CBSE Questions for Class 12 Commerce Maths Three Dimensional Geometry Quiz 11 - MCQExams.com

If $$AB=21,\ B\equiv (-2,1,-8)$$ and the direction cosines of $$AB$$ are $$\dfrac{6}{7},\dfrac{2}{7},\dfrac{3}{7}$$ then the coordinates of $$A$$ are
  • $$(-16,-7,-1)$$
  • $$(-20,-5,17)$$
  • $$(16,7,1)$$
  • $$(20,5,17)$$
The direction ratios of the normal to the plane through $$(1,0,0)$$ and $$(0,1,0)$$ which makes an angle of $$\dfrac{\pi }{4}$$ with the plane $$x + y = 3$$ are-
  • $$1,\sqrt {2},1$$
  • $$1,1,\sqrt {2}$$
  • $$1,1,2$$
  • $$\sqrt {2},1,1$$
If $${l}_{1},{m}_{1},{n}_{1}$$ and $${l}_{2},{m}_{2},{n}_{2}$$ are the d.c.s of the two lines, then $${({l}_{1}{l}_{2}+{m}_{1}{m}_{2}+{n}_{1}{n}_{2})}^{2}+{({l}_{1}{m}_{2}-{l}_{2}{m}_{1})}^{2}+{({m}_{1}{n}_{2}-{m}_{2}{n}_{1})}^{2}+{({n}_{1}{l}_{2}-{n}_{2}{l}_{1})}^{2}=$$
  • $$3$$
  • $$2$$
  • $$1$$
  • $$4$$
If $$P=(0,1,2),Q=(4,-2,1),O=(0,0,0)$$ then $$\angle POQ=$$
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{\pi}{2}$$
If a line makes angles $$\alpha, \beta, \gamma$$ with positive axes, then the range of $$\sin{\alpha}\sin{\beta}+\sin{\beta} \sin{\gamma} +\sin{\gamma} \sin {\alpha}$$ is
  • $$\left(\dfrac{-1}{2},1\right)$$
  • $$\left(\dfrac{1}{2},2\right)$$
  • $$(-1,2)$$
  • $$(-1,2]$$
If the angle between the lines, $$\dfrac { x }{ 2 } =\dfrac { y }{ 2 } =\dfrac { z }{ 1 } $$ and $$\dfrac { 5-x }{ -2 } =\dfrac { 7y-14 }{ P } =\dfrac { z-3 }{ 4 } $$ is $$\cos { ^{ -1 }\left( \dfrac { 2 }{ 3 }  \right)  } $$, then p is equal to: 
  • $$-\dfrac { 4 }{ 7 } $$
  • $$\dfrac { 7 }{ 2 } $$
  • $$-\dfrac { 7 }{ 4 } $$
  • $$\dfrac { 2 }{ 7 } $$
The direction cosines of the line which is perpendicular to the lines whose direction cosines are proportional to ( 1, -1,2 ) and ( 2,1,-1) are:- 
  • $$\frac { 1 }{ \sqrt { 35 } } ,-\frac { 5 }{ \sqrt { 35 } } ,\frac { 3 }{ \sqrt { 35 } } $$
  • $$-\frac { 1 }{ \sqrt { 35 } } ,\frac { 5 }{ \sqrt { 35 } } ,\frac { 3 }{ \sqrt { 35 } } $$
  • $$\frac { 1 }{ \sqrt { 35 } } ,\frac { 5 }{ \sqrt { 35 } } ,\frac { 3 }{ \sqrt { 35 } } $$
  • none of these
If $$(2,1,3)$$ and $$(-1,2,4)$$ are the extermities of a diagonal of a rhombus then the $$d.r's$$ of the other diagonal are
  • $$(2,3,-9)$$
  • $$(-2,3,-9)$$
  • $$(1,-2,4)$$
  • $$(2,3,1)$$
The direction cosines of the line which is perpendicular to the lines whose direction cosines are proportional to $$(1, -1, 2)$$ and $$(2, 1, -1)$$ are
  • $$\dfrac{-1}{\sqrt{35}}, \dfrac{5}{\sqrt{35}}, \dfrac{3}{\sqrt{35}}$$
