Three Dimensional Geometry Questions

We provide three dimensional geometry practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on three dimensional geometry skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

List of three dimensional geometry Questions

Question No Questions Class
1 Perimeter of triangle whose vertices are (0,4,0),(3,4,0) and ( (0,4,4), ) is
A . 10
B. 12
c. 25
D. 15
12
2 87.
(a) 315 (b) 256 (c) 84
The length of the projection of the line segment joining the
points (5,-1, 4) and (4,-1,3) on the plane, x+y+z=
JEE M 2018]
WIN
12
3 Prove that the points ( boldsymbol{A}= ) ( (1,2,3), B(3,4,7), C(-3,-2,-5) ) are
collinear ( & ) find the ratio in which ( B )
divides ( boldsymbol{A C} )
A .2: 5
B . 2: 3
c. 2: 8
D. 2: 7
12
4 In geometry, we take a point, a line and a plane as undefined terms.
A. True
B. False
c. Ambiguous
D. Data Insufficient
12
5 Find the shortest distance between the
skew lines:
( l_{1}: frac{x-1}{2}=frac{y+1}{1}=frac{z-2}{4} )
( l_{2}: frac{x+2}{4}=frac{y-0}{-3}=frac{z+1}{1} )
12
6 The direction ratios of the line joining the points (4,3,-5) and (-2,1,-8) are
A ( cdot frac{6}{7}, frac{2}{7}, frac{3}{7} )
в. 6,2,3
c. 5,8,0
D. 3,7,9
12
7 If a point ( boldsymbol{P} ) from where line drawn cuts
coordinates axes at ( A ) and ( B ) (with ( A ) on
( x-text { axis and } B text { on } y-text { axis }) ) satisfies ( alpha frac{x^{2}}{P B^{2}}+beta frac{y^{2}}{P A^{2}}=1, ) then ( alpha+beta ) is
( A cdot 1 )
B. 2
( c .3 )
D. 4
12
8 The planes ( 2 x-y+4 z=5 ) and ( 5 x- )
( 2.5 y+10 z=6 ) are
A. Parallel
B. Perpendicular
c. Intersect
D. intersect ( x ) axis
12
9 The number of straight line that are
equally inclined to the three dimensional co- ordinate axes, is
12
10 ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}=(mathbf{4}, mathbf{5}, mathbf{7}), boldsymbol{C}= )
( (-4,3,-6), D=(2, k, 2) ) are four
points. If the lines ( A B ) and ( C D ) are
parallel, then ( k= )
( A cdot 0 )
в. -9
( c .9 )
D. 2
12
11 Algebraic sum of intercepts made by the plane ( x+3 y-4 z+6=0 ) on the axes is
A. 7
B. 0
( c cdot frac{13}{2} )
( D cdot-frac{13}{2} )
12
12 The following lines are ( hat{boldsymbol{r}}=(hat{boldsymbol{i}}+hat{boldsymbol{j}})+ )
( lambda(hat{i}+2 hat{j}-hat{k})+mu(-hat{i}+hat{j}-2 hat{k}) )
A. collinear
B. skew-lines
c. co-planar lines
D. parallel lines
12
13 Find the point on Z-axis which are at a distance ( sqrt{21} ) unit from the point
(1,2,3)
12
14 If ( boldsymbol{A}=(mathbf{2},-mathbf{3}, mathbf{1}), boldsymbol{B}=(mathbf{3},-mathbf{4}, mathbf{6}) ) and ( boldsymbol{C} )
is a point of trisection of ( A B, ) then ( C_{y}= )
A ( cdot frac{11}{3} )
B. -11
c. ( frac{10}{3} )
D. ( frac{-11}{3} )
12
15 The distance between the circumcentre
and the ortho centre of the triangle formed by the points (2,1,5),(3,2,3) and (4,0,4) is
A ( cdot sqrt{6} )
B. ( frac{sqrt{6}}{2} )
c. ( 2 sqrt{6} )
D.
12
16 Find the coordinates of a point
equidistant from four points ( boldsymbol{O}(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{A}(ell, mathbf{0}, mathbf{0}), boldsymbol{B}(mathbf{0}, boldsymbol{m}, boldsymbol{0}) ) and
( boldsymbol{C}(mathbf{0}, mathbf{0}, boldsymbol{n}) )
12
17 Find the direction cosines of a line
which makes equal angles with the coordinate axes.
12
18 The projections of a directed line segment on the coordinate axes are
12,4,3 respectively.

What are the direction cosines of the
line segment?
A ( cdot(12 / 13,4 / 13,3 / 13) )
B . ( (12 / 13,-4 / 13,3 / 13) )
c. ( (12 / 13,-4 / 13,-3 / 13) )
D. ( (-12 / 13,-4 / 13,3 / 13) )

12
19 If ( boldsymbol{A}=(mathbf{4}, mathbf{1}, mathbf{5}) ) and ( boldsymbol{B}=(mathbf{3}, mathbf{4}, mathbf{5}) ) The
direction ratios of ( overline{A B} ) are
12
20 x-2 y. 2+2
56. Let the line –
lie in the plane
3 -5 2 .
x+3y-az+B=0. Then (a, b) equals
[2009]
(a) (-6,7)
(b) (5,-15)
(c) (-5,5)
(d) (6,-17)
12
21 If ( boldsymbol{A}=(mathbf{5},-mathbf{1}, mathbf{1}), boldsymbol{B}=(mathbf{7},-mathbf{4}, mathbf{7}), boldsymbol{C}= )
( (1,-6,10), D=(-1,-3,4) . ) Then
( A B C D ) is a
A . square
B. rectangle
c. rhombus
D. none of these
12
22 Find the ratio in which ( 2 x+3 y+5 z= )
1 divides the line joining the points
(1,0,-3) and (1,-5,7)
A .1: 2
B . 2: 1
c. 3: 2
D. 2: 3
12
23 A plane mirror is placed at the origin so that the direction ratios of its normal
( operatorname{are}(1,-1,1) . ) A ray of light, coming along the positive direction of the ( x ) axis, strikes the mirror. The direction ( operatorname{cosines} ) of the reflected ray are
A ( cdot frac{1}{3}, frac{2}{3}, frac{2}{3} )
B. ( -frac{1}{3}, frac{2}{3}, frac{2}{3} )
c. ( -frac{1}{3},-frac{2}{3},-frac{2}{3} )
D. ( -frac{1}{3},-frac{2}{3}, frac{2}{3} )
12
24 Find the square of the distance between
the points whose cartesian coordinates
are:
(-1,1,3),(0,5,6)
12
25 Consider three vectors ( vec{P}=hat{i}+widehat{j}+ ) ( widehat{k} ; overrightarrow{boldsymbol{q}}=2 widehat{hat{boldsymbol{i}}}+4 widehat{boldsymbol{j}}-widehat{boldsymbol{k}} ) and ( overrightarrow{boldsymbol{r}}=boldsymbol{2} hat{boldsymbol{i}}+boldsymbol{4} widehat{boldsymbol{j}}+ )
3 ( widehat{k} ). If ( vec{p}, vec{q} ) and ( vec{r} ) denotes the position vector of three non-collinear points, then the equation of the plane containing these points is
A. ( 2 x-3 y+1=0 )
B. ( x-3 y+2 z=0 )
c. ( 3 x-y+z-3=0 )
D. ( 3 x-y-2=0 )
12
26 The d.r’s of the line of intersection of the
planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}-mathbf{1}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}+ )
( 3 y+4 z-7=0 ) are
A .1,2,-3
в. 2,1,-3
c. 4,2,-6
D. 1,-2,1
12
27 Find the distance of point (1,-2,3)
from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} )
measured parallel to the line whose direction cosines are
proportional to 2,3,-6
12
28 Let two planes ( p_{1}: 2 x-y+z=2, ) and
( boldsymbol{p}_{2}: boldsymbol{x}+boldsymbol{2} boldsymbol{y}-boldsymbol{z}=boldsymbol{3} ) are given. The
image of plane ( P_{1} ) in the plane mirror
( P_{2} ) is
A. ( x+7 y-4 z+5=0 )
B. ( 3 x+4 y-5 z+9=0 )
c. ( 7 x-y+2 z-9=0 )
0
D. ( 7 x+y+9 z+9=0 )
12
29 If the points ( (h, 3,-4),(0,-7,10) ) and ( (1, k, 3) ) are collinear, then ( h+k ) is
( mathbf{A} cdot mathbf{4} )
B.
c. -4
D. 14
12
30 Find the equation of the plane containing the line 2x -y+z
-3 = 0, 3x +y+z= 5 and at a distance of Ta from the point
(2,1,-1).
(2005 – 2 Marks)
12
31 If ( P(x, y, z) ) moves such that ( x=0, z= )
( 0, ) then the locus of ( P ) is the line whose
d.cs are
A . ( y ) -axis
B. 1,0,0
c. 0,1,0
D. 0,0,0
12
32 In three dimensions, the coordinate
axes of a rectangular cartesian coordinate system are
A. three mutually parallel lines
B. three mutually perpendicular lines
c. two mutually perpendicular lines and any two parallel
D. None of these
12
33 If line ( frac{boldsymbol{x}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{y}-mathbf{4}}{mathbf{4}}=frac{boldsymbol{z}+mathbf{2}}{mathbf{1}} ) is
parallel to planes ( mu x+3 y-2 z+d= )
0 and ( x-2 lambda y+z=0, ) then value of ( lambda )
and ( mu ) are
A ( cdotleft(mu=4, lambda=-frac{2}{3}right. )
B. ( mu=-6, lambda=-2 )
c. ( _{mu}=frac{-10}{3}, lambda=frac{-1}{2} )
D. ( mu=frac{-10}{3}, lambda=frac{1}{2} )
12
34 Vectors ( vec{A}, vec{B} ) and ( vec{C} ) are such that ( vec{A} ) ( vec{B}=0 . ) Then the vector parallel to ( vec{A} ) is
A. ( vec{B} ) and ( vec{C} )
в. ( vec{A} times vec{B} )
c. ( vec{B}+vec{C} )
D. ( vec{B} times vec{C} )
12
35 Find the equation of the plane passing through (2,0,1) and (3,-3,4) and perpendicular to ( boldsymbol{x}-mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{6} ) 12
36 Number of points having positive integral co-ordinate lying on the plane ( x+2 y+3 z=15 ) is ( n, ) then ( frac{n}{2} ) is equal
to
( mathbf{A} cdot mathbf{6} )
B. 8
c. 9
D.
12
37 The vector equation of the plane passes through the points ( A & B ) with position vector ( 2 hat{i}+hat{j}-hat{k} &-hat{i}+3 hat{j}+4 hat{k} )
respectively ( & ) Ler to the plane ( bar{r} cdot(hat{i}-2 hat{j}+4 hat{k})=10 ) is
A ( cdot bar{r} cdot(18 hat{i}+17 hat{j}-3 hat{k})=49 )
B . ( bar{r} .(18 hat{i}-17 hat{j}-3 hat{k})+22=0 )
c. ( bar{r} .(18 hat{i}+17 hat{j}+4 hat{k})=25 )
D・ ( bar{r} .(18 hat{i}+17 hat{j}+4 hat{k})=24 )
12
38 The vector equation of the line ( frac{x-2}{2}= ) ( frac{2 y-5}{-3}, z=-1 ) is ( vec{r}= )
( left(2 hat{i}+frac{5}{2} hat{j}-hat{k}right)+lambdaleft(2 hat{i}-frac{3}{2} hat{j}+x hat{k}right) )
where ( x ) is equal to
( mathbf{A} cdot mathbf{0} )
B.
c. 2
D. 3
12
39 If ( P(x, y, z) ) is a point on the line segment joining ( Q(2,2,4) ) and ( R(3,5,6) ) such that the projection of ( overrightarrow{O P} ) on the axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} )
respectively, then ( P ) divides ( Q R ) in ratio
A .1: 3
B. 2: 3
c. 3: 2
D. 3: 1
12
40 f ( (p, q, r) ) is equidistant from
(1,2,-3),(2,-3,1) and ( (-3,1,2), ) then
( boldsymbol{p}+boldsymbol{q}+boldsymbol{r}= )
A . -1
B.
c. 0
D.
12
41 The acute angle between two lines such
that the direction cosines ( l, boldsymbol{m}, boldsymbol{n} ) of each of them satisfy the equation ( l+ )
( boldsymbol{m}+boldsymbol{n}=mathbf{0} ) and ( l^{2}+boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} ) is
( A cdot 30 )
B . 45
( c cdot 60 )
D. ( 15^{circ} )
12
42 Plane ( a x+b y+c z=1 ) intersect axes ( operatorname{in} A, B, C ) respectively. If ( Gleft(frac{1}{6},-frac{1}{3}, 1right) )
is a centroid of ( triangle A B C ) then ( a+b+ )
( 3 c=-1 )
A ( cdot frac{4}{3} )
B. 4
( c cdot 2 )
D.
12
43 The equation of the plane passing through the straight line ( frac{x-1}{2}= ) ( frac{boldsymbol{y}+mathbf{1}}{mathbf{- 1}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{4}} ) and perpendicular to
plane ( boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{1 2} ) is:
A. ( 9 x+2 y-5 z+8=0 )
в. ( 9 x+2 y-5 z+10=0 )
c. ( 9 x-2 y+5 z+6=0 )
D. ( 9 x-2 y-5 z+4=0 )
12
44 Derive the equation of the locus of a point equivalent from the points (1,-2,3) and (-3,4,2) 12
45 If vector ( vec{a}=4 hat{i}+5 hat{j}-3 hat{k} ) and ( vec{b}=5 hat{i}+ )
( 3 hat{j}+8 hat{k} ) then value of ( frac{text { projection of vector b on a }}{text { projection of vector a on b }} ) is :
A ( cdot frac{7}{5} )
B. ( frac{2}{7} )
( c cdot frac{5}{8} )
D. None of these.
12
46 Find ( a, b, c ) if ( a(1,3,2)+b(1,-5,6)+ )
( c(2,1,-2)=(4,10,-8) )
12
47 A variable plane at a distance of 1 unit from the origin cuts the co-ordinate axes at ( A, B ) and ( C . ) If the centroid ( D(x, y, z) ) of triangle ( A B C ) satisfies the relation ( frac{1}{x^{2}}+ ) ( frac{1}{y^{2}}+frac{1}{z^{2}}=k, ) then the value of ( k ) is
A . 3
B. 1
c. ( 1 / 3 )
D.
