MCQ Questions for Class 11 Engineering Maths Introduction To Three Dimensional Geometry Quiz 11

$P\left( x,y,z \right)$ is a point on the line segment joining

$Q\left( 2,2,4 \right)$ and

$R\left( 3,5,6 \right)$ such that the projections of

$OP$ on the axis are

$\cfrac { 13 }{ 5 } ,\cfrac { 19 }{ 5 } ,\cfrac { 26 }{ 5 }$ respectively, then

$P$ divides

$QR$ in the ratio

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$1:2$
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$3:2$
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$2:3$
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$1:3$

Explanation

Since,

$\overline { OP }$ has projections

$\cfrac { 13 }{ 5 } ,\cfrac { 19 }{ 5 } ,\cfrac { 26 }{ 5 }$ on the coordinate axes

$\therefore \overline { OP } =\cfrac { 13 }{ 5 } \hat { i } +\cfrac { 19 }{ 5 } \hat { j } +\cfrac { 26 }{ 5 } \hat { k }$
Suppose

$P$ divides the line segment joining of

$Q(2,2,4)$ and

$R(3,5,6)$ in the ratio

$\lambda:1$
then the position vector of

$P$ is

$\left( \cfrac { 3\lambda +2 }{ \lambda +1 } \right) \hat { i } +\left( \cfrac { 5\lambda +2 }{ \lambda +1 } \right) \hat { j } +\left( \cfrac { 6\lambda +4 }{ \lambda +1 } \right) \hat { k }$

$\therefore \cfrac { 13 }{ 5 } \hat { i } +\cfrac { 19 }{ 5 } \hat { j } +\cfrac { 26 }{ 5 } \hat { k } =\left( \cfrac { 3\lambda +2 }{ \lambda +1 } \right) \hat { i } +\left( \cfrac { 5\lambda +2 }{ \lambda +1 } \right) \hat { j } +\left( \cfrac { 6\lambda +4 }{ \lambda +1 } \right) \hat { k }$

$\Rightarrow \lambda =3/2$

$\therefore$ Required ratio in which

$P$ divides

$QR$ is

$3:2$

The points

$A(1,2,3); B-(-1,-2,-1); C(2,3,2)$ and

$D(4,7,6)$ form

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Square
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Rectangle
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Parallelogram
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Rhombus

Find the coordinates of the points which trisect the line segment AB, given that A =( 2,1 , - 3 ) and B =( 5 , - 8,3 ).

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$( 3,2 , - 1 ) ; ( 4,5 , - 1 )$
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$( 3 , - 2 , - 1 ) ; ( 4 , - 5,1 )$
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$( 3 , - 2,1 ) ; ( - 4,5 , - 1 )$
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$( 3 , - 2 , - 1 ) ; ( 4,5 , - 1 )$

30 consider at three dimensional figure represented by

$xy{z^2} = 2$ , then its minimum distance from origin is

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2
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4
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3
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1

The distances of the point

$P(1,2,3)$ from the coordinates axes are:

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$\sqrt {13} ,\sqrt {10} ,\sqrt 5$
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$\sqrt {11} ,\sqrt {10} ,\sqrt 5$
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$\sqrt {13} ,\sqrt {20} ,\sqrt {15}$
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$\sqrt {23} ,\sqrt {10} ,\sqrt 5$

If O and O' are circumcenter and orthocenter of a

$\Delta ABC$ where

$\overline{OA} + \overline{OB} + \overline{OC}$ is

$\lambda \overline{OO'}$ then the value of

$\lambda$ is

0%
$1$
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$2$
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$3$
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$4$

The distance between the orthocentre and circumcentre of the triangle formed by the points

$(1, 2, 3), (3, -1, 5), (4, 0, -3)$ is

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$\dfrac {\sqrt {66}}{2}$
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$\dfrac {\sqrt {17}}{2}$
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$\dfrac {\sqrt {55}}{2}$
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$\dfrac {\sqrt {37}}{2}$

In the

$xy-plane$ , the length of the shortest from

$(0, 0)$ to

$(12, 16)$ that does not go inside the circle

$(x - 6)^{2} + (y + 8)^{2} = 25$ is

0%
$10\sqrt {3}$
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$10\sqrt {5}$
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$10\sqrt {3} + \dfrac {5\pi}{3}$
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$10 + 5\pi$

