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CBSE Questions for Class 12 Commerce Maths Three Dimensional Geometry Quiz 4 - MCQExams.com

The point collinear with (4,2,0) and (6,4,6) among the following is
  • (0,4,6)
  • (8,6,8)
  • (1,4,6)
  • None of these
Equation of the line which passes through the point with position vector (2,1,0) and perpendicular to the plane containing the vectors i+j and j+k is
  • r=(2,1,0)+t(1,1,1)
  • r=(2,1,0)+t(1,1,1)
  • r=(2,1,0)+t(1,1,1)
  • r=(2,1,0)+t(1,1,1)
If O is the origin and A is the point (a,b,c), then the equation of the plane through A and at right angles to OA is
  • a(xa)b(yb)c(zc)=0
  • a(x+a)+b(y+b)+c(z+c)=0
  • a(xa)+b(yb)+c(zc)=0
  • None of the above
The values of a for which point (8,7,a),(5,2,4) and (6,1,2) are collinear.
  • 4
  • 2
  • 0
  • 2
The equation of the perpendicular from the point (α,β,γ) to the plane ax+by+cz+d=0 is
  • a(xα)+b(yβ)+c(zγ)=0
  • xαa=yβb=zyc
  • a(xα)+b(yβ)+c(zγ)=abc
  • None of these
The straight line x33=y21=z10 is
  • parallel to x-axis
  • parallel to y-axis
  • parallel to z-axis
  • perpendicular to z-axis
The equation of altitude through B to side AC is
  • r=k+t(7i10+2k)
  • r=k+t(9i+6j2k)
  • r=k+t(7i10j2k)
  • r=k+t(7i+10j+2k)
The equation of the plane through (1,2,3) and parallel to the plane 2x+3y4z=0 is
  • 2x+3y+4z=4
  • 2x+3y+4z+4=0
  • 2x3y+4z+4=0
  • 2x+3y4z+4=0
If M denotes the mid-point of the line segment joining A(4ˆi+5ˆj10ˆk) and B(ˆi+2ˆj+ˆk), then the equation of the plane through M and perpendicular to AB, is
  • r.(5ˆi3ˆj+11ˆk)+1352=0
  • r.(32ˆi+72ˆj92ˆk)+1352=0
  • r.(4ˆi+5ˆj10ˆk)+4=0
  • r.(ˆi+2ˆj+ˆk)+4=0
If the points a(1,2,1),B(2,6,2) and c(λ,2,4) are collinear then λ is
  • 0
  • 2
  • 2
  • 1
Given A(1,-1,0); B(3,1,2); C(2,-2,4) and D(-1,1,-1) which of the following points neither lie on AB nor on CD?
  • (2,2,4)
  • (2,-2,4)
  • (2,0,1)
  • (0,-2,-1)
The projections of a directed line segment on the coordinate axes 12, 4, 3. The direction cosines of the line are
  • \dfrac{12}{13}, -\dfrac{4}{13}, \dfrac{3}{13}
  • -\dfrac{12}{13}, -\dfrac{4}{13}, \dfrac{3}{13}
  • \dfrac{12}{13}, \dfrac{4}{13}, \dfrac{3}{13}
  • none of these
Given A(1,-1,0); B(3,1,2);C(2,-2,4) and D(-1,1,-1) which of the following points neither lie on AB nor on CD
  • (2,2,4)
  • (2,-2,4)
  • (2,0,1)
  • (0,-2,-1)
The vector equation of the line \displaystyle \frac { x-2 }{ 2 } =\frac { 2y-5 }{ -3 } ,z=-1 is \displaystyle \overrightarrow { r } =\left( 2\hat { i } +\frac { 5 }{ 2 } \hat { j } -\hat { k }  \right) +\lambda \left( 2\hat { i } -\frac { 3 }{ 2 } \hat { j } +x\hat { k }  \right) , where x is equal to
  • 0
  • 1
  • 2
  • 3
The equation of the right bisector plane of the segment joining (2,3,4) and (6,7,8) is
  • x+y+z+15=0
  • x+y+z-15=0
  • x-y+z-15=0
  • None of these
