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
  • $$\vec{r}= (2, 1, 0) + t(1, -1, 1)$$
  • $$\vec{r} = (2, 1, 0) + t(-1, 1, 1)$$
  • $$\vec{r}= (2, 1, 0) + t(1, 1, -1)$$
  • $$\vec{r} = (2, 1, 0) + t(1, 1, 1)$$
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
  • $$a (x - a) - b (y - b) - c (z - c) = 0$$
  • $$a(x + a) + b (y + b) + c (z + c) = 0$$
  • $$a(x - a) + b (y - b) + c (z - c) = 0$$
  • 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 $$(\alpha, \beta, \gamma)$$ to the plane $$ax + by + cz + d = 0$$ is
  • $$a(x-\alpha) + b (y - \beta) + c ( z - \gamma) = 0$$
  • $$\displaystyle \frac{x - \alpha}{a} = \frac{y - \beta}{b} = \frac{z - y}{c}$$
  • $$a (x - \alpha) + b(y - \beta) + c (z - \gamma) = abc$$
  • None of these
The straight line $$\displaystyle \frac{x - 3}{3} = \frac{y - 2}{1} = \frac{z - 1}{0}$$ 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
  • $$r= k+t(7 i - 10 + 2k)$$
  • $$r = k + t (- 9 i + 6 j - 2k)$$
  • $$r = k + t (7i - 10 j - 2k)$$
  • $$r = k + t (7i + 10 j + 2k)$$
The equation of the plane through $$(1, 2, 3)$$ and parallel to the plane $$2x + 3y - 4z = 0$$ is
  • $$2x + 3y + 4z = 4$$
  • $$2x + 3y + 4z + 4 = 0$$
  • $$2x - 3y + 4z + 4 = 0$$
  • $$2x + 3y - 4z + 4 = 0$$
If $$M$$ denotes the mid-point of the line segment joining $$ A( 4\widehat{i} + 5\widehat{j}- 10 \widehat{k})$$ and $$ B(-\widehat{i} + 2 \widehat{j} + \widehat{k})$$, then the equation of the plane through $$M$$ and perpendicular to $$AB$$, is
  • $$\vec{r}. ( -5 \widehat{i}- 3 \widehat{j} + 11 \widehat{k}) + \dfrac{135}{2}= 0 $$
  • $$\vec{r}.\left  ( \dfrac{3}{2} \widehat{i}+\dfrac{7}{2} \widehat{j} - \dfrac{9}{2} \widehat{k}\right) + \dfrac{135}{2}= 0 $$
  • $$\vec{r}. ( 4 \widehat{i}+ 5\widehat{j} - 10 \widehat{k}) + 4= 0 $$
  • $$\vec{r}. ( - \widehat{i}+ 2 \widehat{j} + \widehat{k}) + 4= 0 $$
If the points $$a(1, 2, -1), B(2, 6, 2)$$ and $$c(\lambda, -2, -4)$$ are collinear then $$\lambda$$ 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
  • $$ 2x+y+z+1= 0 $$
  • $$ 3x+8y-2z-17= 0$$
  • $$ 2x+y-z+1= 0 $$
  • $$ x-2y+z+1= 0 $$
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