$R$ divides the line segment joining

$P(2,3,4)$ and

$Q(4,5,6)$ in the ratio

$-3:2$ , then the parameter which represent

$R$ is

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$3$
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$2$
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$1$
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$-1$

Explanation

$x=\cfrac { -3\times 4+2\times 2 }{ -1 } \quad \quad y=\cfrac { -3\times 5+2\times 3 }{ -1 }$

$x=\cfrac { -12+4 }{ -1 } \quad \quad$

$y =\cfrac { -15+6 }{ -1 }$

$x=8\quad \quad y=9$

$z=\cfrac { -3\times 6+2\times 4 }{ -1 } =\cfrac { -10 }{ -1 }$

$z=10$

$R=\left( 8,9,10 \right)$

1126182_1037717_ans_e18118a9f026412ab072f59cbe36b268.JPG

Consider at three dimensional figure represented by

$xy{ z }^{ 2 }=2$ , then its minimum distance from origin is

0%
2
0%
4
0%
6
0%
8

The point which is equidistant from the points

$(-1,1,3),(2,1,2),(0,5,6)$ and

$(3,2,2)$ is

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$(-1,3,4)$
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$(3,1,4)$
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$(1,3,4)$
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$$(4,1,3)$4

Minimum distance between the curves

$y^{2}=4x$ &

$x^{2}+y^{2} -12x+31=0$ is -

0%
$\sqrt {21}$
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$\sqrt {26}-\sqrt {5}$
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$\sqrt {20}-\sqrt {5}$
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$\sqrt {21}-\sqrt {5}$

Let

$A(2,3,5),B(-1,3,2)$ and

$C(\lambda ,5,\mu )$ are the vertices of a triangle and its median through A meets side BC at D. AD is equally inclined with the axes. If E is the point on BC such that

$BE:EC=1:2.$

Project of BA on BC

0%
$\dfrac { 23 }{ \sqrt { 33 } }$
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$\dfrac { \sqrt { 33 } }{ 24 }$
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$\dfrac { 24 }{ \sqrt { 33 } }$
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$None$ $of$ $these$

If A = (2,-3,1), B = (3,-4,6) and C is a point of trisection of AB, then

${ C }_{ y }$

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$\frac { 11 }{ 3 }$
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-11
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$\frac { 10 }{ 3 }$
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$\frac { -11 }{ 3 }$

The coordinates of the orthocentre of the triangle that has the coordinates of mid points of its sides as (0 , 0) (1 , 2) and ( -6 , 3) is :

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(0 , 0)
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(-4 , 5)
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(-5 , 5)
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(-4 , 4)

Let

$A(2,3,5),B(-1,3,2)$ and

$C(\lambda ,5,\mu )$ are the vertices of a triangle and its median through A meets side BC at D. AD is equally inclined with the axes. If E is the point on BC such that

$BE:EC=1:2.$

Equation of plane containing triangle ABC

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$x+y+3=0$
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$x-z-3=0$
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$x-z+3=0$
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$x-y+3=0$

Perpendicular distance from the origin to the line joining the points

$(a\cos{\theta},a\sin{\theta})(a\cos{\theta},a\sin{\theta})$ is

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$2a\cos{(\theta-\phi)}$
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$a\cos { \left( \cfrac { \theta -\phi }{ 2 } \right) }$
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$4a\cos { \left( \cfrac { \theta -\phi }{ 2 } \right) }$
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$a\cos { \left( \cfrac { \theta +\phi }{ 2 } \right) }$

Consider a variable plane

$lx+my+nz=k(k>0)and\quad l,m,n$ are direction cosines of normal of the plane. Let the given plane intersects the co-ordinate axes at A,B and C, then the minimum area of

$\triangle ABC$ is _______.

0%
$\dfrac { 3\sqrt { 3k } ^{ 2 } }{ 2 }$
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$\dfrac { 3\sqrt { 3k } ^{ 2 } }{ 4 }$
0%
$3\sqrt { 3 } { k }^{ 2 }$
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$12\sqrt { 3 } { k }^{ 2 }$

The shortest distance between the point

$\left( \dfrac { 3 }{ 2 } ,0 \right)$ and the curve

$y=\sqrt { x }$ ,

$(x>0)$ , is:

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$\dfrac { 3 }{ 2 }$
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$\dfrac { \sqrt { 3 } }{ 2 }$
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$\dfrac { \sqrt { 5 } }{ 2 }$
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$\dfrac { 5 }{ 4 }$

Q, R, S are the points

$(-2, -1), (0, 3) (4, 0)$ respectively. Then the coordinates of P such that PQRS is a parallelogram is ________________.