12
48 The points (-5,12),(-2,-3),(9,-10),(6,5)
taken in order, form
A. Parallelogram
B. rectangle
c. rhombus
D. square
12
49 The distance from the origin to the centroid of the tetrahedron formed by
the points ( (0,0,0),(a, 0,0),(0, b, 0),(0,0, c) ) is:
A ( cdot frac{sqrt{a+b+c}}{4} )
B. ( frac{sqrt{a+b+c}}{3} )
c. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{16} )
D. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{4} )
12
50 A point ( P ) lies on a line whose ends are ( A(1,2,3) ) and ( B(2,10,1) . ) If ( z )
component of ( boldsymbol{P} ) is ( boldsymbol{7}, ) then the
coordinates of ( boldsymbol{P} ) are
A ( cdot(-1,-14,7) )
B. (1,-14,7)
c. (-1,14,7)
D. (1,14,7)
12
51 A parallelopiped ‘S’ has base points A, B, C and D and
upper face points A’, B’, C and D’. This parallelopiped is
compressed by upper face A’B’C’D’ to form a new
parallelopiped ‘T” having upper face points A”, B”, C” and
D”. Volume of parallelopiped Tis 90 percent of the volume
of parallelopiped S. Prove that the locus of ‘A”?, is a plane
12
52 If 5,7,6 are the sums of the ( x, y )
intercepts; ( boldsymbol{y}, boldsymbol{z} ) intercepts, ( boldsymbol{z}, boldsymbol{x} )
intercepts respectively of a plane then the perpendicular distance from the origin to that plane is
A ( cdot frac{144}{61} )
в. ( frac{12}{sqrt{61}} )
c. ( frac{sqrt{61}}{12} )
D. ( frac{61}{144} )
12
53 Find the angle between the following pairs of lines:
( frac{x-1}{2}=frac{y-2}{3}=frac{z-3}{-3} ) and ( frac{x+3}{-1}= )
( frac{boldsymbol{y}-mathbf{5}}{mathbf{8}}=frac{boldsymbol{z}-mathbf{1}}{mathbf{4}} )
12
54 The vector equation of the plane which is at a distance of ( frac{3}{sqrt{14}} ) from the origin and the normal from the origin is ( 2 hat{i}- ) ( mathbf{3} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) is
A ( . vec{r} .(2 hat{i}-3 hat{j}+hat{k})=3 )
B . ( vec{r} .(hat{i}+hat{j}+hat{k})=9 )
c. ( vec{r} .(hat{i}+2 hat{j})=3 )
D. ( vec{r} .(2 hat{i}+hat{k})=3 )
12
55 If a line makes angles ( alpha, beta, gamma ) with axes
of co-ordinates, then ( cos 2 alpha+cos 2 beta+ )
( cos 2 gamma ) is equla to
A . -2
B. –
c. 1
D. 2
12
56 Name the octants in which the following
points lie:
( (1,2,3),(4,-2,3)(4,-2,-5),(4,2,-5) )
12
57 The perpendicular distance of ( vec{A}(1,4,-2) ) from the segment BC where ( vec{B} ) (2,1,-2) and ( vec{C}(0,-5,1) ) is
( A cdot frac{3}{7} sqrt{26} )
B ( cdot frac{6}{7} sqrt{26} )
( mathbf{c} cdot frac{4}{7} sqrt{26} )
( D cdot frac{2}{7} sqrt{26} )
12
58 Find the direction cosines of
perpendicular from the origin to the plane ( bar{r}(2 hat{i}+3 hat{j}+6 hat{k})+7=0 )
12
59 Perpendiculars ( A P, A Q ) and ( A R ) are drawn to the ( x-, y- ) and ( z- ) axes, respectively
from the point ( boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{2}) . ) The A.M. of
( A P^{2}, A Q^{2} ) and ( A R^{2} ) is
A .4
B. 5
( c .3 )
D.
12
60 If ( boldsymbol{A}(boldsymbol{6},-mathbf{7}, mathbf{0}), boldsymbol{B}(mathbf{1 6},-mathbf{1 9},-mathbf{4}) )
( C(0,3,-6) ) and ( D(2,-5,10) ) are four points in space, then the point of intersection of the lines ( A B ) and ( C D ) is
A ( cdot(2,1,-1) )
в. (1,1,2)
D. does not exist as the lines are skew
12
61 A plane meet the co-ordinate axes in
( A, B, C ) such that the centroid of
triangle ( A B C ) is ( (a, b, c) . ) If equation of plane ( frac{x}{a}+frac{y}{b}+frac{z}{c}=k(k neq 0), ) then the
value of ( k ) equals
A . 2
B. 3
( c cdot 4 )
D. 5
12
62 Two system of rectangular axes have the same origin. If a plane cuts them at
distances, ( a, b, c ) and ( a_{1}, b_{1}, c_{1} ) from the
origin, then
A ( cdot frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}=frac{1}{a_{1}^{2}}+frac{1}{b_{1}^{2}}+frac{1}{c_{1}^{2}} )
B. ( frac{1}{a^{2}}-frac{1}{b^{2}}+frac{1}{c^{2}}=frac{1}{a_{1}^{2}}-frac{1}{b_{1}^{2}}+frac{1}{c_{1}^{2}} )
C ( cdot a^{2}+b^{2}+c^{2}=a_{1}^{2}+b_{1}^{2}+c_{1}^{2} )
D cdot ( a^{2}-b^{2}+c^{2}=a_{1}^{2}-b_{1}^{2}+c_{1}^{2} )
12
63 ox, oy are positive x-axis, positive ( y ) axis respectively where ( boldsymbol{O}=(mathbf{0}, mathbf{0}, mathbf{0}) )
The ( d . c . s ) of the llne which bisects ( angle x o y )
are
( mathbf{A} cdot 1,1,0 )
B. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, 0 )
c. ( frac{1}{sqrt{2}}, 0, frac{1}{sqrt{2}} )
D. 0,0,1
12
64 Find the equation of the set of points which are equidistant from the points (1,2,3) and (3,2,-1) 12
65 The ratio in which the plane ( bar{r} .(bar{i}-2 bar{j}+ ) ( mathbf{3} bar{k})=17 ) divides the line joining the points ( -2 bar{i}+4 bar{j}+7 bar{k} ) and ( 3 bar{i}-5 bar{j}+8 bar{k} )
is
A. 1: 10
B. 3: 10
( c .3: 5 )
D. 1: 5
12
66 Consider three planes ( 2 x+p y+6 z= )
( mathbf{8}, boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{q} boldsymbol{z}=mathbf{5} ) and ( boldsymbol{x}+boldsymbol{y}+mathbf{3} boldsymbol{z}=mathbf{4} )
These planes do not have any common point of intersection if-
A. ( p=2, q neq 3 )
B . ( p neq 2, q neq 3 )
c. ( p neq 2, q=3 )
D. ( p=2, q=3 )
12
67 The ratio in which the surface ( x^{2}+ )
( y^{2}+z^{2}=25 ) divides the line joining
(0,1,2) and (3,4,5) is ( frac{a pm sqrt{b}}{c} ) then ( boldsymbol{a}+boldsymbol{b}+boldsymbol{c}= )
12
68 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
A ( cdot frac{-2}{3}, frac{1}{3}, frac{2}{3} )
B. ( frac{2}{3}, frac{-1}{3}, frac{2}{3} )
c. ( frac{2}{3}, frac{1}{3}, frac{-2}{3} )
D. ( frac{2}{3}, frac{-1}{3}, frac{-2}{3} )
12
69 Three lines are given by ( vec{r}=lambda hat{i}, lambda epsilon R ) ( overrightarrow{boldsymbol{r}}=boldsymbol{mu}(hat{boldsymbol{i}}+hat{boldsymbol{j}}), boldsymbol{n} boldsymbol{epsilon} boldsymbol{R} ) and ( overrightarrow{boldsymbol{r}}=boldsymbol{v}(hat{boldsymbol{i}}+hat{boldsymbol{j}}+ )
( hat{boldsymbol{k}}), boldsymbol{v} boldsymbol{epsilon} boldsymbol{R} )
Let the lines cut the plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}= )
1 at he points ( A, B ) and ( C ) respectively.
If the area of the triangle ( A B C ) is ( triangle )
then the value of ( (6 triangle)^{2} ) equals
A . 0.75
в. ( 0 . )
( c .0 .85 )
D. 0.65
12
70 Find the distance of a point (3,-5) from the line ( 3 x-4 y-5=0 ) 12
71 The direction cosines of a vector ( hat{boldsymbol{i}}+ ) ( hat{boldsymbol{j}}+sqrt{mathbf{2}} hat{boldsymbol{k}} ) are
A ( cdot frac{1}{2}, frac{1}{2}, 1 )
B. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, frac{1}{2} )
c. ( frac{1}{2}, frac{1}{2}, frac{1}{sqrt{2}} )
D. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, frac{1}{sqrt{2}} )
12
72 Find the equation of plane with intercepts 2,3 and 4 on the ( x, y ) and ( z ) axis respectively. 12
73 If the extremities of a diagonal of a square are (1,-2,3) and (4,2,3) then the area of the square is
A . 25
B. 50
c. ( frac{25}{2} )
D. ( sqrt{50} )
12
74 The points ( (10,7,0),(6,6-1) ) and
(6,9,-4) form a
A. Right -angled triangle
B. Isosceles triangle
( c cdot ) Both (1)( &(2) )
D. Equilateral triangle
12
75 If ( bar{a}, bar{b} ) and ( bar{c} ) are non-zero non collinear vectors and ( theta(neq 0, pi) ) is the angle between ( bar{b} ) and ( bar{c} ) if ( (bar{a} times bar{b}) times bar{c}=frac{1}{2}|bar{b}| bar{c} mid bar{a} )
then ( sin theta= )
A ( cdot sqrt{frac{2}{3}} )
B. ( frac{sqrt{3}}{2} )
( c cdot frac{4 sqrt{2}}{3} )
D. ( frac{2 sqrt{2}}{3} )
12
76 The point which is equidistant from the points (-1,1,3),(2,1,2),(0,5,6) and
(3,2,2) is
A ( cdot(-1,3,4) )
в. (3,1,4)
c. (1,3,4)
D. (4,1,3)
12
77 Find the coordinates of the point where the line ( frac{boldsymbol{x}+mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}+boldsymbol{2}}{mathbf{3}}=frac{boldsymbol{z}+mathbf{3}}{mathbf{4}} ) meets
the plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{4} boldsymbol{z}=boldsymbol{6} )
12
78 Number of lines is space which are equally inclined to three co-ordinate
axes are?
A .2
B. 4
( c .6 )
D. 8
12
79 A cube of side 5 has one vertex at the
point ( (1,0,-1), ) and the three edges from this vertex are, respectively, parallel to the negative ( x ) and ( y ) axes and positive z-axis. Find the coordinates of the other
vertices of the cube.
A. (1,0,1)
B. (0,-1,0)
c. (0,0,-1)
D. (1,0,0)
12
80 45.
The two lines x = ay+b, z= cy+d; and x = a’y+b’,
[2006||
z=c’y+d’ are perpendicular to each other if
(a) aa’+cc’=-1 (b) aa’t.cc’ = 1
(c) 9+ =-1 (d) 9+6=1
12
81 Distance between two parrallel lines, ( overline{boldsymbol{r}}=overline{boldsymbol{a}}_{1}+boldsymbol{lambda} overline{boldsymbol{b}} ) and ( overline{boldsymbol{r}}=overline{boldsymbol{a}}_{2}+boldsymbol{mu} overline{boldsymbol{b}}, ) is given
by
A ( cdot d=mid frac{left(bar{a}_{2}-bar{a}_{1}right)}{hat{b}} )
B . ( d=midleft(bar{a}_{2}-bar{a}_{1}right) times hat{b} )
c. ( d=midleft(bar{a}_{2}+bar{a}_{1}right) times hat{b} )
D . ( d=midleft(bar{a}_{2}-bar{a}_{1}right) )
12
82 If a line ( O P ) of length ( r ) (Where ‘ ( O ) ‘ is the origin) makes an angle ( alpha ) with ( x ) -axis
and lies on the xz-plane, then what are the coordinates of ( P ? )
A ( cdot(r cos alpha, 0, r sin alpha) )
B . ( (0,0, r sin alpha) )
( mathbf{c} cdot(r cos alpha, 0,0) )
D ( cdot(0,0, r cos alpha) )
12
83 Find the equation of the plane passing through the points (2,3,-4) and (1,-1,3) and parallel to the ( x- ) axis. 12
84 Find the angle between the line whose
direction cosines are given by ( l+m+ )
( boldsymbol{n}=mathbf{0} ) and ( l^{2}+boldsymbol{m}^{2}=boldsymbol{n}^{2} )
12
85 63.
If the angle between the line x=>
lin _y-1
Z-3
– and the plane
2
x +2y + 33 =4 is cos” (193), then aequals
x + 2y + 3z=4 is cos-1
, then a equals
12011
[2011]
12
86 The intercepts of the plane ( 2 x-3 y+ )
( mathbf{5} z-mathbf{3 0}=mathbf{0} ) are
A. 15,-10,6
в. 5,10,6
c. ( 1 / 8,-1 / 6,1 / 4 )
D. 3,-4,6
12
87 If ( vec{P}(1,5,4) ) and ( vec{Q}(4,-1,-2), ) find the direction ratio of ( overrightarrow{P Q} ) 12
88 39. Equation of the plane containing the straight line
and perpendicular to the plane containing the
(2010)
x y z
straight lines – ===
3 4 2
(a) x+2y – 2z=0
(c) x-2y+z=0
x y z
is
2
(b) 3x + 2y – 2z=0
(d) 5x + 2y – 4z=0
12
89 ( A ) point ( C ) with position vector ( frac{3 a+4 b-5 c}{3} )
(where ( a, b ) and ( c ) are non co-planar
vectors) divides the line joining ( A ) and ( B ) in the ratio ( 2: 1 . ) If the position vector
of ( A ) is ( a-2 b+3 c, ) then the position
vector of ( boldsymbol{B} ) is
A ( .2 a+3 b-4 c )
B . ( 2 a-3 b+4 c )
c. ( 2 a+3 b+4 c )
D. ( a+3 b-4 c )
12
90 The coordinates of a point which is equidistant from the point ( (0,0,0),(a, 0,0),(0, b, 0) ) and ( (0,0, c) )
are given by
( ^{mathbf{A}} cdotleft(frac{a}{2}, frac{b}{2}, frac{c}{2}right) )
в. ( left(frac{-a}{2}, frac{-b}{2}, frac{c}{2}right) )
( ^{mathrm{c}}left(frac{a}{2}, frac{-b}{2}, frac{-c}{2}right) )
D ( cdotleft(frac{-a}{2}, frac{b}{2}, frac{-c}{2}right) )
12
91 Find the magnitude of the shortage
distance between the lines ( , frac{x-8}{3}= )
( frac{y+9}{-16}=frac{z-10}{7} ; frac{x-15}{3}=frac{y-29}{8}=frac{z-5}{-5} )
12
92 A line making angles ( 45^{circ} ) and ( 60^{circ} ) with the positive direction of ( x- ) axis and
( y- ) axis respectively. Then the angle made by the line with positive direction
of ( z- ) axis is
A ( .60^{circ} )
B. ( 120^{circ} )
( mathbf{c} cdot 60^{circ} ) or ( 120^{circ} )
D. None of these
12
93 The equation to the altitude of the
altitude triangle formed by
( (1,1,1) cdot(1,2,3),(2,-1,1) ) through
(1,1,1) is
A ( cdot bar{r}=(bar{i}+bar{j}+bar{k})+t(bar{i}-bar{j}-2 bar{k}) )
B ( cdot bar{r}=(bar{i}-bar{j}+bar{k})+t(bar{i}+bar{j}-2 bar{k}) )
( mathbf{c} cdot bar{r}=(bar{i}+bar{j}+bar{k})+t(bar{i}-bar{j}+2 bar{k}) )
D ( cdot bar{r}=(bar{i}-bar{j}-bar{k})+t(bar{i}+bar{j}-2 bar{k}) )
12
94 Cartesian equation of the plane ( bar{r}= ) ( (1+lambda-mu) bar{i}+(2-lambda) bar{j}+(3-2 lambda+ )
( 2 mu) bar{k} ) is :
A ( .2 x+y=5 )
в. ( 2 x-y=5 )
c. ( 2 x+z=5 )
D. ( 2 x-mathrm{z}=5 )
12
95 Unit vector perpendicular to the plane passing through the points ( hat{mathbf{i}}-hat{mathbf{j}}+ ) ( 2 hat{k}, 2 hat{i}-hat{k} ) and ( 2 hat{j}+hat{k} ) is
A ( cdot frac{2 hat{i}-hat{j}+hat{k}}{sqrt{6}} )
( ^{text {В } cdot frac{2 hat{i}+hat{j}+hat{k}}{sqrt{6}}} )
c. ( frac{2 hat{i}+hat{j}-hat{k}}{sqrt{6}} )
D. None of these
12
96 Find the direction cosines of two lines
which are connected by the relations ( l+m+n=0 ) and ( m n-2 n l-2 l m= )
( mathbf{0} )
12
97 Line ( overrightarrow{boldsymbol{r}}=(hat{boldsymbol{i}}-hat{boldsymbol{j}}+hat{boldsymbol{k}})+boldsymbol{t}(boldsymbol{2} hat{boldsymbol{i}}-hat{boldsymbol{j}}+hat{boldsymbol{k}}) )
contained in a plane to which vector ( vec{n}=3 hat{i}-2 hat{j}+lambda hat{k} ) is normal. Find the
value of ( lambda ). Also find the vector equation
of the plane.
