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 $$\displaystyle  3\hat{i}-3\hat{j}+\hat{k} $$ & normal to the line joining the points with position vectors $$\displaystyle  3\hat{i}+4\hat{j}-\hat{k} $$ & $$\displaystyle  2\hat{i}-\hat{j}=5\hat{k} $$ is
  • $$\displaystyle \overline{r}.\left ( -\hat{i}-5\hat{j}+6\hat{k} \right )+18=0$$
  • $$\displaystyle \overline{r}.\left ( \hat{i}-5\hat{j}+6\hat{k} \right )=22$$
  • $$\displaystyle \overline{r}.\left ( \hat{i}+5\hat{j}-6\hat{k} \right )+18=0$$
  • $$\displaystyle \overline{r}.\left ( -\hat{i}+5\hat{j}+6\hat{k} \right )+12=0$$
Vector equation of the plane passing through a point having position vector $$\displaystyle 2\hat{i}+3\hat{j}-4\hat{k}$$ and perpendicular to the vector $$\displaystyle 2\hat{i}-\hat{j}+2\hat{k}$$ is
  • $$\displaystyle \overrightarrow{r}\cdot \left ( 2\hat{i}-\hat{j}+2\hat{k} \right )+7=0$$
  • $$\displaystyle \overrightarrow{r}\cdot \left ( 2\hat{i}-\hat{j}+2\hat{k} \right )=7$$
  • $$\displaystyle \overrightarrow{r}\cdot \left ( -3\hat{i}-2\hat{j}-3\hat{k} \right )=0$$
  • $$\displaystyle \overrightarrow{r}\cdot \left ( 2\hat{i}-\hat{j}+2\hat{k} \right )=9$$
A line makes the same angle $$\theta $$ with each of the $$x$$ and $$z$$ axis. 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 
  • $$\displaystyle \dfrac{3}{5}$$
  • $$\displaystyle \frac{1}{5}$$
  • $$\displaystyle \frac{2}{3}$$
  • $$\displaystyle \frac{2}{5}$$
Direction cosines of the vector $$\displaystyle \bar{v}=a_{1}\hat{i}+a_{2}\hat{j}+a_{3}\hat{k}$$ are
  • $$\displaystyle < a_{1},a_{2},a_{3} > $$
  • $$\displaystyle <-a_{1},-a_{2},-a_{3}>$$
  • $$\displaystyle <\frac {a_{1}}{\left|\bar{v}\right|},\frac {a_{2}}{\left|\bar{v}\right|,}\frac {a_{3}}{\left|\bar{v}\right|},>$$
  • none of these
The Equation of the plane through a point $$\displaystyle  2\hat{i}-\hat{j}+4\hat{k} $$ & parallel to the plane $$\displaystyle \overline{r}.\left ( 2\hat{i}+4\hat{j}-7\hat{k} \right )=6$$ is
  • $$\displaystyle \overline{r}.\left ( 2\hat{i}+4\hat{j}-7\hat{k} \right )=21$$
  • $$\displaystyle \overline{r}.\left ( 2\hat{i}+4\hat{j}-7\hat{k} \right )=14$$
  • $$\displaystyle \overline{r}.\left ( 2\hat{i}+4\hat{j}-7\hat{k} \right )=42$$
  • $$\displaystyle \overline{r}.\left ( 2\hat{i}+4\hat{j}-7\hat{k} \right )=28$$
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 $$\beta=$$
  • $$0$$
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{2}$$
  • $$\pi$$
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