • $$\displaystyle \pm\dfrac{1}{\sqrt{3}}$$
$$l = m =n = 1$$ represents the direction cosines of 
  • $$x-$$axis
  • $$y-$$axis
  • $$z-$$axis
  • none of these
 List IList II 
1) d.c's of $$x -$$ axisa) $$(1,1,1)$$ 
2) d.c's of $$y -$$ axisb)$$\left(\displaystyle \frac{]}{\sqrt{3}}\frac{]}{\sqrt{3}},\frac{]}{\sqrt{3}}\right)$$
3) d.c's of $$z -$$ axisc) $$(1,0,0)$$
4) d.c's of a line makes equal angles with axesd) $$(0,1,0)$$
 e) $$(0,0,1)$$
The correct order for 1, 2, 3, 4 is
  • $$c,d,e,b$$
  • $$a,b,c,e$$
  • $$c,d,a,b$$
  • $$b,c,a,e$$
If $$P(x, y, z)$$ moves such that $$x=0, z=0$$, then the locus of $$P$$ is the line whose d.cs are
  • $$y$$-axis
  • $$1, 0, 0$$
  • $$0, 1, 0$$
  • $$0, 0, 0$$
If $$P$$ is a point on the line passing through the point $$A$$ with position vector $$2\overline{i}+\overline{j}-3\overline{k}$$ and parallel to $$\overline{i}+2\overline{j}+\overline{k}$$ such that $$AP=2\sqrt{6}$$ then the position vector of $$P$$ is
  • $$4\overline{i}+5\overline{j}+\overline{k}$$
  • $$3\overline{j}+5\overline{k}$$
  • $$4\overline{i}+5\overline{j}-\overline{k}$$
  • $$3\overline{j}-4\overline{k}$$
If the points $$A(\overline{a}), B(\overline{b}), C(\overline{c})$$ satisfy the relation $$3\mathrm{a}-8\mathrm{b}+5\mathrm{c}=0$$ then the points are
  • Vertices of an equilateral triangle
  • Collinear
  • Vertices of a right angled triangle
  • Vertices of a isosceles triangle
Cosine of the angle between two diagonals of a cube is equal to
  • $$\displaystyle \frac{2}{\sqrt 6}$$
  • $$\displaystyle \frac{1}{3}$$
  • $$\displaystyle \frac{1}{2}$$
  • None of these
Area of $$\displaystyle \Delta ABC$$ is
  • $$45$$ squares units
  • $$55$$ squares units
  • $$65$$ squares units
  • none of these
A vector is equally inclined to the $$x$$-axis, $$y$$-axis and $$z$$-axis respectively, its direction cosines are
  • $$< \dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}>$$
  • $$< -\dfrac{1}{\sqrt{3}}, -\dfrac{1}{\sqrt{3}}, -\dfrac{1}{\sqrt{3}}>$$
  • $$< \dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}>$$ or $$< -\dfrac{1}{\sqrt{3}}, -\dfrac{1}{\sqrt{3}}, -\dfrac{1}{\sqrt{3}}>$$
  • None of these
What are the DR's of vector parallel to $$\left( 2,-1,1 \right) $$ and $$\left( 3,4,-1 \right) $$?