12
98 Three vertices of a tetrahedron are
(0,0,0),(6,-5,-1) and ( (-4,1,3) . ) If the centroid of the tetrahedron be (1,-2,5)
then the fourth vertex is
A. (2,-4,18)
в. (2,-4,-18)
( ^{c} cdotleft(frac{3}{4}, frac{-3}{2}, frac{7}{4}right) )
D. none of these
12
99 Assertion
If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with
( O X, O Y, O Z ) respectively, then ( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma=2 )
Reason
If ( l=cos alpha, m=cos beta, n=cos gamma, ) are
direction cosines of a line, then ( l^{2}+ )
( boldsymbol{m}^{2}+boldsymbol{n}^{2}=mathbf{1} )
A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
C. Assertion is true but Reason is false
D. Assertion is false but Reason is true
12
100 The plane ( x=0 ) divides the joinning of (-2,3,4) and (1,-2,3) in the ratio
A .2: 1
B. 1: 2
c. 3: 2
D. -4: 3
12
101 The image of the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{3}}=frac{boldsymbol{y}-mathbf{3}}{mathbf{1}}= )
( frac{z-4}{-5} ) in the plane ( 2 x-y+z+3=0 )
is the line?
A. ( frac{x-3}{3}=frac{y+5}{1}=frac{z-2}{-5} )
в. ( frac{x-3}{-3}=frac{y+5}{-1}=frac{z-2}{5} )
c. ( frac{x+3}{3}=frac{y-5}{1}=frac{z-2}{-5} )
D. ( frac{x+3}{-3}=frac{y-5}{-1}=frac{z+2}{5} )
12
102 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
( A cdot 3 )
B. – –
( c cdot 2 )
D. –
12
103 ( begin{array}{ll}text { List I } & text { List II } \ text { 1) d.c’s of } x-text { axis } & text { a) }(1,1,1) \ text { 2) d.c’s of } y-text { axis } & text { b) } \ text { 3) d.c’s of } z-text { axis } & left(frac{1}{sqrt{3}} frac{1}{sqrt{3}}, frac{1}{sqrt{3}}right) \ begin{array}{l}text { 4) d.c’s of a line makes equal } \ text { angles with axes }end{array} & text { c) }(1,0,0) \ begin{array}{l}text { d) }(0,1,0) \ text { e) }(0,0,1)end{array}end{array} ) The correct order for 1,2,3,4 is
( mathbf{A} cdot c, d, e, b )
B. ( a, b, c, e )
( mathbf{C} cdot c, d, a, b )
D. ( b, c, a, e )
12
104 If the centroid of the tetrahedron
( O A B C, ) where ( A, B, C ) are given by ( (alpha, 5,6),(1, beta, 4),(3,2, gamma) ) respectively
be ( 1,-1,2, ) then value of ( alpha^{2}+beta^{2}+gamma^{2} )
equals
A ( cdot alpha^{2}+beta^{2} )
B. ( gamma^{2}+beta^{2} )
c. ( alpha^{2}+gamma^{2} )
D. None of these
12
105 The shortest distance between the lines ( frac{boldsymbol{x}-mathbf{5}}{mathbf{4}}=frac{boldsymbol{y}-mathbf{7}}{-mathbf{5}}=frac{boldsymbol{z}+mathbf{3}}{-mathbf{5}} ) and ( frac{boldsymbol{x}-mathbf{8}}{mathbf{4}}= )
( frac{y-7}{-5}=frac{z-5}{-5} ) is
A . 45
B . 46
c. 47
D. 48
12
106 If ( overrightarrow{mathbf{A}} times overrightarrow{mathbf{B}}=overrightarrow{mathbf{B}} times overrightarrow{mathbf{A}}, ) then the angle
between ( A ) and ( B ) is
A . ( pi )
в. ( pi / 3 )
c. ( pi / 2 )
D . ( pi / 4 )
12
107 ( boldsymbol{A}=(-1,2,-mathbf{3}), boldsymbol{B}=(mathbf{5}, mathbf{0},-mathbf{6}), boldsymbol{C}= )
(0,4,-1) are the vertices of a triangle.
The d.c’s of the internal bisector of ( angle mathrm{BAC} )
are?
( ^{mathbf{A}} cdotleft(frac{25}{sqrt{714}}, frac{-8}{sqrt{714}}, frac{-5}{sqrt{714}}right) )
в. ( left(frac{5}{sqrt{74}}, frac{6}{sqrt{74}}, frac{8}{sqrt{74}}right) )
( ^{mathbf{C}} cdotleft(frac{25}{sqrt{714}}, frac{8}{sqrt{714}}, frac{5}{sqrt{714}}right) )
D. ( left(frac{-5}{sqrt{74}}, frac{6}{sqrt{74}}, frac{-8}{sqrt{74}}right) )
12
108 Equation of plane parallel to ( 3 x+4 y+ )
( mathbf{5} boldsymbol{z}-mathbf{6}=mathbf{0}, mathbf{6} boldsymbol{x}+mathbf{8} boldsymbol{y}+mathbf{1 0} boldsymbol{z}-mathbf{1 6}=mathbf{0} )
and equidistant from them is
A. ( 3 x+4 y+5 z=7 )
B. ( 3 x+4 y+5 z=10 )
c. ( 6 x+8 y+10 z=0 )
D. ( 6 x+8 y+10 z=3 )
12
109 Assertion (A): The points ( boldsymbol{A}(mathbf{2}, mathbf{9}, mathbf{1 2}), boldsymbol{B}(mathbf{1}, mathbf{8}, mathbf{8}), boldsymbol{C}(mathbf{2}, mathbf{1 1}, mathbf{8}) boldsymbol{D}(mathbf{1}, mathbf{1 2} )
are the vertices of a rhombus
Reason ( (mathrm{R}): A B=B C=C D=D A )
and ( boldsymbol{A C}=boldsymbol{B D} )
A. Both A and R are individually true and R is the correct explanation of
B. Both A and R individually true but R is not the correct explanation of A
c. ( A ) is true but ( R ) is false
D. Both A and R false
12
110 Two equat ions ( vec{r} . vec{n}_{1}=q_{1} ) and ( vec{r} . vec{n}_{2}=q_{2} )
represent two perpendicular planes, where ( vec{n}_{1} ) and ( vec{n}_{2} ) are two unit vectors.
One of these plane is rotated through an angle 45 about line of intersection of
two given planes then equation of plane in new position can be This question has multiple correct options
A ( cdot vec{r} cdotleft(vec{n}_{1}+vec{n}_{2}right)=q_{1}-q_{2} )
В ( cdot vec{r} cdot(overrightarrow{n_{1}}+overrightarrow{n_{2}})=q_{1}+q_{2} )
c. ( vec{r} .left(vec{n}_{1}-vec{n}_{2}right)=q_{1}+q_{2} )
D ( cdot vec{r} cdot(overrightarrow{n_{1}}-overrightarrow{n_{2}})=q_{1}-q_{2} )
12
111 13.
The shortest distance from the plane 12x+4y+3z =327
to the sphere x2 + y2 + z2 + 4x – 2y – 6z =155 is
(2) 39
(b) 26
(c) 11
(d) 13
12
112 25.
-3
y-k _Z intersect, then
If the lines
x-1 y+1 Z-1
2 3 4 ”
the value of k is
(a) 3/2 (b) 9/2
and
2
1
(c) – 2/9
(2004S)
(d) – 3/2
12
113 ( boldsymbol{P}(mathbf{0}, mathbf{5}, mathbf{6}), boldsymbol{Q}(mathbf{1}, mathbf{4}, mathbf{7}), boldsymbol{R}(mathbf{2}, mathbf{3}, mathbf{7}) ) and
( S(3,5,16) ) are four points in the space.
The point nearest to the origin ( boldsymbol{O}(mathbf{0}, mathbf{0}, mathbf{0}) )
is
A. ( P )
B. ( Q )
( c . R )
D. ( S )
12
114 Show that the lines whose d.c’s are
given by ( 2 l+2 m-n=0, m n+n l+ )
( l m=0 ) are perpendicular to each other
12
115 The coordinates of any point, which lies
on ( x ) axis are
A ( .(0, x, 0) )
в. ( (x, 0,0) )
c. ( (x, x, 0) )
D. ( (x, x, x) )
12
116 If ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}=(mathbf{2}, mathbf{3}, mathbf{4}) ) and ( boldsymbol{A} boldsymbol{B} ) is
produced upto ( C ) such that ( 2 A B=B C )
then ( C= )
A. (5,4,6)
в. (6,2,4)
c. (4,5,6)
D. (6,4,5)
12
117 If ( P(x, y, z) ) is a point on the line
segment joining ( Q(2,2,4) ) and
( boldsymbol{R}(boldsymbol{3}, boldsymbol{5}, boldsymbol{6}) ) such that the projection of
( O P ) on the axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} )
respectively, then ( P ) divides ( Q R ) in the ratio
A .1: 2
B. 3: 2
( c cdot 2: 3 )
D. 1: 3
12
118 Find the intersection of the line ( x- )
( mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}+mathbf{4}=mathbf{0}, boldsymbol{x}+boldsymbol{y}+boldsymbol{z}-mathbf{8}=mathbf{0} )
with the plane ( boldsymbol{x}-boldsymbol{y}+mathbf{2} boldsymbol{z}+mathbf{1}=mathbf{0} )
12
119 The coordinates of a point ( mathrm{P} ) are (3,12,4) w.r.t origin ( 0, ) then the direction cosines of ( O P ) are
( mathbf{A} cdot 3,12,4 )
B. ( frac{1}{4}, frac{1}{3}, frac{1}{2} )
c. ( frac{3}{sqrt{13}}, frac{1}{sqrt{13}}, frac{2}{sqrt{13}} )
D. ( frac{3}{13}, frac{12}{13}, frac{4}{13} )
12
120 The angle between vectors ( (bar{M} times bar{N}) ) and ( (overline{boldsymbol{N}} times overline{boldsymbol{M}}) ) is then
( mathbf{A} cdot 0^{circ} )
B . ( 60^{circ} )
( c .90^{circ} )
D. ( 180^{circ} )
12
121 If the plane ( 3 x+2 y+6 z=6 )
intersects the coordinate axes at
( A, B, C ) then the area of the ( Delta A B C ) is
( mathbf{A} cdot 49 )
B. 7
( c cdot frac{7}{2} )
D. ( frac{11}{2} )
12
122 Arrange the points: ( mathbf{A}(1,2- ) ( mathbf{3}), mathbf{B}(-mathbf{1}, mathbf{2},-mathbf{3}), mathbf{C}(-mathbf{1},-mathbf{2}-mathbf{3}) ) and
( mathbf{D}(mathbf{1},-mathbf{2},-mathbf{3}) ) in the increasing order of
their octant numbers:
A. ( A, B, C, D )
в. ( B, C, D, A )
c. ( C, D, A, B )
D. ( D, C, B, A )
12
123 Distance of the point ( boldsymbol{P}(overrightarrow{boldsymbol{p}}) ) from the line ( vec{r}=vec{a}+lambda vec{b} ) is –
A ( cdot(vec{a}-vec{p})+frac{((vec{p}-vec{a}) cdot vec{b}) vec{b}}{|vec{b}|^{2}} mid )
B. ( (vec{b}-vec{p})+frac{((vec{p}-vec{a}) cdot vec{b}) vec{b}}{|vec{b}|^{2}} mid )
c. ( quadleft|(vec{a}-vec{p})+frac{((vec{p}-vec{b}) cdot vec{b}) vec{b}}{|vec{b}|^{2}}right| )
D. None of these.
12
124 A line makes angle ( theta_{1}, theta_{2}, theta_{3}, theta_{4} ) with the
diagonals of the cube. Show that ( cos ^{2} theta_{1}+cos ^{2} theta_{2}+cos ^{2} theta_{3}+cos ^{2} theta_{4}= )
( frac{4}{3} ? )
12
125 if a line makes angles ( alpha, beta, gamma, delta ) with
four diagonals a cube then value of
( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma+sin ^{2} delta ) equals
( A cdot 2 )
B. ( frac{4}{3} )
( c cdot frac{8}{3} )
D.
12
126 Derive the coordinates of the points ( R(x, y, z) ) dividing the line joining the
points ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}, boldsymbol{z}_{1}right) ) and ( boldsymbol{Q}left(boldsymbol{x}_{2}, boldsymbol{y}_{2}, boldsymbol{z}_{2}right) )
internally in the ratio ( m: n )
12
127 Find the value of ( lambda ) for which the four
points ( A, B, C, D ) with position vectors ( -widehat{boldsymbol{j}}-widehat{boldsymbol{k}} ; boldsymbol{4} hat{boldsymbol{i}}+boldsymbol{5} hat{boldsymbol{j}}+boldsymbol{lambda} hat{boldsymbol{k}} ; boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{9} hat{boldsymbol{j}}+boldsymbol{4} widehat{boldsymbol{k}} ) and
( -4 hat{i}+4 widehat{j}+4 widehat{k} ) are coplanar.
12
128 If direction cosines of two lines are
proportional to (2,3,-6) and (3,-4,5) then the acute angle between them is
( ^{mathbf{A}} cdot cos ^{-1}left(frac{49}{36}right) )
B. ( cos ^{-1}left(frac{18 sqrt{2}}{35}right) )
( c cdot 96^{circ} )
D. ( cos ^{-1}left(frac{18}{35}right) )
12
129 A non-zero vector ( vec{a} ) is parallel to the line
of intersection of the plane determined by the vectors ( hat{i}, hat{i}+hat{j} ) and the plane determined by the vectors ( hat{i}-hat{j}, hat{i}+hat{k} ) The angle between ( vec{a} ) and ( hat{i}-2 hat{j}+2 hat{k} ) is
A.
в.
c.