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$(2, -6)$
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$(-6, 2)$
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$(2, -4)$
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$(-3, 2)$

$A=(1, 2, 3)$ and

$B(3, 5, 7)$ and P, Q are the points on AB such that AP

$=$ PQ

$\neq$ QB, then the mid point of PQ is?

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$(2, 3, 5)$
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$\left(2, \dfrac{7}{2}, 5\right)$
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$(2, 4, 5)$
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$(4, 7, 0)$

$A=(1, -2, -1), B=(4, 0, -3); C=(1, 2, -1)$ and

$D=(2, -4, -5)$ , then the distance between AB and CD is?

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$\dfrac{2}{3}$
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$\dfrac{4}{3}$
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$\dfrac{3}{2}$
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$\dfrac{5}{3}$

The equation of plane which is passing through the point

$(1,2,3)$ and which is at maximum distance from the point

$(-1,0,2)$ is

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$2x+2y+z=9$
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$2x+z=5$
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$3x+y-z=2$
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none of these

The distance of the point

$(2,1,-1)$ from the line

$\dfrac{x-1}{2}=\dfrac{y+1}{1}=\dfrac{z-3}{-3}$ measured parallel to the plane

$x+2y+z=4$ is

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$\sqrt{10}$
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$\sqrt{20}$
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$\sqrt{5}$
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$\sqrt{30}$

A line passes through two points A(2, -3, -1) and B(8, -1, 2) the coordinates of a point on this line nearer to the origin at a distance of 14 units from A are

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(14 , 1, 5)
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(-10, -7, -7)
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(10 , 7, 7)
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(-14, -1, -5)

$\lambda$ is the length of any edge of a regular tetrahedron, then the distance of any vertex form the opposite face is-

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$\dfrac{2}{3}\lambda^{2}$
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$\sqrt{\dfrac{2}{3}}\lambda$
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$\dfrac{\sqrt{2}}{3}\lambda$
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$None of these$

Explanation

$\begin{array}{l} In\, \, { 30^{ 0 } }-{ 60^{ 0 } }-{ 90^{ 0 } }-side\, \, is\, \, in\, \, ratio,\, \\ 1:\sqrt { 3 } :2 \end{array}$

In regular tetraheron, height (h)

$\begin{array}{l} 5\frac { { \sqrt { 2 } } }{ 3 } \\ and\, \, side\, \, ,s=\lambda \\ \therefore \lambda =\sqrt { \frac { 2 }{ 3 } } \lambda \end{array}$

1218909_1319837_ans_54acc5016e9342939f49e1e00387951a.PNG

The distance of the point

$(2,3)$ form the line

$x-2y+5=0$ measured in a direction parallel to the line

$x-3y=0$ is

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$2\sqrt{10}$
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$\sqrt{10}$
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$2\sqrt{5}$
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$None\ of\ these$

$A=(1,-2,-1), B=(4,0,-3), C=(1,2,-1), D=(2,-4,-5)$ , then distance between A B and CD is

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$\frac{1}{3}$
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$\frac{2}{3}$
0%
1
0%
$\frac{4}{3}$

$A(1, -1, -3)$ ,

$B(2,1,-2)$ &

$(-5, 2, -6)$ are the position vectors of the vertices of a triangle ABC. The length of the bisector of its internal angle at A is:

0%
$\sqrt { 10 } /4$
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$3\sqrt { 10 } /4$
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$\sqrt { 10 }$
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None

If x-coordinates of a point P on the joining the points

$Q(2, 2,1)$ and

$R(5, 1, -2)$ is 4, then the z-coordinates of P is

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-2
0%
-1
0%
1
0%
2

CBSE Questions for Class 11 Engineering Maths Introduction To Three Dimensional Geometry Quiz 11 - MCQExams.com

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Practice Class 11 Engineering Maths Quiz Questions and Answers

CBSE Questions for Class 11 Engineering Maths Introduction To Three Dimensional Geometry Quiz 11 - MCQExams.com

0:0:1

Practice Class 11 Engineering Maths Quiz Questions and Answers

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