  • $$\left( 1,5,-2 \right) $$
  • $$\left( -2,-5,2 \right) $$
  • $$\left( -1,5,2 \right) $$
  • $$\left( -1,-5,-2 \right) $$
The direction cosines of the vectors $$2\vec {i} + \vec {j} - 2\vec {k}$$ is equal to
  • $$\dfrac {2}{3}, \dfrac {1}{3}, -\dfrac {2}{3}$$
  • $$\dfrac {2}{3}, \dfrac {1}{3}, \dfrac {2}{3}$$
  • $$\dfrac {1}{3}, \dfrac {2}{3}, -\dfrac {2}{3}$$
  • $$\dfrac {2}{3}, \dfrac {2}{3}, \dfrac {1}{3}$$
The Cartesian equation of the plane passing through the point $$(3, -2, -1)$$ and parallel to the vectors $$\overline {b} = \overline {i} - 2\overline {j} + 4\overline {k}$$ and $$\overline {c} = 3\overline {i} + 2\overline {j} - 5\overline {k}$$ is
  • $$2x - 17y - 8z + 63 = 0$$
  • $$3x + 17y + 8z - 36 = 0$$
  • $$2x + 17y + 8z + 36 = 0$$
  • $$3x - 16y + 8z - 63 = 0$$
A straight line is equally inclined to all the three coordinate axes. Then an angle made by the line with the y-axis is
  • $$\cos^{1}\left (\dfrac {1}{3}\right )$$
  • $$\cos^{1}\left (\dfrac {1}{\sqrt {3}}\right )$$
  • $$\cos^{1}\left (\dfrac {2}{\sqrt {3}}\right )$$
  • $$\dfrac {\pi}{4}$$
If the foot of the perpendicular from $$(0, 0, 0)$$ to a plane is $$(1, 2, 3)$$, then the equation of the plane is
  • $$2x + y + 3z = 14$$
  • $$x + 2y + 3z = 14$$
  • $$x + 2y + 3z + 14 = 0$$
  • $$x + 2y - 3z = 14$$
A line makes angles $$\alpha, \beta, \gamma, \delta$$ with four diagonals of a cube. then  $$\displaystyle \sum_{r\epsilon \{\alpha, \beta, \gamma, \delta\}} cos^2(r)=$$
  • $$\dfrac{-4}{3}$$
  • $$\dfrac{4}{3}$$
  • $$\dfrac{3}{4}$$
  • $$\dfrac{3}{5}$$
Find vector equation for the line passing through the points $$3\overline i+4\overline j-7\overline k,\overline i-\overline j+6\overline k$$.
  • $$\overline r=(3-2\lambda)\overline i+(4-5\lambda)\overline j+(-7+13\lambda)\overline k$$
  • $$\overline r=(2\lambda)\overline i+(4+5\lambda)\overline j+(-7-13\lambda)\overline k$$
  • $$\overline r=(3-2\lambda)\overline i-(4-5\lambda)\overline j+(-7+13\lambda)\overline k$$
  • $$\overline r=(3-2\lambda)\overline i+(4-5\lambda)\overline j-(-7+13\lambda)\overline k$$
Find vector equation of line passing through  $$(1,4,1),(1,2,2)$$.
  • $$\overline r=\overline i+(4-2\lambda)\overline j+(1+\lambda)\overline k$$
  • $$\overline r=\overline i+(4+2\lambda)\overline j+(1+\lambda)\overline k$$
  • $$\overline r=\overline i-(4-2\lambda)\overline j+(1+\lambda)\overline k$$
  • $$\overline r=\overline i+(4-2\lambda)\overline j+(1-\lambda)\overline k$$
The projections of a directed line segment on the coordinate axes are $$12, 4, 3$$ respectively.
What are the direction cosines of the line segment?
  • $$(12/13, 4/13, 3/13)$$
  • $$(12/13, -4/13, 3/13)$$
  • $$(12/13, -4/13, -3/13)$$
  • $$(-12/13, -4/13, 3/13)$$
From the point $$P(3, -1, 11)$$, a perpendicular is drawn on the line $$L$$ given by the equation $$\dfrac {x}{2} = \dfrac {y - 2}{3} = \dfrac {z - 3}{4}$$. Let $$Q$$ be the foot of the perpendicular.
What are the direction ratios of the line segment $$PQ$$?