D. ( frac{pi}{2} )
12
130 The intercept made by the plane ( vec{r} cdot vec{n}= ) ( q ) on the ( x ) -axis is
A ( cdot frac{q}{hat{i} cdot vec{n}} )
в. ( frac{hat{i} cdot vec{n}}{q} )
( c cdot frac{hat{i} cdot q}{n} )
D. ( frac{q}{|vec{n}|} )
12
131 The point ( P ) is the intersection of the
straight line joining the points ( Q(2,3,5) ) and ( R(1,-1,4) ) with the plane
( 5 x-4 y-z=1 . ) If ( S ) is the foot of the
perpendicular drawn from the point
( T(2,1,4) ) to ( Q R, ) then the length of the
line segment ( P S ) is
A ( cdot frac{1}{sqrt{2}} )
B. ( sqrt{2} )
( c cdot 2 )
D. ( 2 sqrt{2} )
12
132 Equation of the plane containing the ( operatorname{lines} overline{boldsymbol{r}}=(overline{boldsymbol{i}}-boldsymbol{2} overline{boldsymbol{j}}+overline{boldsymbol{k}})+boldsymbol{t}(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}-overline{boldsymbol{k}}) )
( boldsymbol{boldsymbol { r }}=(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}-overline{boldsymbol{k}})+boldsymbol{s}(overline{boldsymbol{i}}+overline{boldsymbol{j}}+mathbf{3} overline{boldsymbol{k}}) ) is
A. ( bar{r}(7 bar{i}-4 bar{j}-bar{k})=14 )
В. ( bar{r}(bar{i}+2 bar{j}-bar{k})=10 )
c. ( bar{r}(bar{i}+bar{j}+3 bar{k})=20 )
D. ( bar{r}(bar{i}-2 bar{j}+bar{k})=27 )
12
133 18.
A tetrahedron has vertices at O(0, 0, 0), A(1,2,1) B(2,1,3)
and C(-1,1,2). Then the angle between the faces OAB and
ABC will be
[2003]
(a)
90°
(b) cos-
(c) cos-1( 37
(d) 30°
31
12
134 Find the length and foot of the perpendicular from the point ( (mathbf{7}, mathbf{1 4}, mathbf{5}) ) to the plane ( 2 x+4 y-z=2 ) 12
135 The circum centre of the triangle formed by the points (2,5,1),(1,4,-3) and (-2,7,-3) is
A. (6,0,1)
()
В. (0,6,-1)
c. (-1,6,2)
D. (6,1,-2)
12
136 Graph ( x^{2}+y^{2}=4 ) in ( 3 D ) looks like
A . Circle
B. Cylinder
c. Hemisphere
D. sphere
12
137 Write the equations for the ( x ) -and ( y )
axes.
12
138 16.
The image of the point (-1,3,4) in the plane x-2y = 0 is
(
17
19
41
(b) (15,11,4)
(2006)
10
U
(d) None of these
12
139 Find the image of :
(-2,3,4) in the ( y z ) -plane
12
140 The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
parallel to the line ( frac{x}{2}=frac{y}{3}=frac{z-1}{-6} ) is
A .
B . 2
( c cdot 4 )
D. None of these
12
141 ff ( left(x_{1}, y_{1}, z_{1}right) ) and ( Bleft(x_{2}, y_{2}, z_{2}right) ) are
two points such that the direction ( operatorname{cosines} ) of ( A B ) are ( l, m, n ) then
( l=frac{x_{2}-x_{1}}{|A B|}, m=frac{y_{2}-y_{1}}{|A B|}, n=frac{z_{2}-z_{1}}{|A B|} )
12
142 Four vertices of a tetrahedron are
(0,0,0),(4,0,0),(0,-8,0) and
( (0,0,12) . ) Its centroid has the
coordinates
A ( cdotleft(frac{4}{3},-frac{8}{3}, 4right) )
В. (2,-4,6)
c. (1,-2,3)
a 5
D. none of these
12
143 If ( boldsymbol{A}=(-2,3,4), B=(1,2,3) ) are two
points and ( P ) is the point of intersection
of ( A B ) and ( z x ) -plane, then ( P_{x}+P_{y}+ )
( boldsymbol{P}_{z}= )
( mathbf{A} cdot mathbf{6} )
B. -8
c. 8
D.
12
144 If the points ( (1,1, p) ) and (-3,0,1) be equidistant from the plane ( vec{r} .(3 hat{i}+ ) ( 4 hat{j}-12 hat{k})+13=0, ) then find the value
of ( p )
12
145 Find the ratio in which YZ-plane divides the line joining ( A(2,4,5) ) and ( B(3,5,-4) . ) Also find the point of
intersection.
12
146 ( operatorname{Points} boldsymbol{A}(boldsymbol{3}, boldsymbol{2}, boldsymbol{4}), boldsymbol{B}left(frac{boldsymbol{3} boldsymbol{3}}{boldsymbol{5}}, frac{boldsymbol{2} boldsymbol{8}}{boldsymbol{5}}, frac{boldsymbol{3} boldsymbol{8}}{boldsymbol{5}}right), ) and
( C(9,8,10) ) are given. The ratio in which
( B ) divides ( overline{A C} ) is
A . 5: 3
B . 2: 1
c. 1: 3
D. 3: 2
12
147 The image of the point (-1,3,4) in the
plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}=mathbf{0} ) is
A ( cdot(15,11,4) )
в. ( left(frac{9}{5},-frac{13}{5}, 4right) )
( c cdot(8,4,4) )
D. None of these
12
148 28. P, and P, are planes passing through origin. L, and L, are
two line on P, and P2 respectively such that their
intersection is origin. Show that there exists points A, B, C,
whose permutation A’, B’, C can be chosen such that (i) Ais
on L,, B on P, but not on L, and C not on P, (ii) A’ is on L,,
B’ on P, but not on L and C not on P2
12
149 26.
Find the equation of plane passing through (1,1,1) & parallel
to the lines L,, L, having direction ratios (1,0,-1), (1,-1,0)
Find the volume of tetrahedron formed by origin and the
points where these planes intersect the coordinate axes.
(2001.2 Mau
12
150 6.
A line makes the same angle , with each of the x and z axis.
If the angle ß, which it makes with y-axis, is such that
sin B = 3 sin?e, then cos2e equals
[2004]
т.
12
151 Find the coordinates of the point ( boldsymbol{P} )
which divides the join of ( boldsymbol{A}(-2,5) ) and ( B(3,-5) ) in the ratio 2: 3
12
152 If a line makes an angle of ( frac{pi}{4} ) with the positive direction of each of ( x ) -axis and
( boldsymbol{y} ) -axis, then the angle that the line
makes with the positive direction of ( z )
axis is-
A ( cdot frac{pi}{3} )
в.
c. ( frac{pi}{2} )
D.
12
153 The eartesian equations of the line are
( mathbf{3} boldsymbol{x}+mathbf{1}=mathbf{6} boldsymbol{y}-mathbf{2}=mathbf{1}-boldsymbol{z} . ) Find its
equation in vector form and find direction ratios of the line.
12
154 The projections of a line segment on ( x, y ) and ( z ) axes are respectively ( sqrt{2}, 3,5 ) The length of the line segment is
( mathbf{A} cdot mathbf{6} )
B. 11
c. 8
D. 5
12
155 ( operatorname{Given} boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{0}) ; boldsymbol{B}(mathbf{3}, mathbf{1}, mathbf{2}) )
( C(2,-2,4) ) and ( D(-1,1,-1) ) which of
the following points neither lie on ( boldsymbol{A B} )
nor on ( C D ? )
A ( .(2,2,4) )
В. (2,-2,4)
c. (2,0,1)
D. (0,-2,-1)
12
156 Directions ratio of two lines are ( 3,-2, k )
and ( -2, k, 4 . ) Find ( k ) if the lines are perpendicular to each other.
12
157 Find the distance of the point ( P(3,4,4) ) from the point, where the line joining the point ( A(3,-4,-5) ) and ( B(2,-3,1) ) intersects the plane ( 2 x+y+z=7 ) 12
158 Two vectors ( vec{A} ) and ( vec{B} ) inclined at an
angle ( theta ) have a resultant ( vec{R} ) which makes an angle ( alpha ) with ( vec{A} ) and angle ( beta ) with ( vec{B} ). Let the magnitudes of the vectors ( vec{A}, vec{B} ) and ( vec{R} ) be represented by ( A ) B and R respectively. Which of the following relations is not correct?
A . Asinalpha ( =B sin beta )
B. ( R sin alpha=B sin (alpha+beta) )
c. ( R sin beta=operatorname{Asin}(alpha+beta) )
D. None of these
12
159 ( A=(2,3,0) ) and ( B=(2,1,2) ) are two
points. If the points ( P, Q ) are on the line
( A B ) such that ( A P=P Q=Q B, ) then
( boldsymbol{P Q}= )
( A cdot 2 sqrt{2} )
B. ( 6 sqrt{2} )
( c cdot sqrt{frac{8}{9}} )
D. ( sqrt{2} )
12
160 Point, Plane: ( (0,0,0), 3 x-4 y+12 z= )
3
12
161 The angle between two diagonals of a cube is.
A ( .30^{circ} )
B . ( 45^{circ} )
c. ( cos ^{-1}left(frac{1}{3}right) )
D. ( cos ^{-1}left(frac{1}{sqrt{3}}right) )
12
162 If direction numbers of two lines are
( a, b, c ) and ( b-c, c-a, a-b ) prove that
they are perpendicular to each other.
12
163 The direction cosines of a line equally
inclined to three mutually perpendicular lines having D.C.’s as
( ell_{1} m_{1} n_{1}: ell_{2} m_{2} n_{2}: ell_{3} m_{3} n_{3} ) are
A. ( l_{1}+l_{2}+l_{3}, m_{1}+m_{2}+m_{3}, n_{1}+n_{2}+n_{3} )
B ( cdotleft(pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}right) )
( ^{mathbf{c}} cdotleft(pm frac{1}{sqrt{2}}, pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{4}}right) )
D. none of these
12
164 The direction cosines of the line which
is perpendicular to the lines with direction cosines proportional to
(1,-2,-2)( &(0,2,1) ) are
( ^{mathrm{A}} cdotleft(frac{2}{3},-frac{1}{3}, frac{2}{3}right) )
в. ( left(frac{2}{3}, frac{1}{3}, frac{2}{3}right) )
c. ( left(frac{2}{3}, frac{1}{3}, frac{-2}{3}right) )
D. ( left(frac{-2}{3}, frac{1}{3}, frac{2}{3}right) )
12
165 A plane ( pi ) makes intercept 3 and 4
respectively on z-axis and x-axis. If ( pi ) is
parallel to y-axis, then its equation is
A. ( 3 x+4 z=12 )
B. ( 3 z+4 x=12 )
c. ( 3 y+4 z=12 )
D. ( 3 z+4 y=12 )
12
166 Distance between ( boldsymbol{A}(mathbf{4}, mathbf{5}, mathbf{6}) ) from origin
( boldsymbol{O} ) is
A ( cdot 25 sqrt{3} )
B. ( sqrt{77} )
c. ( 3 sqrt{5} )
D. Data Insufficient
12
167 If ( vec{A}=-4 hat{i}+3 hat{j} ) and ( vec{B}=2 hat{i}+5 hat{j} ) and
( vec{C}=vec{A} times vec{B} ) then ( vec{C} ) makes an angle of :
A ( cdot 45^{0} ) with ( mathrm{x} ) -axis
B. ( 180^{circ} ) with ( Y ) -axis
( c cdot 0^{0} ) with ( mathrm{z} ) -axis
D. ( 180^{circ} ) with ( z ) -axis
12
168 Show that ( A(3,-2) ) is a point trisection of the line segment joining the points
(2,1) and (5,-8)
Also find the co-ordinates of the other
points of trisections.
12
169 A point on the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{2}}{mathbf{2}}= )
( frac{z+1}{3} ) at a distance ( sqrt{6} ) from the origin
is
This question has multiple correct options
( mathbf{A} cdotleft(frac{-5}{7}, frac{-10}{7}, frac{13}{7}right) )
в. (1,2,-1)
( ^{mathbf{C}} cdotleft(frac{5}{7}, frac{10}{7}, frac{-13}{7}right) )
D. (-1,-2,1)
12
170 Let the co – ordinates of the point where the line joining the points
(2,-3,1),(3,-4,-5) cuts the plane
( mathbf{2} boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{7} ) be ( (boldsymbol{x}, boldsymbol{y}, boldsymbol{z}) . ) Find ( boldsymbol{x}+ )
( boldsymbol{y}+boldsymbol{z} ? )
12
171 The plane ( X O Z ) divides the join of (1,-1,5) and (2,3,4) in the ratio ( lambda: 1 )
then ( lambda ) is
A . -3
B. ( -1 / 3 )
( c cdot 3 )
D. ( 1 / 3 )
12
172 The distance of the point ( ,(-1,-5,-10) ) from the point intersection of the line,
( frac{x-2}{3}=frac{y+1}{4}=frac{z-2}{12} ) and the plane ( x- )
( boldsymbol{y}+boldsymbol{z}=mathbf{5}, ) is
( A cdot 13 )
B. 1
c. 12
D. none of these
12
173 If ( overrightarrow{P O}+overrightarrow{O Q}=overrightarrow{Q O}+overrightarrow{O R}, ) prove that
the points ( P, Q, R ) are collinear.
12
174 17. The radius of the circle in which the sphere
x2 + y2 + 2? + 2x – 2y – 42 -19 = 0 is cut by the plane
x +2y + 2z+ 7 = 0 is
[2003]
(a) 4 (6) 1 (c) 2 (d) 3
12
175 Let the vector ( vec{a}, vec{b}, vec{c} ) and ( vec{d} ) be such ( operatorname{that}(overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}}) times(overrightarrow{boldsymbol{c}} times overrightarrow{boldsymbol{d}})=mathbf{0} cdot operatorname{Let} boldsymbol{P}_{1} )
and ( P_{2} ) be planes determined by the pairs of vectors ( vec{a}, vec{b} ) and ( vec{c}, vec{d} )
respectively then the angle between ( P_{1} )
and ( P_{2} ) is
A. 0
в. ( frac{pi}{4} )
c. ( frac{pi}{3} )
D. ( frac{pi}{2} )
12
176 Find the shortest distance between the
skew lines ( r=(6 i+2 j+2 k)+t(i- )
( 2 j+2 k) ) and ( F=(-4 i-k)+s(3 i- )
( 2 j-2 k) ) where s,t are scalars.
12
177 Find the direction cosines ( l, m, n ) of a line which are connected by the relation ( l+m-n=0 ) and ( 2 m l-2 m n+n l= )
0
This question has multiple correct options
A ( cdot frac{-2}{sqrt{6}}, frac{1}{sqrt{6}}, frac{-1}{sqrt{6}} )
в. ( frac{2}{sqrt{6}}, frac{-1}{sqrt{6}}, frac{1}{sqrt{6}} )
c. ( frac{-2}{sqrt{6}}, frac{-1}{sqrt{6}}, frac{-1}{sqrt{6}} )
D. ( frac{2}{sqrt{6}}, frac{1}{sqrt{6}}, frac{1}{sqrt{6}} )
12
178 Verify the following
(i) (0,7,-10),(1,6,-6) and (4,9,-6) are the vertices of an isosceles triangle
(ii) (0,7,10),(-1,6,6) and (-4,9,6) are the vertices of a right angled triangle
(iii) (-1,2,1),(1,-2,5),(4,-7,8) and (2,-3,4) are the vertices of a parallelogram
12
179 Show that the points ( boldsymbol{A}(mathbf{0}, mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{2},-mathbf{1}, mathbf{3}) ) and ( boldsymbol{C}(mathbf{1},-mathbf{3}, mathbf{1}) )
are vertices of an isosceles right-angled triangle.