  • $$\dfrac{13}{\sqrt{35}}, \dfrac{-1}{\sqrt{35}}, \dfrac{1}{\sqrt{35}}$$
  • $$\dfrac{2}{\sqrt{3}}, \dfrac{5}{\sqrt{3}}, \dfrac{7}{\sqrt{3}}$$
  • $$\dfrac{3}{\sqrt{35}}, \dfrac{5}{\sqrt{35}}, \dfrac{7}{\sqrt{35}}$$
A mirror and source of light are kept at the origin and the positive x-axis respectively a ray a light from the sources strikes the mirror and is reflected. If (1,-1,1) are the Dr's of a normal to the plane. Then the D.C's of the reflected ray are 
  • $$(\frac{-1}{3},\frac{-2}{3},\frac{2}{3})$$
  • $$(\frac{1}{3},\frac{2}{3},\frac{2}{3})$$
  • $$(\frac{1}{3},\frac{-2}{3},\frac{2}{3})$$
  • $$(\frac{-1}{3},\frac{2}{3},\frac{2}{3})$$
If the incident ray and normal have the directions of the vectors $$\left(1,-3,1\right),\left(1,1,1\right)$$ respectively, then direction of the reflected ray is-
  • $$\left(4,-8,4\right)$$
  • $$\left(5,-7,5\right)$$
  • $$\left(6,-6,6\right)$$
  • $$\left(-5,7,5\right)$$
The direction cosines of the line which is perpendicular to the lines whose direction cosines are proportional to (1,-1,2) and (2,1,-1) are:- 
  • $$\frac { 1 }{ \sqrt { 35 } } ,-\frac { 5 }{ \sqrt { 35 } } ,\frac { 3 }{ \sqrt { 35 } } $$
  • $$-\frac { 1 }{ \sqrt { 35 } } ,\frac { 5 }{ \sqrt { 35 } } ,\frac { 3 }{ \sqrt { 35 } } $$
  • $$\frac { 1 }{ \sqrt { 35 } } ,\frac { 5 }{ \sqrt { 35 } } ,\frac { 3 }{ \sqrt { 35 } } $$
  • None of these
If from the point $$P ( f , g , h )$$ perpendiculars $$PL , PM$$ be drawn to $$y z$$ and $$zx$$ planes then the equation to the plane $$OLM$$ is -
  • $$\frac { x } { f } + \frac { y } { g } + \frac { z } { h } = 0$$
  • $$\frac { x } { f } + \frac { y } { g } - \frac { z } { h } = 0$$
  • $$- \frac { x } { f } + \frac { y } { g } + \frac { z } { h } = 0$$
  • $$\frac { x } { f } - \frac { y } { g } + \frac { z } { h } = 0$$
The equation of the plane passing through $$(1, 1, 1)$$ and $$(1, -1, -1)$$ and perpendicular to $$2x - y + z + 5 = 0$$ is
  • $$2x + 5y + z - 8 = 0$$
  • $$x + y - z - 1 = 0$$
  • $$2x + 5y + z + 4 = 0$$
  • $$x - y + z - 1 = 0$$
The direction ratios of two lines are $$(4,3,5)$$ and $$(\lambda, -1, 2)$$. If the angle between them is $$45^{o}$$, a value of $$\lambda$$ is
  • $$0$$
  • $$2$$
  • $$3$$
  • $$-1$$
The st lines whose direction cosines satisfy:
$$al+bm+cn=0$$ and $$fmn+gnl+hlm=0$$ are perpendicular if: 
  • $$\dfrac {f}{a}+\dfrac {g}{b}+\dfrac {h}{c}=0$$
  • $$\dfrac {a^{2}}{f}+\dfrac {b^{2}}{g}+\dfrac {c^{2}}{h}=0$$
  • $$\sqrt {af}+\sqrt {bg}+\sqrt {ch}=0$$
  • $$a^{2}f+b^{2}g+c^{2}h=0$$.