  • $$(1, 6, 4)$$
  • $$(-1, 6, -4)$$
  • $$(-1, -6, 4)$$
  • $$(2, -6, 4)$$
Find the vector equation of line joining the points $$ (2,1,3)$$ and $$(-4,3,-1)$$
  • $$\overline r=2(1-3\lambda)\overline i-(1+2\lambda)\overline j-(3-4\lambda)\overline k$$
  • $$\overline r=2(1-3\lambda)\overline i-(1+2\lambda)\overline j+(3-4\lambda)\overline k$$
  • $$\overline r=2(1-3\lambda)\overline i+(1+2\lambda)\overline j+(3-4\lambda)\overline k$$
  • $$\overline r=2(1+3\lambda)\overline i+(1+2\lambda)\overline j+(3+4\lambda)\overline k$$
The vector equation of line passing through two points $$A(x_1,y_1,z_1),B(x_2,y_2,z_2) $$ is
  • $$\vec r=\vec a-\vec b$$
  • $$\vec r=\vec a +\lambda(\vec b-\vec a)$$
  • $$\vec r=\vec a+\lambda\vec b$$
  • $$\vec r=\vec a+\vec b$$
Which of the following represents direction cosines of the line:
  • $$0,\cfrac { 1 }{ \sqrt { 2 } } ,\cfrac { 1 }{ 2 } $$
  • $$0,\cfrac { -\sqrt { 3 } }{ 2 } ,\cfrac { 1 }{ \sqrt 2  } $$
  • $$0,\cfrac { \sqrt { 3 } }{ 2 } ,\cfrac { 1 }{  { 2 } } $$
  • $$\cfrac { 1 }{ 2 } ,\cfrac { 1 }{ 2 } ,\cfrac { 1 }{ 2 } $$
A line passes through the points (6, -7, -1) and (2, -3, 1). What are the direction ratios of the line ? 
  • $$(4, -4, 2)$$
  • $$( 4, 4, 2 )$$
  • $$( -4, 4, 2 )$$
  • $$( 2, 1, 1 )$$
The length of perpendicular from the origin to the plane which makes intercepts $$\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5}$$ respectively on the coordinate axes is 
  • $$\dfrac{1}{\sqrt[5]{2}}$$
  • $$\dfrac{1}{10}$$
  • $$\displaystyle\sqrt[5]{2}$$
  • 5
ABCD is a trapezium in which AB and CD are parallel.that the mid points of the sides AB ,CD and the intersection of the diagonals are collinear.
  • True
  • False
If points (1,2), (3 , 5) and (0 , b ) are collinear the value of b is  
  • $$\dfrac{1}{2}$$
  • $$\dfrac{7}{2}$$
  • 2
  • -1
The direction angles of the line $$x = 4z + 3, y = 2 - 3z$$ are $$\alpha, \beta$$ and $$\gamma$$, then $$\cos \alpha + \cos \beta + \cos \gamma =$$ ________.
  • $$\dfrac {2}{\sqrt {26}}$$
  • $$\dfrac {8}{\sqrt {26}}$$
  • $$1$$
  • $$2$$
The angle between the lines whose direction cosines are $$\left( \dfrac {\sqrt{3}}{4}, \dfrac {1}{4}, \dfrac {\sqrt{3}}{2} \right)$$ and $$\left( \dfrac {\sqrt{3}}{4}, \dfrac {1}{4}, -\dfrac {\sqrt{3}}{2} \right)$$ is :
  • $$ \pi$$
  • $$ \dfrac {\pi}{2} $$
  • $$ \dfrac {\pi}{3} $$
  • $$ \dfrac {\pi}{4} $$
If a line makes the angles $$ \alpha , \beta$$ and $$\gamma$$ with the axes, then what is the value of $$1+\cos 2\alpha +\cos 2\beta+\cos 2\gamma$$ equal to ?