12
180 47. The equation of the plane passing through the point (1,1,1)
and perpendicular to the planes 2x+y-2z=5 and 3x – 6y-
2z=7, is
(JEE Adv. 2017)
(a) 14x +2y-15z=1 (b) 14x – 2y + 15z=27
(c) 14x +2y+ 15z=31 (d) -14x + 2y + 15z=3
12
181 The distance of point ( A(-2,3,1) ) from
the PQ through ( P(-3,5,2), ) which makes equal angles with the axes is-
A ( cdot frac{2}{sqrt{3}} )
в. ( sqrt{frac{14}{3}} )
c. ( frac{16}{sqrt{3}} )
D. ( frac{5}{sqrt{3}} )
12
182 The cartesian from of equation a line passing through the point position vector ( 2 hat{i}-hat{j}+2 hat{k} ) and is in the direction of ( -2 hat{i}+hat{j}+hat{k}, ) is
A ( frac{x-2}{-2}=frac{y+1}{1}=frac{z-2}{1} )
B. ( frac{x+4}{-2}=frac{y-1}{1}=frac{z+2}{1} )
c. ( frac{x+2}{4}=frac{y-1}{-1}=frac{z-1}{2} )
D. None of these
12
183 Find the vector equation of the plane whose cartesian form of equation is ( 3 x- ) ( 4 y+2 z=5 ) 12
184 Column I shows some vector equations. Match Column I with the value of angle between ( vec{A} ) and ( vec{B} ) given in Column II
Column I Column I
12
185 The point ( P(x, y, z) ) lies in the first octant and its distance from the origin
is 12 units. If the position vector of ( P )
make ( 45^{circ} ) and ( 60^{circ} ) with the ( x ) -axis and ( y )
axis respectively, then the coordinates of ( boldsymbol{P} ) are
A ( cdot(3 sqrt{3}, 6,3 sqrt{2}) )
B. ( (4 sqrt{3}, 8,4 sqrt{2}) )
c. ( (6 sqrt{2}, 6,6,) )
D. ( (6,6,6 sqrt{2}) )
E ( .(4 sqrt{2}, 8,4 sqrt{3}) )
12
186 The projection of a line segment joining
the points ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}, boldsymbol{z}_{1},right) ) and
( Qleft(x_{1}, y_{1}, z_{1},right) ) on another line whose
DC’s are ( l, m, n ) is given by This question has multiple correct options
A ( cdot lleft(x_{1}+x_{2}right)+mleft(y_{2}+y_{2}right)+nleft(z_{1}+z_{2}right) )
B. ( 2left[frac{left(l x_{2}+m y_{2}+n z_{2}right)}{2}-frac{left(l x_{1}+m y_{1}+n z_{1}right)}{2}right. )
c. ( lleft(x_{2}-x_{1}right)+mleft(y_{2}-y_{1}right)+nleft(z_{2}-z_{1}right) )
D. ( frac{x_{2}-x_{1}}{l}+frac{y_{2}-y_{1}}{m}+frac{z_{2}-z_{1}}{n} )
12
187 The point ( P ) is on the ( y ) -axis. If ( P ) is equidistant from (1,2,3) and (2,3,4)
then ( boldsymbol{P}_{boldsymbol{y}}= )
A ( cdot frac{15}{2} )
B. 15
c. 30
D. ( frac{3}{2} )
12
188 If the planes ( boldsymbol{x}-boldsymbol{b} boldsymbol{z}=mathbf{0}, boldsymbol{c} boldsymbol{x}-boldsymbol{y}+=mathbf{0} )
and ( b x+a y-z=0, ) pass through a
line, then find the value of ( a^{2}+b^{2}+ )
( c^{2}+2 a b c )
A.
B.
c. -1
D. ( frac{1}{2} )
12
189 The vector ( vec{P} ) makes ( 120^{circ} ) with the ( x- )
axis and the vector ( vec{Q} ) makes ( 30^{circ} ) with ( boldsymbol{y}- ) axis. What is the resultant vector?
A. ( P+Q )
в. ( P-Q )
c. ( sqrt{P^{2}+Q^{2}} )
D. ( sqrt{P^{2}-Q^{2}} )
12
190 90.
The equation of the line passing through (-4, 2,
to the plane x + 2y – Z-5 = 0 and intersecting the
passing through (-4, 3, 1), parallel
-2.
x+l
-3 1 |
y-3
2..
(JEEM 2019-9 Jan (M)
-1
)
12
1
4
() 4 +3 2+1 (Jet 2
() | 13 1
12
191 The variable plane ( (2 lambda+1) x+ )
( (3-lambda) y+z=4 ) always passes
through the line
A ( cdot frac{x}{0}=frac{y}{0}=frac{x+4}{1} )
в. ( frac{x}{1}=frac{y}{2}=frac{z}{-3} )
c. ( frac{x}{1}=frac{y}{2}=frac{z-4}{-7} )
D. none of these
12
192 Find the point where the line of
intersection of the planes ( x-2 y+z= )
1 and ( x+2 y-2 z=5 ) intersects the
plane ( 3 x+2 y+z+6=0 )
A ( cdot P(1,-2,-4) )
B . ( P(1,2,-4) )
c. ( P(1,-2,4) )
D. None of these
12
193 26. The intersection of the spheres
x2 + y2 + z2 + 7x-2y-z = 13 and
x2 + y2 + 22 – 3x + 3y + 4z = 8
is the same as the intersection of one of the sphere and the
plane
[2004]
(a) 2x – y – z=1 (b) x-2y-z=1
(©) x-y–2z=1 (d) x-y-z = 1
12
194 The d.rs of the lines ( boldsymbol{x}=boldsymbol{a} boldsymbol{y}+boldsymbol{b}, boldsymbol{z}= )
( boldsymbol{c} boldsymbol{y}+boldsymbol{d} ) are:
A. ( 1, a, c )
B. ( a, 1, c )
c. ( b, 1, c )
D. ( c, a, 1 )
12
195 The circum radius of the triangle
formed by the points (1,2,-3),(2,-3,1) and (-3,1,2) is:
A ( cdot sqrt{14} )
B. 14
c. ( sqrt{13} )
D.
12
196 If ( hat{i}, hat{j} ) and ( hat{k} ) represents unit vectors
along the ( x, y ) and ( z- ) axes respectively
then find the value of angle ( theta ) between the vectors ( hat{i}+hat{j}+hat{k} ) and ( hat{i}+hat{j} )
12
197 Find the shortest distance between the
line ( boldsymbol{x}=mathbf{1}+boldsymbol{t}, boldsymbol{y}=mathbf{1}+boldsymbol{6} boldsymbol{t}, boldsymbol{z}=boldsymbol{2} boldsymbol{t}, boldsymbol{t} in boldsymbol{R} )
and ( boldsymbol{x}=mathbf{1}+mathbf{2 k}, boldsymbol{y}=mathbf{5}+mathbf{1 5 k}, boldsymbol{z}= )
( -2+6 k, k in R )
12
198 29
(0) Tu, ,
A plane which is perpendicular to two planes 2x – 2y+z=0
and x-y + 2z= 4, passes through (1, -2, 1). The distance of
the plane from the point (1,2,2) is (2006 – 3M, -1)
(a) o (b) 1 (c) √ (d) 252
12
199 A line makes an angle ( theta ) with each of the
( x- ) and ( z^{-} ) axes. If the angle ( beta, ) which it makes with the ( y ) -axis, is such that
( sin ^{2} beta=3 sin ^{2} theta, ) then ( cos ^{2} theta ) equals-
A ( cdot frac{2}{3} )
B. ( frac{1}{5} )
( c cdot frac{3}{5} )
D. ( frac{2}{5} )
12
200 Find the vector equation of line joining the points (2,1,3) and (-4,3,-1)
( mathbf{A} cdot bar{r}=2(1-3 lambda) bar{i}-(1+2 lambda) bar{j}-(3-4 lambda) bar{k} )
B . ( bar{r}=2(1-3 lambda) bar{i}-(1+2 lambda) bar{j}+(3-4 lambda) bar{k} )
C ( . bar{r}=2(1-3 lambda) bar{i}+(1+2 lambda) bar{j}+(3-4 lambda) bar{k} )
D. ( bar{r}=2(1+3 lambda) bar{i}+(1+2 lambda) bar{j}+(3+4 lambda) bar{k} )
12
201 Show that LHS=RHS i.e. ( frac{2}{9} times 3=frac{2}{9} times ) ( frac{3}{1}=frac{2 times 3}{9 times 1}=frac{6}{9}=frac{2}{3} ) 12
202 A plane meets the axes in ( A, B ) and ( C )
such that centroid of the ( triangle A B C ) is
( (1,2,3) . ) The equation of the plane is
A ( cdot x+frac{y}{2}+frac{z}{3}=1 )
В ( cdot frac{x}{3}+frac{y}{6}+frac{z}{9}=1 )
c. ( x+2 y+3 z=1 )
D. None of these
12
203 If the projections of the line segment ( A B ) on the coordinate axes are 2,3,6
then the square of the sine of the angle made by ( boldsymbol{A B} ) with ( boldsymbol{x}=mathbf{0}, ) is
A ( cdot frac{3}{7} )
B. ( frac{3}{49} )
( c cdot frac{4}{7} )
D. ( frac{40}{49} )
12
204 Find the equation of the plane in scalar product form ( hat{r}=(2 hat{i}+hat{k})+lambda hat{i}+ )
( mu(hat{i}+2 hat{j}-3 hat{k}) )
12
205 If ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}=(mathbf{2}, mathbf{1 0}, mathbf{1}), boldsymbol{Q} ) are
collinear points and ( Q_{x}=-1, ) then
( boldsymbol{Q}_{z}= )
A . -3
B. 7
( c cdot-14 )
D. –
12
206 The extremities of a diagonal of a rectangular parallelopiped whose faces are parallel to the reference planes are (-2,4,6) and ( (3,16,6) . ) The length of the base diagonal is
A . 13
B. ( sqrt{13} )
c. ( 2 sqrt{13} )
D. 169
12
207 The coordinates of the points in which the line joining the points (2,5,-7) and (-3,-1,8) are intersected by the ( y-z )
plane are
A ( cdotleft(0, frac{13}{5},-1right) )
B ( cdotleft(0, frac{-13}{5},-2right) )
D. ( left(0, frac{13}{5}, frac{2}{5}right) )
12
208 Find the vector and cartesian equations of the plane passing through the points ( A(1,1,-2), B(1,2,1) ) and ( C(2,-1,1) ) 12
209 The projections of a directed line segment on the coordinate axes 12,4,3 The direction cosines of the line are
A ( cdot frac{12}{13},-frac{4}{13}, frac{3}{13} )
B. ( -frac{12}{13},-frac{4}{13}, frac{3}{13} )
c. ( frac{12}{13}, frac{4}{13}, frac{3}{13} )
D. none of these
12
210 The direction cosines to two lines at
right angles are (1,2,3) and ( left(-2, frac{1}{2}, frac{1}{3}right), ) then
the direction cosine perpendicular to both given lines are:
A ( cdot sqrt{frac{25}{219}} cdot sqrt{frac{19}{2198}} cdot sqrt{frac{729}{21988}} )
B. ( sqrt{frac{24}{2198} cdot sqrt{frac{38}{2198}}} sqrt{frac{730}{21988}} )
c. ( frac{1}{3},-2, frac{-7}{2} )
D. None of the above
12
211 If the points whose position vectors are ( mathbf{2} overline{mathbf{i}}+overline{boldsymbol{j}}+overline{boldsymbol{k}}, mathbf{6} overline{mathbf{i}}-overline{boldsymbol{j}}+mathbf{2} overline{mathbf{k}} ) and ( mathbf{1 4} overline{mathbf{i}}-mathbf{5} overline{mathbf{j}}+ )
( p bar{k} ) are collinear then the value of ( mathbf{p} ) is
( A cdot 2 )
B. 4
( c cdot 6 )
D. 8
12
212 A point ( P ) lies on the line whose end
points are ( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{3}) ) and ( boldsymbol{B}(mathbf{2}, mathbf{1 0}, mathbf{1}) )
If ( z ) -co-ordinate of ( P ) is ( 7, ) find sum of its
other co – ordinates.
12
213 ( A=(2,4,5) ) and ( B=(3,5,-4) ) are two
points. If the ( x y ) -plane, ( y z ) -plane divide ( A B ) in the ratios ( a: b, p: q ) respectively then ( frac{a}{b}+frac{p}{q}= )
A. ( frac{7}{15} )
в. ( frac{-7}{12} )
c. ( frac{7}{12} )
D. ( frac{22}{25} )
12
214 The ratio in which the line joining points (2,4,5) and (3,5,-4) divide YZ -plane is
( mathbf{A} cdot-2: 3 )
B. 2: 3
c. -3: 2
D. 3: 2
12
215 On a plane are two points ( A ) and ( B ) at a
distance of 5 units apart. The number of straight lines in this plane which are at distance of 2 units from ( A ) and 3 units
from ( mathrm{B} ) are:
A . 1
B. 2
( c .3 )
D. 4
12
216 Find the equation of the plane passing through the point (1,-2,1) and perpendicular to the line joining the points ( boldsymbol{A}(mathbf{3}, mathbf{2}, mathbf{1}) ) and ( boldsymbol{B}(mathbf{1}, mathbf{4}, mathbf{2}) ) 12
217 Find the shortest distance between the line ( frac{x-3}{3}=frac{y-8}{-1}=frac{z-3}{1} ) and the
line of intersection of the planes ( 2 x+ )
( 5 y-z+47=0 ) and ( 2 x+y+z+7= )
( mathbf{0} )
12
218 The projection of a directed line segment on the co-ordinate axes are
( 12,4,3, ) the DC’s of the line are
A ( cdot frac{-12}{13}, frac{-4}{13}, frac{-3}{13} )
B. ( frac{12}{13}, frac{4}{13}, frac{3}{13} )
c. ( frac{12}{13}, frac{-4}{13}, frac{3}{13} )
D. ( frac{12}{13}, frac{4}{13}, frac{-3}{13} )
12
219 The plane ( 2 x-(1+lambda) y+3 z=0 )
passes through the intersection of the
planes
A. ( 2 x-y=0 ) and ( y+3 z=0 )
в. ( 2 x-y=0 ) and ( y-3 z=0 )
c. ( 2 x+3 z=0 ) and ( y=0 )
D. None of the above
12
220 If ( x y ) -plane and ( y z ) -plane divides the line segment joining ( A(2,4,5) ) and ( B(3,5,-4) ) in the ratio a:b and p:q respectively then value of ( left(frac{a}{b}, frac{p}{q}right) ) may be
A ( cdot frac{23}{12} )
B. ( frac{7}{5} )
( c cdot frac{7}{12} )
D. ( frac{21}{10} )
12
221 The vector equation of the plane through the point (1,-2,-3) and
parallel to the vectors (2,-1,3) and
(2,3,-6) is ( bar{r}= )
A ( cdot(1+2 t+2 s) bar{i}-(2+t-3 s) bar{j}-(3-3 t+6 s) bar{k} )
B. ( (1+2 t+2 s) bar{i}+(2+t+3 s) bar{j}-(3+3 t+6 s) bar{k} )
c. ( (1+2 t+2 s) bar{i}+(2+t+3 s) bar{j}+(3+3 t+6 s) bar{k} )
D. ( (1+2 t+2 s) bar{i}+(2+t-3 s) bar{j}+(3+3 t+6 s) bar{k} )
12
222 If ( O ) is origin ( O P=3 ) with direction
ratios proportional to -1,2,-2 then what are the coordinates of ( P ? )
12
223 Direction ratios of the line which is perpendicular to the lines with direction ratios -1,2,2 and 0,2,1 are
( mathbf{A} cdot 1,1,2 )
B. 2,-1,2
c. -2,1,2
D. 2,1,-2
12
224 ( boldsymbol{A}=(mathbf{1},-mathbf{2}, mathbf{3}), boldsymbol{B}=(2,1,3), boldsymbol{C}=(4,2, )
1) and ( G=(-1,3,5) ) is the centroid of
the tetrahedron ( A B C D . ) Then the fourth
coordinate is
A. (11,11,13)
В. (-11,11,45)
c. (-11,11,13)
D. (11,13,11)
12
225 A line d.c’s proportional to (2,1,2)
meets each of the lines ( boldsymbol{x}=boldsymbol{y}+boldsymbol{a}=boldsymbol{z} )
and ( x+a=2 y=2 z . ) Then the
coordinates of each of the points of
intersection are given by
A . ( (3 a, 2 a, 3 a) ;(a, a, 2 a) )
в. ( (3 a, 2 a, 3 a) ;(a, a, a) )
C. ( (3 a, 3 a, 3 a) ;(a, a, a) )
D. ( (2 a, 3 a, 3 a) ;(2 a, a, a) )
12
226 If the plane a ( 2 x-3 y+5 z-2=0 )
divides the line segment joining (1,2,3) and ( (2,1, k) ) in the ratio 9: 11
then ( k ) is
A .