If $$l_1$$, $$m_1$$, $$n_1$$ and $$l_2$$, $$m_2$$, $$n_2$$ are the direction cosines of two perpendicular lines, then the direction cosine of the line which is perpendicular to both the lines , will be
  • ($$m_1$$$$n_2$$ - $$m_2$$$$n_1$$), ($$n_1$$$$l_2$$ - $$n_2$$$$l_1$$), ($$l_1$$$$m_2$$ - $$l_2$$$$m_1$$)
  • ($$l_1$$$$l_2$$ - $$m_1$$$$m_2$$), ($$m_1$$$$m_2$$ - $$n_1$$$$n_2$$), ($$n_1$$$$n_2$$ - $$l_1$$$$l_2$$)
  • $$\dfrac{1} {\sqrt {l^{2}_1+m^{2}_1+n^{2}_1}}$$, $$\dfrac{1} {\sqrt {l^{2}_2+m^{2}_2+n^{2}_2}}$$, $$\dfrac{1} {\sqrt3}$$
  • $$\dfrac{1} {\sqrt3}$$, $$\dfrac{1} {\sqrt3}$$, $$\dfrac{1} {\sqrt3}$$
If two straight lines having directions cosines $$\lambda, m, n$$ and $$f, g, h$$ satisfy $$\lambda+m+n=0$$ and $$fmn+gn\lambda+h\lambda m=0$$ and are perpendicular then $$f+g+h$$ is equal to
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$2$$
The Dr's of two lines are 1, -2, -2 and 0, 2, 1 the Dc's of the line perpendicular to the above lines are :-
  • $$\frac { 2 }{ 3 } ,-\frac { 1 }{ 3 } ,\frac { 2 }{ 3 } $$
  • $$-\frac { 1 }{ 3 } ,\frac { 2 }{ 3 } ,\frac { 2 }{ 3 } $$
  • $$\frac { 1 }{ 14 } ,\frac { 3 }{ 4 } ,\frac { 2 }{ 3 }$$
  • None of these
In a plane there are 10 points, no three are in same straight line except 4 points which are collinear, then the number of straight lines are
  • 39
  • 41
  • 45
  • 40
The point collinder with (1,-2,-3) and (2,0,0) amoung the following is 
  • (0,4,6)
  • (0, -4, -5)
  • (0, -4, -6)
  • non of these
A line with direction ratio 2,7,-5 is drawn to intersect the lines $$\frac { x-y }{ 3 } =\frac { y-7 }{ -1 } =\frac { z+2 }{ 1 } $$ and $$\frac { x+3 }{ -3 } =\frac { y-3 }{ 2 } =\frac { z-6 }{ 4 } $$ at P and Q respectively, then length of PQ is-
  • $$\sqrt { 78 } $$
  • $$\sqrt { 77 } $$
  • $$\sqrt { 54 } $$
  • $$\sqrt { 74 } $$
$${ L }_{ 1 }$$ and $${ L }_{ 2 }$$ are two lines whose vector equations are $${ L }_{ 1 }:\vec { r } =\lambda \left[ \left( cos\theta +\sqrt { 3 }  \right) \hat { i } +\left( \sqrt { 2 } sin\theta  \right) \hat { j } +\left( cos\theta -\sqrt { 3 }  \right) \hat { k }  \right] $$
$${ L }_{ 2 }:\vec { r } =\mu \left( a\hat { i } +b\hat { j } +c\hat { k }  \right) ,$$ where$$\lambda$$ and $$\mu $$ are scalars and$$\alpha $$ is the acute angle between $${ L }_{ 1 }$$and$${ L }_{ 2 }$$ If the angle$$ '\alpha '$$ is independent of $$\theta $$ then the value of $$\alpha $$ is
  • $$\dfrac { \pi }{ 6 } $$
  • $$\dfrac { \pi }{ 4 } $$
  • $$\dfrac { \pi }{ 3 } $$
  • $$\dfrac { \pi }{ 2 } $$
Direction ratio of two lines are $$l_{1}, m_{1},n_{1}$$ and  $$l_{2},m_{2},n_{2}$$ then direction ratios of the line perpendicular to both the lines are
  • $$l_{1}-l_{2}, m_{1}-m_{2}, n_{1}-n_{2}$$
  • $$l_{1}+l_{2}, m_{1}+m_{2}, n_{1}+n_{2}$$
  • $$m_{1}n_{2}-n_{1}m_{2}, n_{1}l_{2}-n_{2}l_{1}, l_{1}m_{2}-m_{1}l_{2}$$
  • $$m_{1}n_{2}-n_{1}m_{2}, n_{1}l_{2}-n_{1}l_{1}, l_{1}m_{2}-m_{1}l_{2}$$
The equation to the altitude of the triangle formed by $$(1, 1, 1)$$, $$(1, 2, 3)$$, $$(2, -1, 1)$$ through $$(1, 1, 1)$$.
  • $$\bar{r}=(\bar{i}+\bar{j}+\bar{k})+t(\bar{i}-3\bar{j}-2\bar{k})$$
  • $$\bar{r}=(\bar{i}+\bar{j}+\bar{k})+t(3\bar{i}+\bar{j}+2\bar{k})$$
  • $$\bar{r}=(\bar{i}+\bar{j}+\bar{k})+t(\bar{i}-\bar{j}+2\bar{k})$$
  • $$|\bar{r}|=5$$
The plane passing through $$(1, 1, 1), (1, -1, 1)$$ and $$(-7, -3, -5)$$ is parallel to 
  • $$X-axis$$.
  • $$Y-axis$$.
  • $$Z-axis$$.
  • None of these
Each group from the alternatives represents lengths of sides of a triangleStare which group does not represent a right-angled triangle.