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$2$$
Direction cosines of the line $$\cfrac { x+2 }{ 2 } =\cfrac { 2y-5 }{ 3 } ,z=-1$$ are ____
  • $$\cfrac { 4 }{ 5 } ,\cfrac { 3 }{ 5 } ,0$$
  • $$\cfrac { 3 }{ 5 } ,\cfrac { 4 }{ 5 } ,\cfrac { 1 }{ 5 } $$
  • $$-\cfrac { 3 }{ 5 } ,\cfrac { 4 }{ 5 } ,0$$
  • $$\cfrac { 4 }{ 5 } ,-\cfrac { 2 }{ 5 } ,\cfrac { 1 }{ 5 } $$
The following lines are $$\hat { r } =\left( \hat { i } +\hat { j }  \right) +\lambda \left( \hat { i } +2\hat { j } -\hat { k }  \right) +\mu \left( -\hat { i } +\hat { j } -\hat { 2k }  \right) $$
  • collinear
  • skew-lines
  • co-planar lines
  • parallel lines
$$L$$ and $$M$$ are two points with position vectors $$2\overline { a } -\overline { b } $$ and $$a+2\overline { b } $$ respectively. The position vector of the point $$N$$ which divides the line segment $$LM$$ in the ratio $$2:1$$ externally is
  • $$3\overline { b } $$
  • $$4\overline { b } $$
  • $$5\overline { b } $$
  • $$3\overline { a } +4\overline { b } $$
If the normal of the plane makes an angles $$\dfrac {\pi}{4}, \dfrac {\pi}{4}$$ and $$\dfrac {\pi}{2}$$ with positive X-axis, Y-axis and Z-axis respectively and the length of the perpendicular line segment from origin to the plane is $$\sqrt {2}$$, then the equation of the plane is ________.
  • $$x + y + z = \sqrt {2}$$
  • $$x + y + z = 1$$
  • $$x + y = 2$$
  • $$x = \sqrt {2}$$
Direction cosines of ray from $$P(1, -2, 4)$$ to $$Q(-1, 1, -2)$$ are
  • $$-2, 3, -6$$
  • $$2, -3, 6$$
  • $$2, 3, 6$$
  • $$\dfrac{-2}{7}, \dfrac{3}{7}, \dfrac{-6}{7}$$
If the lines $$x=1+a,y=-3-\lambda a,z=1+\lambda a$$ and $$x=\cfrac { b }{ 2 } ,y=1+b,z=2-b$$ are coplanar, then $$\lambda$$ is equal to
  • $$-3$$
  • $$2$$
  • $$1$$
  • $$-2$$
Vector equation of the line $$6x - 2 = 3y + 1 = 2z - 2$$ is 
  • $$\vec r = \hat i + \hat j + 3\hat k + \lambda (\hat i - 2\hat j + 3\hat k)$$
  • $$\vec r = \hat i - 2\hat j + 3\hat k + \lambda \left( {\frac{1}{3}\hat i + \frac{1}{3}\hat j + \hat k} \right)$$
  • $$\vec r = \frac{1}{3}\hat i - \frac{1}{3}\hat j\, + \hat k + \lambda (\hat i + 2\hat j + 3\hat k)$$
  • $$\vec r = - 2\hat i + \hat j\, - 2\hat k + \lambda (6\hat i + 3\hat j + 2\hat k)$$
The points with position vectors $$60\hat{i}+3\hat{j}$$, $$40\hat{i}-8\hat{j}$$, $$a\hat{i}-52\hat{j}$$  are collinear if
  • $$a=-40$$
  • $$a=40$$
  • $$a=20$$
  • $$None\ of\ these$$
Direction cosines $$l, m, n$$ of two lines are connected by the equation $$l-5m+3n=0$$ and $$7l^{2}+5m^{2}-3n^{2}=0$$. The direction cosines of one of the lines are
  • $$-1/\sqrt{6},1/\sqrt{6},2/\sqrt{6}$$
  • $$2/\sqrt{6},1/\sqrt{6},-1/\sqrt{6}$$
  • $$1/\sqrt{6},2/\sqrt{6},-1/\sqrt{6}$$
  • $$1/\sqrt{6},1/\sqrt{6},2/\sqrt{6}$$
The direction ratios of the line joining the points $$A(4,-3,7)$$ and $$B(1,3,5)$$ are:
  • $$(5,0,12)$$
  • $$(3,-6,2)$$
  • $$(5,6,2)$$
  • $$\left(\dfrac{5}{2},0,6\right)$$
Can $$\dfrac{1}{\sqrt{3}}, \dfrac{2}{\sqrt{3}}, \dfrac{-2}{\sqrt{3}}$$ be the direction cosines of any directed line?
  • Yes
  • No
  • Cannot say
  • None of these
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