в. -2
c. -10
D. ( -frac{1}{2} )
12
227 37. A line with positive direction cosines passes through
point P(2,-1,2) and makes equal angles with the coordinata
axes. The line meets the plane
2x+y+z=9
at point Q. The length of the line segment PQ equals
(2009)
(a) 1 (b) √ (c) √3 (d) 2
12
228 The distance of origin from the image of (1,2,3) in plane ( x-y+z=5 ) is
A ( cdot sqrt{17} )
B. ( sqrt{29} )
c. ( sqrt{34} )
D. ( sqrt{41} )
12
229 If ( |vec{A} times vec{B}|=sqrt{3} vec{A} cdot vec{B} ) then the value of
( |vec{A}+vec{B}| ) is:
( ^{A} cdotleft(A^{2}+B^{2}+frac{A B}{sqrt{3}}right)^{1 / 2} )
в. ( A+B )
c. ( left(A^{2}+B^{2}+sqrt{3} A Bright)^{1 / 2} )
D. ( left(A^{2}+B^{2}+A Bright)^{1 / 2} )
12
230 Find the vector equation of the line through ( A(3,4,-7) ) and ( B(6,-1,1) ) 12
231 If the orthocentre, circumcentre of a
triangle are (-3,5,2),(6,2,5)
respectively then the centroid of the triangle is
( mathbf{A} cdot(3,3,4) )
В. ( left(frac{3}{2}, frac{7}{2}, frac{9}{2}right) )
c. (9,9,12)
D. ( left(frac{9}{2} frac{-3}{2}, frac{3}{2}right) )
12
232 The Cartesian equation of line ( 6 x- ) ( mathbf{2}=mathbf{3} boldsymbol{y}+mathbf{1}=mathbf{2} z-mathbf{2} ) is given by
( ^{text {A } cdot frac{3 x-1}{3}}=frac{3 y+1}{6}=frac{z-1}{3} )
B. ( frac{3 x+1}{3}=frac{3 y-1}{6}=frac{z-1}{3} )
c. ( frac{3 x-1}{3}=frac{3 y-1}{6}=frac{z-1}{3} )
D. ( frac{3 x-1}{6}=frac{3 y-1}{3}=frac{z-1}{3} )
12
233 The direction ratios of the line ( boldsymbol{x}-boldsymbol{y}+ )
( z-5=0=x-3 y-6 ) are
A. 3,1,-2
в. 2,-4,1
c. ( frac{3}{sqrt{14}}, frac{1}{sqrt{14}}, frac{-2}{sqrt{14}} )
D. ( frac{2}{sqrt{14}}, frac{-4}{sqrt{14}}, frac{1}{sqrt{14}} )
12
234 A plane is at a distance of 5 units from the origin and perpendicular to the vector ( 2 hat{i}+hat{j}+2 hat{k} . ) The equation of the
plane is
A ( . vec{r} .(2 hat{imath}+hat{j}-2 hat{k})=15 )
в. ( vec{r} .(2 hat{i}+hat{j}-hat{k})=15 )
c. ( vec{r} .(2 hat{i}+hat{j}+2 hat{k})=15 )
D. ( vec{r} .(hat{i}+hat{j}+2 hat{k})=15 )
E ( . vec{r} .(2 hat{i}-hat{j}+2 hat{k})=15 )
12
235 Find vector equation for the line passing through the points ( 3 bar{i}+4 bar{j}- ) ( mathbf{7} bar{k}, overline{boldsymbol{i}}-overline{boldsymbol{j}}+mathbf{6} overline{boldsymbol{k}} )
A ( . bar{r}=(3-2 lambda) bar{i}+(4-5 lambda) bar{j}+(-7+13 lambda) bar{k} )
B. ( bar{r}=(2 lambda) bar{i}+(4+5 lambda) bar{j}+(-7-13 lambda) bar{k} )
c. ( bar{r}=(3-2 lambda) bar{i}-(4-5 lambda) bar{j}+(-7+13 lambda) bar{k} )
D. ( bar{r}=(3-2 lambda) bar{i}+(4-5 lambda) bar{j}-(-7+13 lambda) bar{k} )
12
236 (-1,-5,-7) lies in Octant
( A )
B. VII
( c cdot v )
D. II
12
237 If ( boldsymbol{A}(cos boldsymbol{alpha}, sin boldsymbol{alpha}, boldsymbol{0}), boldsymbol{B}(cos boldsymbol{beta}, sin beta, boldsymbol{0}) )
( C(cos gamma, sin gamma, 0) ) are vertices of ( Delta A B C )
and let
[
begin{array}{l}
cos alpha+cos beta+cos gamma=3 a, sin alpha+ \
sin beta+sin gamma=3 b, text { then correct }
end{array}
]
matching of the following is:
List:1
begin{tabular}{ll}
A. Circumcentre & ( 1 .(3 a, 3 b, 0) ) \
cline { 0 }
end{tabular}
B. Centroid
[
2 .(0,0,0)
]
c. Ortho centre
[
3 .(a, b, 0)
]
A . 432
в. 231
c. 123
D. 234
12
238 If ( bar{a}, bar{b} ) are the position vectors of ( A ) and
( B ) then one of the following points lie on
( A B )
A ( cdot frac{2(bar{a}+bar{b})}{3} )
в. ( frac{(bar{a}-bar{b})}{3} )
c. ( frac{(bar{a}+bar{b})}{3} )
( ^{mathrm{D}} cdot frac{2 bar{a}+2 bar{b}}{3} )
E. None of these
12
239 The perimeter of triangle with vertices at (1,0,0),(0,1,0) and (0,0,1) is :
( mathbf{A} cdot mathbf{3} )
B . 2
( c cdot 2 sqrt{2} )
D. ( 3 sqrt{2} )
12
240 Show that the points are collinear
(1)
( boldsymbol{A}(mathbf{3}, mathbf{2},-mathbf{4}), boldsymbol{B}(mathbf{9}, mathbf{8},-mathbf{1 0}), boldsymbol{C}(-mathbf{2},-mathbf{3}, mathbf{1}) )
12
241 The points ( A ) and ( B ) have co-ordinate (1,2,4) and (-1,3,5) respectively. Find ( A B ) and its magnitude. 12
242 Prove the ( boldsymbol{A}(-mathbf{5}, mathbf{4}), boldsymbol{B}(-mathbf{1},-mathbf{2}), boldsymbol{C}(mathbf{5}, mathbf{2}) )
are the vertices of ( n ) isosceles right
angled triangle?
12
243 If ( P(x, y, z) ) is a point on the line segment joining ( A(2,2,4) ) and ( B(3,5,6) ) such that projection of ( overline{O P} ) on axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} ) respectively, then ( P ) divide ( A B ) in the ratio
A .3: 2
B. 2: 3
c. 1: 2
D. 1: 3
12
244 The direction cosine of a line equally inclined to the axes are
A ( cdot frac{1}{3}, frac{1}{3}, frac{1}{3} )
B. ( -frac{1}{3},-frac{1}{3},-frac{1}{3} )
c. ( frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}} )
D. none of these
12
245 If ( l, m, n ) are d.c’s of vector ( overline{O P} ) then
maximum value of ( l ) mn is
A ( cdot frac{1}{sqrt{3}} )
B. ( frac{1}{2 sqrt{3}} )
c. ( frac{1}{3 sqrt{3}} )
D. ( frac{2}{sqrt{3}} )
12
246 The lines ( frac{x-2}{1}=frac{y-3}{1}=frac{z-4}{-k} & )
( frac{boldsymbol{x}-mathbf{1}}{boldsymbol{k}}=frac{boldsymbol{y}-boldsymbol{4}}{boldsymbol{2}}=frac{boldsymbol{z}-mathbf{5}}{mathbf{1}} ) are coplanar if
A ( . k=0 ) or -1
B. ( k=1 ) or -1
c. ( k=0 ) or -3
D. ( k=3 ) or -3
12
247 If ( A B perp B C, ) then the value of ( lambda ) equal
where ( boldsymbol{A}(mathbf{2 k}, mathbf{2}, mathbf{3}), boldsymbol{B}(boldsymbol{k}, mathbf{1}, mathbf{5}), boldsymbol{C}(boldsymbol{3}+ )
( k, 2,1) )
( mathbf{A} cdot mathbf{3} )
B.
( c .-3 )
D. ( -frac{1}{3} )
12
248 The equation of altitude through ( B ) to
side ( A C ) is
A. ( r=k+t(7 i-10+2 k) )
в. ( r=k+t(-9 i+6 j-2 k) )
c. ( r=k+t(7 i-10 j-2 k) )
D. ( r=k+t(7 i+10 j+2 k) )
12
249 Equation of the plane passing through the point of intersection of ( x+2 y+=0 ) and ( 2 x+y=0 ) and which is perpendicular to
( 5 x+6 z=0 )
This question has multiple correct options
A. ( x=0 )
B. ( y=0 )
( c cdot z=0 )
D. x+y=0
E. ( x+z=0 )
12
250 Solve: ( sqrt{mathbf{2}} boldsymbol{x}+sqrt{mathbf{3}} boldsymbol{y}=mathbf{0} )
( sqrt{mathbf{3}} boldsymbol{x}-sqrt{mathbf{8}} boldsymbol{y}=mathbf{0} )
12
251 Find the vector equation of the plane through the points (2,1,-1) and (-1,3,4) and perpendicular to the
plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}=mathbf{1 0} )
12
252 Write the direction cosines of the line
whose cartesian equations are ( 2 x= )
( mathbf{3} boldsymbol{y}=-boldsymbol{z} )
12
253 If the angles made by a straight line with the coordinate axes are ( alpha, frac{pi}{2}- ) ( boldsymbol{alpha}, boldsymbol{beta} ) then ( boldsymbol{beta}= )
A . 0
в. ( frac{pi}{6} )
c. ( frac{pi}{2} )
( D )
12
254 Find the coordinates of the point which
divides the line segment joining the points (-2,3,5) and (1,-4,6) in the ratio
(i) 2: 3 internally
(ii) 2: 3 externally
12
255 If the distance between a point ( boldsymbol{P} ) and the point (1,1,1) on the line ( frac{x-1}{3}= ) ( frac{boldsymbol{y}-mathbf{1}}{mathbf{4}}=frac{boldsymbol{z}-mathbf{1}}{mathbf{1 2}} ) is ( mathbf{1 3}, ) then the
coordinates of ( boldsymbol{P} ) are
A. (3,4,12)
в. ( left(frac{3}{13}, frac{4}{13}, frac{12}{13}right) )
c. (4,5,13)
D. (40,53,157)
12
256 If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with
positive directions of ( mathrm{X}, mathrm{Y}, mathrm{Z} ) -axes, what is the value of ( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma )
( ? )
12
257 ( l=m=n=1 ) represents the direction
cosines of
A. ( x ) -axis
B. ( y ) -axis
c. ( z ) -axis
D. none of these
12
258 The coordinates of the foot of the
perpendicular drawn from of the origin to a plane are ( (12,-4,3) . ) Find the equation of the plane.
12
259 What is the angle between ( vec{P} times vec{Q} ) and ( vec{P}+vec{Q} ? )
( mathbf{A} cdot mathbf{0} )
в. ( frac{pi}{2} )
( c . pi )
D. ( frac{3 pi}{2} )
12
260 If 0 is the origin and the coordinates of ( P ) is ( (1,2,-3), ) then find the equation of the plane passing through P and perpendicular to OP.
A. ( x-2 y-3 z=-15 )
в. ( x+2 y-3 z=14 )
c. ( x-2 y+3 z=15 )
D. ( x-2 y-3 z=15 )
12
261 If ( 4 x+4 y-k z=0 ) is the equation of
the plane through the origin that contains the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}+mathbf{1}}{mathbf{3}}=frac{boldsymbol{z}}{mathbf{4}} )
then ( boldsymbol{k}= )
A . 1
B. 3
( c .5 )
D.
12
262 The sum of the intercepts on the coordinate axes of the plane passing through the point (-2,-2,2) and containing the line joining the points (1,-1,2) and (1,1,1) is?
A . 12
B. -8
( c .-4 )
D. 4
12
263 If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with
positive axes, then the range of ( sin alpha sin beta+sin beta sin gamma+sin gamma sin alpha ) is
( ^{mathrm{A}} cdotleft(frac{-1}{2}, 1right) )
в. ( left(frac{1}{2}, 2right) )
c. (-1,2)
D. (-1,2]
12
264 Let the line ( frac{boldsymbol{x}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{y}-mathbf{1}}{-mathbf{5}}=frac{boldsymbol{z}+mathbf{2}}{mathbf{2}} ) lie
in the plane ( x+3 y-alpha z+beta=0 . ) Then
( (alpha, beta) ) equals:
A ( cdot(-6,7) )
B ( cdot(5,-15) )
c. (-5,5)
D. (6,-17)
12
265 Find the ratio in which the ( X Y ) – plane
divides ( A B ) if is (1,2,3) and ( B ) is (-3,4,-5)
Also find the positive vector of the point of division.
12
266 Find the measure of the angle between two lines if their direction cosines
( ell, boldsymbol{m}, boldsymbol{n} ) satisfy ( ell+boldsymbol{m}-boldsymbol{n}=mathbf{0}, ell^{2}+ )
( boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} )
12
267 What are the DR’s of vector parallel to
(2,-1,1) and (3,4,-1)( ? )
A ( cdot(1,5,-2) )
B ( cdot(-2,-5,2) )
( mathbf{c} cdot(-1,5,2) )
D ( cdot(-1,-5,-2) )
12
268 The equation of motion of a rocket are:
( boldsymbol{x}=mathbf{2} boldsymbol{t}, boldsymbol{y}=-boldsymbol{4} boldsymbol{t}, boldsymbol{z}=boldsymbol{4} boldsymbol{t}, ) where the time
( t ) is given in seconds and the coordinate
of a moving point in kilometers. At what distance will the rocket be from the
starting point ( O(0,0,0) ) in 10 seconds?