Equation of plane passing through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x+ 6y + 6z-1= 0 is
  • 3x+ 4y+ 5z= 9
  • 3x+ 4y- 5z+ 9= 0
  • 3x+ 4y- 5z- 9=0
  • None of these
Vector equation of the plane \vec{r}=\widehat{i}-\widehat{j}+\lambda (\widehat{i}+\widehat{j}+\widehat{k})+ \mu (\widehat{i}-2\widehat{j}+3\widehat{k}) in the scalar dot product form is
  • \vec{r}.(5\widehat{i}+2\widehat{j}-3\widehat{k})=7
  • \vec{r}.(5\widehat{i}-2\widehat{j}+3\widehat{k})=7
  • \vec{r}.(5\widehat{i}-2\widehat{j}-3\widehat{k})=7
  • \vec{r}.(5\widehat{i}+2\widehat{j}+3\widehat{k})=17
If the points (0, 1, -2), (3, \lambda, 1) and (\mu, 7, 4) are collinear, the point on the same line is
  • (5, 6, 3)
  • (1, -1, -2)
  • (-5, -6, -3)
  • (0, 0, 0)
The equation of the plane passing through the line \displaystyle \frac{x - 1}{2} = \frac{y + 1} {-1} = \frac{z}{3} and parallel to the direction whose direction numbers are 3, 4, 2 is
  • \displaystyle 14x - 5y - 11z = 19
  • \displaystyle 3x + 4y + 2z + 1 = 0
  • \displaystyle 2x - y + 3z = 3
  • none of these
A line makes angles \alpha, \beta , \gamma with the positive directions of the axes of reference. The value of \cos 2\alpha +\cos 2\beta +\cos 2\gamma is
  • 1
  • 2
  • -1
  • 0
If the points A(1,2,-1), B(2,6,2) and \displaystyle C\left ( \lambda,-2,-4 \right ) are collinear, then \displaystyle \lambda is
  • 0
  • 2
  • -2
  • 1
The direction cosines of the line joining the points (2,3,-1) and (3,-2,1) are
  • -1,5,-2
  • \dfrac{1}{\sqrt{30}},-\sqrt{\dfrac{5}{6}},\sqrt{\dfrac{2}{15}}
  • \dfrac{-1}{30 },\dfrac{1}{6 },-\dfrac{1}{15 }
  • none of these
The direction cosines of the perpendicular from the origin to the plane \displaystyle 3x  - y + 4z = 5 are
  • \displaystyle 4, \: -1, \: 3
  • \displaystyle 3, \: -1, \: 4
  • \displaystyle \frac{3}{\sqrt{26}}, \: \frac{-1}{\sqrt{26}}, \: \frac{4}{\sqrt{26}}
  • \displaystyle \frac{4}{\sqrt{26}}, \: \frac{-1}{\sqrt{26}}, \: \frac{3}{\sqrt{26}}
The angle between the lines \displaystyle \frac{x+2}{2}=\frac{y+3}{2}=\frac{z-4}{1} and \displaystyle \frac{x-1}{0}=\frac{y}{3}=\frac{z}{0} is
  • \cos^{-1}\left(\dfrac{2}{3}\right)
  • \cos^{-1}\left(\dfrac{1}{3}\right)
  • \cos^{-1}\left(\dfrac{1}{\sqrt{2}}\right)
  • \cos^{-1}\left(1\right)
The direction cosines of the normal to the plane \displaystyle 5\left ( x - 2 \right ) = 3\left ( y - z \right ) are
  • \displaystyle 5, \: -3, \: 3
  • \displaystyle \frac{5}{\sqrt{43}}, \: \frac{-3}{\sqrt{43}}, \: \frac{3}{\sqrt{43}}
  • \displaystyle \frac{1}{2}, \: \frac{-3}{10}, \: \frac{3}{10}
  • \displaystyle 1, \: \frac{-3}{5}, \: \frac{3}{5}
If the direction cosines of the line joining the origin and a point at unit distance from the origin are \displaystyle \frac{1}{\sqrt{3}}, -\displaystyle \frac{1}{2}, \lambda    then value of \lambda is?