  • $$ ( 8,40,41 ) $$
  • $$ ( 20,25,30 ) $$
  • $$ ( 8,15,17 ) $$
  • $$ ( 6,8,10 ) $$
there are 20 points in the plane on three of which are collinear. the number of straight lines by joining them is
  • 190
  • 200
  • 40
  • 500
The lines $$\vec{r}=i-j+\lambda(2i+k)$$ and $$\vec{r}=(2i-j)+\mu(i+j-k)$$ intersect for
  • $$\lambda=1, \mu =1$$
  • $$\lambda=2, \mu =1$$
  • All values of $$\lambda$$ and $$\mu$$
  • No value of $$\lambda$$ and $$\mu$$
The value of p so that the lines $$\frac { 1-x }{ 3 } =\frac { 7y-14 }{ 2p } =\frac { z-3 }{ 2 } $$ and $$\frac { 7-7x }{ 3p } =\frac { y-5 }{ 1 } =\frac { 6-z }{ 5 } $$ are at right angles are
  • 70/11
  • 7/11
  • 10/7
  • 17/11
What is the area of the triangle with vertices $$(0,2,2),\,(2,0,-1)$$ and $$(3,4,0)$$ ?
  • $$\frac{{15}}{2}sq$$ unit
  • $$15sq$$ unit
  • $$\frac{{7}}{2}sq$$ unit
  • $$7sq$$ unit
In the three points with position vectors (1, a. b) : (a, b, 3) are collinear in space, then the value of a + b is 
  • 3
  • 4
  • 5
  • none
The number of straight lines that can be drawn through any two points out of $$10$$ points, of which $$7$$ are collinear.
  • $$25$$
  • $$30$$
  • $$35$$
  • $$45$$
If the vectors $$2\hat{i} + 3\hat{j} , ~5\hat{i} + 6\hat{j} ,$$ and $$ 8\hat{i} +\lambda{\hat{j}}$$ have their initial points at $$(1 , 1)$$, then the value of $$\lambda$$ so that the vectors terminate on one straight line is
  • $$0$$
  • $$3$$
  • $$6$$
  • $$9$$
If  $$ ( 0,0 ) , ( a , 0 )$$  and  $$( 0 , b )$$  are collinear, then
  • $$a b = 0$$
  • $$a = b$$
  • $$a = - b$$
  • $$a - b = c$$
If points $$(a - 2, a - 4); (a, a + 1)$$ and $$(a + 4, 16)$$ are collinear, then $$a$$ is equal to
  • $$5$$
  • $$-5$$
  • $$7$$
  • $$-7$$
The direction cosines of the normal to the plane $$2x - y + 2z = 3 $$ are 
  • $$\dfrac{2}{3},\dfrac{-1}{3},\dfrac{2}{3}$$
  • $$\dfrac{-2}{3},\dfrac{1}{3},\dfrac{-2}{3}$$
  • $$\dfrac{2}{3},\dfrac{1}{3},\dfrac{2}{3}$$
  • $$\dfrac{2}{3},\dfrac{-1}{3},\dfrac{-2}{3}$$
The plane passing through the point $$\left ( 5,1,2 \right )$$ perpendicular to the line $$2\left ( x - 2 \right ) = y - 4 = z - 5$$ will meet the line in the point
  • $$\left ( 1,2,3 \right )$$
  • $$\left ( 2,3,1 \right )$$
  • $$\left ( 1,3,2 \right )$$
  • $$\left ( 3,2,1 \right )$$
The equation of the plane passing through the points $$\left ( 3,2,-1 \right ), \left ( 3,4,2 \right )$$ and $$\left ( 7,0,6 \right )$$ is $$5x + 3y - 2z =\lambda$$ where $$\lambda$$ is
  • $$23$$
  • $$21$$
  • $$19$$
  • $$27$$
The vector equation of line 2x - 1 = 3 y + 2 = z - 2 is 
  • $$\bar{r}=\left ( \dfrac{1}{2}\hat{i}-\dfrac{2}{3}\hat{j}+2\hat{k} \right )+\lambda \left ( 3\hat{i}+2\hat{j}+6\hat{k} \right )$$
  • $$\bar{r}=\hat{i}-> j+(2\hat{i}+\hat{j}+\hat{k})$$
  • $$\bar{r}=\left ( \dfrac{1}{2}\hat{i}-\hat{j}\right )+\lambda \left ( \hat{i}+2\hat{j}+6\hat{k} \right )$$
  • $$\bar{r}=\left ( \hat{i}+\hat{j} \right )+\lambda \left ( \hat{i}-2\hat{j}+6\hat{k} \right )$$
If P, Q  R are collinear points such that P( 7, 7) Q( 3, 4) and PR = 10 then R is 
  • (1, 1)
  • ( 1, -1)
  • ( -1, 1)
  • ( -1, -1)
0:0:1


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