( mathbf{A} cdot 60 mathrm{km} )
B. ( 30 mathrm{km} )
c. ( 45 mathrm{km} )
D. None of these
12
269 Point ( D ) has coordinates as (3,4,5)
Referring to the given figure, find the
coordinates of point ( boldsymbol{E} )
( mathbf{B} cdot(0,4,5) )
C. (0,5,4)
D. (0,3,4)
12
270 The point (0,-2,5) lies on the
( A cdot z ) axis
B. x axis
c. xy plane
D. yz plane
E . xz plane
12
271 ( left(cos ^{-1} l+cos ^{-1} m+cos ^{-1} nright) ) is equal
to
A . ( 90^{circ} )
B. ( 50^{circ} )
( c cdot 180^{circ} )
D. None of these
12
272 Find the distance of the point (2,3,5) from the ( x y- ) plane 12
273 Find the equation of the plane passing through the point (2,-1,1) and through the line of intersection of the planes ( vec{r} ). ( (2 hat{i}-3 hat{j}+hat{k})=3 ) and ( vec{r} cdot(hat{i}+5 hat{j}- )
( hat{boldsymbol{k}})=mathbf{4 . 4} )
12
274 Find the direction cosines of the line:
( frac{x-1}{2}=-y=frac{z+1}{2} )
12
275 Find the equation to the plane through the point (-1,3,2) and perpendicular to the planes ( x+2 y+2 z=11 ) and
( 3 x+3 y+2 z=15 )
12
276 figure bounded by non coplanar triangular planes. So, a tetrahedron has four non-coplanar points as its vertices. Suppose a tetrehedron has points
A,B,C,D as its vertices which have
coordinates ( left(boldsymbol{x} mathbf{1}, boldsymbol{y} mathbf{1}, boldsymbol{z}_{1}right)left(boldsymbol{x}_{2}, boldsymbol{y}_{2}, boldsymbol{z}_{2}right) )
( left(x_{3}, y_{3}, z_{3}right) ) and ( left(x_{4}, y_{4}, z_{4}right), ) respectively
in a rectangular three dimensional
space. Then, the coordinates of
its centroid are ( left[frac{boldsymbol{x}_{1}+boldsymbol{x}_{2}+boldsymbol{x}_{3}+boldsymbol{x}_{4}}{boldsymbol{4}}, frac{boldsymbol{y}_{1}+boldsymbol{y}_{2}+boldsymbol{y}_{3}+boldsymbol{y}_{4}}{boldsymbol{4}}right. )
Let a tetrahedron have three of its
vertices represented by the points (0,0,0),(6,5,1) and (4,1,3) and
its centroid lies at the point (1,2,5) Now, answer the following question. The coordinate of the fourth vertex of the
tetrahedron is:
A ( cdot(-6,2,16) )
B. (1,-2,13)
c. (-2,4,-2)
D. (1,-1,1)
12
277 The distance of the point ( 3 hat{i}+5 hat{k} ) from the line parallel to the vector ( 6 hat{i}+hat{j}- ) ( 2 hat{k} ) and passing through the point ( 8 hat{i}+ ) ( mathbf{3} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) is
A . 1
B . 2
( c .3 )
D.
12
278 If ( z=cos frac{pi}{6}+i sin frac{pi}{6}, ) then
A ( cdot|z|=1, arg z=frac{pi}{4} )
в ( cdot|z|=1, arg z=frac{pi}{6} )
с. ( quad|z|=frac{sqrt{3}}{2}, arg z=frac{5 pi}{24} )
D. ( quad|z|=frac{sqrt{3}}{2}, arg z=tan ^{-1} frac{1}{sqrt{2}} )
12
279 The vertices of a triangle are ( (2,3,5),(-1,3,2),(3,5,-2), ) then the
angles are
( ^{mathrm{B}} cos ^{-1}left(frac{1}{sqrt{5}}right), 90^{circ}, cos ^{-1}left(frac{sqrt{5}}{sqrt{3}}right) )
c. ( 30^{circ}, 60^{circ}, 90^{circ} )
D ( cdot cos ^{-1}left(frac{1}{sqrt{3}}right), 90^{circ}, cos ^{-1}(sqrt{frac{2}{3}}) )
12
280 An equation of a plane containing the
lines ( r=a_{1}+t b_{1} ) and ( r=a_{2}+t b_{2} )
where ( left[boldsymbol{a}_{2}-boldsymbol{a}_{1}, boldsymbol{b}_{1}, boldsymbol{b}_{2}right]=boldsymbol{0} ) is
A ( cdotleft[r-a_{1}, b_{1}, b_{2}right]=0 )
B . ( left[r-a_{2}, b_{1}, b_{2}right]=0 )
c. ( left[r-a_{2}, a_{1}, b_{2}right]=0 )
D . ( left[r-a, a_{2}, b_{2}right]=0 )
12
281 f ( A(2,1,3), B(2,-3,4), C(-1,2,7) ) are the
three points ; find a point D such that ( |overrightarrow{A B}|=frac{1}{2}|overrightarrow{C D}| ) and ( A B|| C D )
12
282 The distance between the line ( r=2 hat{i}- ) ( 2 hat{j}+3 hat{k}+lambda(hat{i}-hat{j}+4 hat{k}) ) and the plane
( r cdot(hat{i}+5 hat{j}+hat{k})=5 ) is
A ( cdot frac{10}{9} )
в. ( frac{10}{3 sqrt{3}} )
c. ( frac{10}{3} )
D. None of these
12
283 The name of the figure formed by the points (-1,-3,4),(5,-1,1),(7,-4,7)
and (1,-6,10) is a
A. square
B. rhombus
c. parallelogram
D. rectangle
12
284 The ratio in which the plane ( 2 x+3 y- )
( mathbf{2} z+mathbf{7}=mathbf{0} ) divides the line segment
joining the points (-1,1,3),(2,3,5) is
A .3: 5
B. 7: 5
( mathbf{c} cdot 9: 11 )
D. 1: 5 externally
12
285 State the following statement is True or
False
If two distinct lines are intersecting each other in a plane then they cannot
have more than one point in common.
A. True
B. False
12
286 The direction cosines of the ray
( boldsymbol{P}(1,-2,4) ) and ( Q(-1,1,-2) ) are
A ( cdot(-2,-3,-6) )
B ( cdot(2,-3,-6) )
( ^{C} cdotleft(frac{2}{7}, frac{3}{7}, frac{6}{7}right) )
D ( cdotleft(-frac{2}{7}, frac{3}{7},-frac{6}{7}right) )
12
287 Find the equation of the plane through the line
( boldsymbol{P}=boldsymbol{a} boldsymbol{x}+boldsymbol{b} boldsymbol{y}+boldsymbol{c} boldsymbol{z}+boldsymbol{d}=mathbf{0} )
( boldsymbol{Q}=boldsymbol{a}^{prime} boldsymbol{x}+boldsymbol{b}^{prime} boldsymbol{y}+boldsymbol{c}^{prime} boldsymbol{z}+boldsymbol{d}^{prime}=mathbf{0} )
and parallel to the line ( frac{x}{l}=frac{y}{m}=frac{z}{n} )
A ( cdot Pleft(a^{prime} l+b^{prime} m+c^{prime} nright)+Q(a l+b m+c n)=0 )
B – ( Pleft(a^{prime} l+b^{prime} m+c^{prime} nright)-Q(a l+b m+c n)=0 )
c. ( Qleft(a^{prime} l+b^{prime} m+c^{prime} nright)-P(a l+b m+c n)=0 )
D. ( Qleft(a^{prime} l+b^{prime} m+c^{prime} nright)+P(a l+b m+c n)=0 )
12
288 69.
t-3
y
=
and
If the line X-1 y +1 Z-1
1 2 3 4
intersect, then k is equal to:
[2012]
(a)
-1
12
289 find the equation of a line passing
through the point (1,2,-4) and
perpendicular to two lines. ( vec{r}= ) ( (8 hat{i}-19 hat{j}+10 hat{k})+ )
( boldsymbol{lambda}(boldsymbol{3} hat{boldsymbol{i}}-boldsymbol{1 6} hat{boldsymbol{j}}+boldsymbol{7} hat{boldsymbol{k}}) ) and ( overrightarrow{boldsymbol{r}}= )
( (15 hat{i}+29 hat{j}+5 hat{k})+ )
( boldsymbol{mu}(mathbf{3} hat{boldsymbol{i}}+quad boldsymbol{8} hat{boldsymbol{j}}-boldsymbol{5} hat{boldsymbol{k}}) )
12
290 The line passing through the points ( (5,1, a) ) and ( (3, b, 1) ) crosses the ( y z ) plane at the point ( left(0, frac{17}{2}, frac{-13}{2}right) . ) Then
A ( cdot a=2, b=8 )
В. ( a=4, b=6 )
c. ( a=6, b=4 )
D. ( a=8, b=2 )
2
12
291 ff ( y ) varies directly as ( x ) and ( y=12 )
when ( x=4, ) then find the linear
equation.
12
292 Cartesian equation of a line is ( frac{x-5}{3}= ) ( frac{boldsymbol{y}+boldsymbol{4}}{mathbf{7}}=frac{boldsymbol{z}-boldsymbol{6}}{boldsymbol{2}} . ) Write it in vector form 12
293 Name three undefined terms.
A. Point
B. Line
c. Plane
D. All of the above
12
294 Find a unit vector normal to the plane
through the points (1,1,1),(-1,2,3) and (2,-1,3)
12
295 If the line joining the points (-1,2,3),(2,-1,4) is perpendicular to the line joining the points ( (x,-2,4),(1,2,3) ) then ( x= )
( mathbf{A} cdot mathbf{3} )
B. 10
( c cdot frac{-3}{10} )
D. ( frac{-10}{3} )
12
296 The equation of the plane passing through the straight line ( frac{x-1}{2}= ) ( frac{boldsymbol{y}+mathbf{1}}{mathbf{- 1}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{4}} ) and perpendicular to
the plane ( boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{1 2} ) is
A. ( 9 x+2 y-5 z+4=0 )
В. ( 9 x-2 y-5 z+4=0 )
c. ( 9 x+2 y+5 z+4=0 )
D. None of these
12
297 92.
x-1 y+1 z-2
If the line,
meets the plane, x+2y+
2 3 4
3z= 15 at a point P, then the distance of P from the origin in
[JEEM 2019-9 April (M)
(a) V5/2
(b) 215
(c) 9/2
(d) 7/2
12
298 Let ( boldsymbol{O} ) be the origin and ( boldsymbol{A} ) be the point ( (64,0) . ) If ( P ) and ( Q ) divide ( O A ) in the ratio
( mathbf{1}: mathbf{2}: mathbf{3}, ) then the point ( boldsymbol{P} ) is
( ^{mathbf{A}} cdotleft(frac{32}{3}, 0right) )
в. (32,0)
( ^{mathbf{c}} cdotleft(frac{64}{3}, 0right) )
D. (16,0)
E ( cdotleft(frac{16}{3}, 0right) )
12
299 Calculate the distance between the
points (-3,6,7) and (2,-1,4) in ( 3 D )
space.
A . 4.36
в. 5.92
c. 7.91
D. 9.11
E . 22.25
12
300 Find the equation of the plane which
passes through the point (3,2,0) and contains the line ( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{6}}{mathbf{5}}= )
( frac{z-4}{4} ? )
12
301 The direction ratios of the line, given by the planes ( x-y+z-5=0, x-3 y-6=0 )
are
A ( .(3,1,-2) )
в. (2,-4,1)
c. (1,-1,1)
D. (0,2,1)
12
302 Using section formula prove that the point ( (0.7 .-7),(1,4,-5) ) and
(-1,10,-9) are collinear.
12
303 Find the vector equation of the line joining points ( 2 hat{i}+hat{j}+3 hat{k} ) and ( -4 hat{i}+ ) ( mathbf{3} hat{boldsymbol{j}}-hat{boldsymbol{k}} ) 12
304 Assertion
The point ( boldsymbol{A}(boldsymbol{3}, boldsymbol{1}, boldsymbol{6}) ) is the mirror image
of the point ( B(1,3,4) ) in the plane ( x- )
( boldsymbol{y}+boldsymbol{z}=mathbf{5} )
Reason
The plane ( x-y+z=5 ) bisects the line
segment joining ( boldsymbol{A}(mathbf{3}, mathbf{1}, boldsymbol{6}) ) and
( boldsymbol{B}(mathbf{1}, boldsymbol{3}, boldsymbol{4}) )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
305 Show that the plane whose vector equation is ( vec{r} .(hat{i}+2 hat{j}-hat{k})=3 ) contains the line ( vec{r}=hat{i}+hat{j}+lambda(2 hat{i}+hat{j}+4 hat{k}) ) 12
306 A unit vector parallel to the intersection of the planes ( vec{r} cdot(hat{i}-hat{j}+hat{k})=5 ) and ( vec{r} )
( (2 hat{i}+hat{j}-3 hat{k})=4 ) can be
This question has multiple correct options
( frac{2 hat{i}+5 hat{j}+3 hat{k}}{sqrt{38}} )
B. ( frac{2 hat{i}-5 hat{j}+3 hat{k}}{sqrt{38}} )
c. ( frac{-2 hat{i}-5 hat{j}-3 hat{k}}{sqrt{38}} )
D. ( frac{-2 hat{i}+5 hat{j}-3 hat{k}}{sqrt{38}} )
12
307 The length of the perpendicular from the
vertex ( D ) on the opposite face is
A ( cdot frac{14}{sqrt{6}} )
в. ( frac{2}{sqrt{6}} )
c. ( frac{3}{sqrt{6}} )
D. none of these
12
308 The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
parallel to the line ( frac{x}{2}=frac{y}{3}=frac{z-1}{-6} ) is
A .
B . 2
( c cdot 4 )
D. none of these
12
309 The circum radius of the triangle formed by the points (0,0,0),(0,0,12) and (3,4,0) is
A. ( sqrt{156} )
B. 13
c. ( frac{13}{2} )
D.