  • \displaystyle \frac{1}{2}\sqrt{\frac{5}{3}}
  • \displaystyle \frac{2}{\sqrt{3}}
  • \displaystyle \frac{-1}{2}\sqrt{\frac{5}{3}}
  • none of these
If \theta is an angle given by \cos \theta   = \displaystyle  \frac{\cos ^{2}\alpha+\cos ^{2}\beta +\cos ^{2}\gamma}{\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma } where  \alpha , \beta , \gamma are the angles made by a line with the positive directions of the axes of reference then the measure of \theta is
  • \displaystyle \frac{\pi }{4}
  • \displaystyle \frac{\pi }{6}
  • \displaystyle \frac{\pi }{2}
  • \displaystyle \frac{\pi }{3}
If a line makes angles of 60^{\circ} and 45^{\circ} with the positive directions of the x-axis and y-axis respectively, then the acute angle between the line and the z-axis is
  • 60^{\circ}
  • 45^{\circ}
  • 75^{\circ}
  • 15^{\circ}
ABC is a triangle where A= \left ( 2,3,5 \right ),B= \left ( -1,3,2 \right ) and C= \left (\lambda ,5,\mu   \right ). If the median through A is equally inclined with the axes, then
  • \lambda = 14,\mu = 20
  • \lambda = 9,\mu = 6
  • \lambda = \displaystyle \frac{7}{2},\mu = 20
  • \lambda = 10,\mu = 7
If the image of the point \displaystyle \left ( 1, \: 1, \: 1 \right ) by a plane \displaystyle \left ( 3, \: -1, \: 5 \right ) then the equation of the plane is
  • \displaystyle x - y + 2z = 8
  • \displaystyle x - y + 2z = 16
  • \displaystyle x - y + 2z = 14
  • None of these
Which of the triplet can not represent direction cosine of a line
  • \displaystyle \left( \frac { 1 }{ \sqrt { 3 }  } ,\frac { 1 }{ \sqrt { 3 }  } ,\frac { 1 }{ \sqrt { 3 }  }  \right)
  • \displaystyle \left( \frac { 3 }{ \sqrt { 50 }  } ,\frac { 4 }{ \sqrt { 50 }  } ,\frac { 5 }{ \sqrt { 50 }  }  \right)
  • \displaystyle \left( \frac { 4 }{ \sqrt { 77 }  } ,\frac { 5 }{ \sqrt { 77 }  } ,\frac { 6 }{ \sqrt { 77 }  }  \right)
  • \displaystyle \left( \frac { 2 }{ \sqrt { 25 }  } ,\frac { 3 }{ \sqrt { 25 }  } ,\frac { 4 }{ \sqrt { 25 }  }  \right)
The direction ratios of a normal to the plane through \left( 1,0,0 \right) ,\left( 0,1,0 \right) , which makes an angle of \displaystyle \frac { \pi  }{ 4 } with the plane x+y=3 are 
  • \displaystyle 1,\sqrt { 2 } ,1
  • \displaystyle 1,1,\sqrt { 2 }
  • \displaystyle 1,1,2
  • \displaystyle \sqrt { 2 } ,1,1
let P(4,1,\lambda) and Q(2,-1,\lambda) be two points. A line having direction ratios 1,-1,6 is perpendicular to the plane passing through the origin, P and Q, then \lambda equals
  • -\dfrac{1}{2}
  • \dfrac{1}{2}
  • 1
  • none of these
The position vectors of three points are 2\vec{a}-\vec{b}+3\vec{c}, \vec{a}-2\vec{b}+\lambda \vec{c} and \mu \vec{a}-5\vec{b} where \vec{a}, \vec{b}, \vec{c} are non coplanar vectors, then the points are collinear when
  • \displaystyle \lambda =-2, \mu =\dfrac{9}{4}
  • \displaystyle \lambda =-\dfrac{9}{4}, \mu =2
  • \displaystyle \lambda =\dfrac{9}{4}, \mu =-2
  • None of these
The angle between two lines whose direction cosines satisfy the equations n= l+m and m= 2l+3n is
  • 180^{\circ}
  • 90^{\circ}
  • 60^{\circ}
  • none of these
Let the direction - cosines of the line which is equally inclined to the axis be \displaystyle \pm \frac{1}{\sqrt{k}}. Find k ?
  • 2
  • 3
  • 5
  • 6
If the direction ratios of a line are 1+\lambda ,1-\lambda ,2, and it makes an angle of 60^{o} with the y-axis then \lambda is
  • 1+\sqrt{3}
  • 2+\sqrt{5}
  • 1-\sqrt{3}
  • 2-\sqrt{5}
Let A= \left ( 1,2,2 \right ), B=\left ( 2,3,6 \right )and C= \left ( 3,4,12 \right ). The direction cosines of a line equally inclined with OA,OB and OC , where O is the origin, are
  • \displaystyle \frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0
  • \displaystyle \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0
  • \displaystyle \frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}
  • \displaystyle \frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{-1}{\sqrt{3}}
A st. line which makes angle of \displaystyle 60^{\circ} with each of y - and z - axes, is inclined with x - axis at an angle
  • \displaystyle 45^{\circ}
  • \displaystyle 30^{\circ}
  • \displaystyle 75^{\circ}
  • \displaystyle 60^{\circ}
The acute angle between two lines whose direction ratios are 2,3,6 and 1,2,2 is
  • \displaystyle \cos ^{ -1 }{ \left( \frac { 20 }{ 21 }  \right)  }
  • \displaystyle \cos ^{ -1 }{ \left( \frac { 18 }{ 21 }  \right)  }
  • \displaystyle \cos ^{ -1 }{ \left( \frac { 8 }{ 21 }  \right)  }
  • None of these
Find the equations of the plane through the point \displaystyle \left ( x_{1},y_{1},z_{1} \right ) and perpendicular to the straight line \displaystyle \frac{x-\alpha }{l}=\frac{y-\beta }{m}=\frac{z-\gamma }{n}
  • \displaystyle l\left ( x-x_{1} \right )+m\left ( y-y_{1} \right )+n\left ( z-z_{1} \right )=0.
  • \displaystyle l\left ( x-x_{1} \right )+m\left ( y-y_{1} \right )-n\left ( z-z_{1} \right )=0.
  • \displaystyle l\left ( x-x_{1} \right )-m\left ( y-y_{1} \right )+n\left ( z-z_{1} \right )=0.
  • none of these
Which of the following triplets give the direction cosines of a line ?
  • \displaystyle 1, 1, 1
  • \displaystyle 1, -1, 1
  • \displaystyle 1, 1, -1
  • \displaystyle \frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}
If a point P in the space such that \overline{OP} is inclined to OX at 45 and OZ to 60 then \overline{OP} inclined to OY is
  • 75^{\circ}
  • 75^{\circ} \text{or} 105^{\circ}
  • 60^{\circ} \text{or} 120^{\circ}
  • None of these
The equation of the plane which bisect the join of point (7, 2, 3) and (-1, -4, 3) perpendicularly is
  • x + 2y + 3= 0
  • 2x -y + 3 = 0
  • x + y +2z + 7= 0
  • 4x + 3y =9
If a line makes angles \displaystyle \alpha ,\beta ,\gamma with axes of co-ordinates, then \displaystyle \cos 2\alpha +\cos 2\beta +\cos 2\gamma is equla to
  • \displaystyle -2
  • \displaystyle -1
  • 1
  • 2
If a line makes an angle \displaystyle \theta_1, \theta_2, \theta_3 which the axis respectively, then \displaystyle cos 2\theta_1 + cos 2 \theta_2 + cos 2 \theta_3 = ?
  • -4
  • 2
  • 3
  • -1
\bar a,\bar b,\bar c are three non-zero vectors such that any two of them are non-collinear. If  \bar a+\bar b is collinear with  \bar c and  \bar b+\bar c is collinear with \bar a, then what is their sum?
  • -1
  • 0
  • 1
  • 2
The equation of plane through (1, 2, 3) and parallel to the plane \bar{r}.\left ( \hat{3i}+\hat{4j}+\hat{5k} \right )=0
  • \bar{r}.\left ( \hat{3i}+\hat{4j}+\hat{5k} \right )=26
  • \bar{r}.\left ( \hat{3i}-\hat{4j}+\hat{5k} \right )+4
  • \bar{r}.\left ( \hat{3i}+\hat{4j}-\hat{5k} \right )+4=0
  • None of these
  • Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
  • Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
  • Assertion is true but Reason is false
  • Assertion is false but Reason is true
Equation of the plane which passes through the point (-1, 3, 2) & is  \perp   to each of the planes p_{1} & p_{2}  is
  • 2x+y+z+1= 0
  • 3x+8y-2z-17= 0
  • 2x+y-z+1= 0
  • x-2y+z+1= 0
0:0:2


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