12
310 What is the equation of the plane containing the parallel lines
( boldsymbol{r}=boldsymbol{a}+boldsymbol{t c}, boldsymbol{r}=boldsymbol{b}+boldsymbol{p} boldsymbol{c} )
( mathbf{A} cdot r cdot(b-a) times c=[a b c] )
B cdot ( r cdot(a-b) times c=[a b c] )
C ( cdot r cdot(b-a) times c=-[a b c] )
D ( cdot r c times(b-a) times c=[a b c] )
12
311 The direction cosines of the line joining the points (2,3,-1) and (3,-2,1) are
в. ( frac{1}{sqrt{30}},-sqrt{frac{5}{6}} cdot sqrt{frac{2}{15}} )
c. ( frac{-1}{30}, frac{1}{6},-frac{1}{15} )
D. none of these
12
312 The ratio in which the line joining (3,4,-7) and (4,2,1) is dividing the ( x y ) plane
A .3: 4
B . 2: 1
( c cdot 7: 1 )
D. 4: 3
12
313 ( G(1,1,-2) ) is the centroid of the
triangle ( A B C ) and ( D ) is the mid point of
( boldsymbol{B C} cdot ) If ( boldsymbol{A}=(-1,1,-4), ) then ( boldsymbol{D}= )
( ^{mathbf{A}} cdotleft(frac{1}{2}, 1, frac{-5}{2}right) )
в. (5,1,2)
c. (-5,-1,-2)
D. (2,1,-1)
12
314 Let ( S ) be the set of all real values of ( lambda )
such that a plane passing through the points ( left(-lambda^{2}, 1,1right),left(1,-lambda^{2}, 1right) ) and
( left(1,1,-lambda^{2}right) ) also passes through the point ( (-1,-1,1) . ) Then ( S ) is equal to:
A ( cdot(sqrt{3}) )
B. ( {sqrt{3}-sqrt{3}} )
begin{tabular}{l}
c. {1,-1} \
hline
end{tabular}
D. {3,-3}
12
315 Find the Cartesian equation of ( vec{r} ) ( (2 hat{i}+3 hat{j}-4 hat{k})=1 ) 12
316 A point on XOZ-plane divides the join of (5,-3,-2) and (1,2,-2) at
( ^{mathrm{A}} cdotleft(frac{13}{5}, 0,-2right) )
в. ( left(frac{13}{5}, 0,2right) )
c. (5,0,2)
D. (5,0,-2)
12
317 A swimmer can swim ( 2 mathrm{km} ) in 15
minutes in a lake and in a river he can
swim a distance of ( 4 mathrm{km} ) in 20 minutes
along the stream. If a paper boat is put in the river, then the distance covered by it in ( 2 frac{1}{2} 2 ) hours will be
( mathbf{A} cdot 18 mathrm{km} )
B. ( 12 mathrm{km} )
c. ( 8 mathrm{km} )
D. ( 10 mathrm{km} )
12
318 The points ( A(-1,3,0), B(2,2,1) ) and
( C(1,1,3) ) determine a plane. The distance of the plane ( A, B, C ) from the
point ( D(5,7,8) ) is
A . ( sqrt{66} )
B. ( sqrt{71} )
c. ( sqrt{73} )
D. ( sqrt{76} )
12
319 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
A ( cdot 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}} )
в. ( 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}} )
c. ( 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}} )
D. ( 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}} )
12
320 Find the equation of the line in vector and in cartesion form that passes through the point with position vector ( mathbf{2} hat{mathbf{i}}-hat{mathbf{j}}+mathbf{4} hat{boldsymbol{k}} ) and is in the direction ( hat{mathbf{i}}+ )
( mathbf{2} hat{boldsymbol{j}}-hat{boldsymbol{k}} )
12
321 The number of octants in which ( Z )
coordinate is positive is
A . 2
B. 3
( c cdot 4 )
D. 1
12
322 The length of the normal from origin to
the plane ( x+2 y-2 z=9 ) is equal to
A . 2 units
B. 3 units
c. 4 units
D. 5 units
12
323 16.
Consider three points
P=(-sin(-a), -cos ),
Q = (cos(-a), sin B)
and
R=(cos(B-a +0), sin(6-0)), where
0<a,ß,e<I
(2008
Then,
(a) Plies on the line segment RQ
(b) Q lies on the line segment PR
(C) R lies on the line segment OP
(d) P, Q, R are non-collinear
12
324 A plane passes through (1,-2,1) and is
perpendicular to the planes ( 2 x-2 y+ )
( z=0 ) and ( x-y+2 z=4 . ) Then the
distance of the plane from the point
(1,2,2) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot sqrt{2} )
( D cdot 2 sqrt{2} )
12
325 If the distance between the plane,
( 23 x-10 y-2 z+48=0 ) and the plane containing the lines ( frac{boldsymbol{x}+mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}-boldsymbol{3}}{boldsymbol{4}}= )
( frac{z+1}{3} ) and ( frac{x+3}{2}=frac{y+2}{6}= )
( frac{z-1}{lambda}(lambda epsilon R) ) is equal to ( frac{k}{sqrt{633}} ) then ( k ) is
equal to
12
326 Prove that ( (vec{A} times vec{B})^{2}=A^{2} B^{2}-(vec{A} times vec{B})^{2} ) 12
327 A line makes an angle ( alpha, beta, gamma ) with the
( X, Y, Z ) axes. Then ( sin ^{2} alpha+sin ^{2} beta+ )
( sin ^{2} gamma= )
A .
B. 2
( c cdot frac{3}{2} )
D.
12
328 Show that the points (1,2,3),(7,0,1)
and (-2,3,4) are collinear.
12
329 Find the direction cosines of the sides
of the triangles whose vertices are (3,5,-4),(-1,1,2) and (-5,-5,-2)
12
330 The image of the point ( boldsymbol{P}(boldsymbol{alpha}, boldsymbol{beta}, gamma) ) by
the plane ( l x+m y+n z=0 ) is the point ( boldsymbol{Q}left(boldsymbol{alpha}^{prime}, boldsymbol{beta}^{prime}, boldsymbol{gamma}^{prime}right) . ) Then
A ( cdot alpha^{2}+beta^{2}+gamma^{2}=l^{2}+m^{2}+n^{2} )
B . ( alpha^{2}+beta^{2}+gamma^{2}=alpha^{2}+beta^{2}+gamma^{2} )
( mathbf{c} cdot alpha alpha^{prime}+beta beta^{prime}+gamma gamma^{prime}=0 )
D ( cdot lleft(alpha-alpha^{prime}right)+mleft(beta-beta^{prime}right)+nleft(gamma-gamma^{prime}right)=0 )
12
331 Find the vector equation of the plane passing through the points (2,5,-3),(-2,-3,5) and (5,3,-3) 12
332 The distance of the point (1,-2,3) from
the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
parallel to the line ( frac{x}{2}=frac{y}{3}=frac{z-1}{-6} ) is
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 4 )
D. None of these
12
333 Find the vector equation of the plane passing through (1,2,3) and perpendicular to the plane
( vec{r} cdot(hat{i}+2 hat{j}-5 hat{k})+9=0 )
12
334 The distance of the point (1,1,1) from
the plane passing through the points (2,1,1),(1,2,1) and (1,1,2) is
A ( cdot frac{1}{sqrt{3}} )
B.
( c cdot sqrt{3} )
D. None of these
12
335 If the d.c’s of a line are ( left(frac{1}{c}, frac{1}{c}, frac{1}{c}right), ) find ( c . ) 12
336 The position vectors ( vec{a}, vec{b}, vec{c} ) of three points satisfy the relation ( 2 vec{a}+7 vec{b}+ )
( mathbf{5} vec{c}=overrightarrow{0} . ) Are these points collinear?
12
337 If a line makes angles ( 90^{circ} ) and ( 60^{circ} )
respectively with the positive directions of ( x ) and ( y ) axes, find the angle which it
makes with the positive direction of ( z- )
axis.
12
338 ( operatorname{Given} boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{0}) ; boldsymbol{B}(mathbf{3}, mathbf{1}, mathbf{2}) ; boldsymbol{C}(mathbf{2},-mathbf{2}, mathbf{4}) )
and ( D(-1,1,-1) ) which of the following
points neither lie on ( A B ) nor on ( C D )
A ( .(2,2,4) )
В. (2,-2,4)
c. (2,0,1)
D. (0,-2,-1)
12
339 Find the direction cosines of the line
which is perpendicular to the lines which direction cosines proportional to
1,-2,-2 and 0,2,1
12
340 If ( A, B ) are the feet of the perpendiculars
from (2,4,5) to the ( x ) -axis, ( y ) -axis
respectively, then the distance ( A B ) is
A ( .2 sqrt{5} )
B. ( sqrt{29} )
c. ( sqrt{41} )
D. ( 3 sqrt{5} )
12
341 Find the distance of the point (2,12,5) from the point of intersection of the line ( vec{r}=2 hat{i}-4 hat{j}+2 hat{k}+lambda(3 hat{i}+4 hat{j}+2 hat{k}) )
and the plane ( vec{r}(hat{i}-2 hat{j}+hat{k})=0 )
12
342 The distance of point (-1,-5,-10) from the point of intersection of ( frac{x-2}{3}=frac{y+1}{4}=frac{-2}{12} ) and plane ( x )
( boldsymbol{y}+boldsymbol{z}=mathbf{5} ) is :
A . 10
B. 8
( c cdot 2 )
D. 13
12
343 The line ( frac{x-2}{3}=frac{y+1}{2}=frac{z-1}{-1} )
intersects the curve ( boldsymbol{x} boldsymbol{y}=boldsymbol{c}^{2}, boldsymbol{z}=mathbf{0}, ) if
( boldsymbol{c}= )
A. ( 5 sqrt{5} )
B. ( 4 sqrt{5} )
c. ( sqrt{5} )
D. ( 2 sqrt{5} )
12
344 Find the ratio in which the YZ-plane divides the line segment formed by joining the points (-2,4,7) and (3,-5,8) 12
345 Find the direction cosines of the vector
( overrightarrow{boldsymbol{r}}=(boldsymbol{6} hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}-boldsymbol{3} hat{boldsymbol{k}}) )
12
346 Four vertices of a tetrahedron are
(0,0,0),(4,0,0),(0,-8,0) and
( (0,0,12), ) Its centroid has the coordinates
A ( cdotleft(frac{4}{3},-frac{8}{3}, 4right) )
В. (2,-4,6)
c. (1,-2,3)
a 5
D. none of these
12
347 Find the equation of a plane which is
parallel to the plane ( x-2 y+2 z=5 )
and whose distance from the point (1,2,3) is 1
12
348 Let the direction – cosines of the line
which is equally inclined to the axis be ( pm frac{1}{sqrt{k}} . ) Find ( k ? )
A .2
B. 3
c. 5
D. 6
12
349 If the extremities of a diagonal of a square are (1,-2,3) and (2,-3,5) then the length of its side is:
A ( cdot sqrt{6} )
B. ( sqrt{3} )
c. ( sqrt{5} )
D. ( sqrt{7} )
12
350 ( operatorname{can} frac{1}{sqrt{3}}, frac{2}{sqrt{3}}, frac{-2}{sqrt{3}} ) be the direction
cosines of any directed line?
A. Yes
B. No
c. cannot say
D. None of these
12
351 The centroid of Tetraheadron
(3,4,5),(2,5,9),(5,2,8),(2,5,2)
12
352 The distance between (0,1,-1) and the point of intersection of the line ( frac{x}{2}= ) ( frac{boldsymbol{y}-mathbf{1}}{mathbf{3}}=frac{boldsymbol{z}+mathbf{1}}{mathbf{4}} ) and the plane ( boldsymbol{x}+boldsymbol{y}+ )
( z=9 ) is
A. ( sqrt{29} )
в. ( frac{1}{2} sqrt{29} )
( c cdot frac{4}{9} )
D. ( frac{2}{9} sqrt{29} )
12
353 If the d.rs of two lines are 1,-2,3 and ( 2,0,1, ) then the d.rs of the line
perpendicular to both the given lines is
A. -2,5,4
в. 2,-5,4
c. 2,5,-4
D. 1,5,-4
12
354 The direction cosines of ( A B ) are -2,2,1
If ( boldsymbol{A} equiv(mathbf{4}, mathbf{1}, mathbf{5}) ) and ( l(boldsymbol{A} boldsymbol{B})=mathbf{6 u n i t s} )
find the coordinates of ( boldsymbol{B} )
12
355 Match the statements/expressions in
List 1 with the values given in List 2
12
356 If ( O A B C ) is a tetrahedron such that the
( boldsymbol{O} boldsymbol{A}^{2}+boldsymbol{B} boldsymbol{C}^{2}=boldsymbol{O} boldsymbol{B}^{2}+boldsymbol{C} boldsymbol{A}^{2}=boldsymbol{O} boldsymbol{C}^{2}+ )
( A B^{2}, ) then which of the following is/are
correct
A. ( A B perp O C )
в. ( O B neq C A )
c. ( O C=A B )
D. ( A B perp B C )
12
357 Line through origin and parallel to ( Y ) –
axis is ( dots dots )
A ( cdot frac{x}{1}=frac{y}{0}=frac{2}{0} )
B. ( frac{x}{0}=frac{y}{1}=frac{z}{0} )
( mathbf{c} cdot frac{x}{1}=frac{y}{0}=frac{z}{1} )
D ( cdot frac{x}{1}=frac{y}{1}=frac{z}{0} frac{z}{0} )
12
358 A lines makes angles ( frac{boldsymbol{alpha}}{2}, frac{boldsymbol{beta}}{2}, frac{gamma}{2} ) with
positive direction of coordinate axes, then ( cos alpha+cos beta+cos gamma ) is equal to
A . -1
B. 1
( c cdot 2 )
D. 3
12
359 The cartesian equation of a line is ( frac{boldsymbol{x}+mathbf{3}}{mathbf{2}}=frac{boldsymbol{y}-mathbf{5}}{mathbf{4}}=frac{boldsymbol{z}+mathbf{6}}{mathbf{2}} ) find the vector
equation of the line?
12
360 Let ( A(-1,0) ) and ( B(2,0) ) be two points.
A point ( M ) moves in the plane in such a
way that ( angle M B A=2 angle M A B ). Then the
point ( M ) moves along
A. A straight line
B. A parabola
c. An ellipse
D. A hyperbola
12
361 The equation of the plane passing through the intersection of the planes ( mathbf{3} boldsymbol{x}-boldsymbol{y}+mathbf{2} boldsymbol{z}-mathbf{4}=mathbf{0} ) and ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}- )
( mathbf{2}=mathbf{0} ) and passing through the point
( A(2,2,1) ) is given by?
A. ( 7 x+5 y-4 z-8=0 )
в. ( 7 x-5 y+4 z-8=0 )
c. ( 5 x-7 y+4 z-8=0 )
D. ( 5 x+7 y-4 z+8=0 )
12
362 Show that ( boldsymbol{A}(mathbf{1}, mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{2}, mathbf{1}, mathbf{3}) ) and
( C(1,3,5) ) are not collinear.
12
363 Find the coordinates of the point, which divides the line segment joining the points ( boldsymbol{A}(mathbf{2},-mathbf{6}, mathbf{8}) ) and ( boldsymbol{B}(-mathbf{1}, mathbf{3},-mathbf{4}) )
externally in the ratio 1: 3
12
364 Name the octants in which the following
points lie: ( (mathbf{5}, mathbf{2}, mathbf{3}) )
12
365 The ratio in which the plane ( vec{r} cdot(vec{i}- ) ( mathbf{2} overrightarrow{boldsymbol{j}}+boldsymbol{3} overrightarrow{boldsymbol{k}})=17 ) divides the line joining
the points ( -2 vec{i}+4 vec{j}+7 vec{k} ) and ( 3 vec{i}- ) ( mathbf{5 j}+mathbf{8 k} ) is
A .1: 5
B. 1: 10
( c .3: 5 )
D. 3: 10
12
366 Find the distance between the point (7,2,4) and the plane determined by
the points ( boldsymbol{A}(mathbf{2}, mathbf{5},-mathbf{3}), boldsymbol{B}(-mathbf{2},-mathbf{3}, mathbf{5}) )
and (5,3,-3)
12
367 The distance from the origin to the centroid of the tetrahedron formed by
the points (0,0,0),(3,0,0),(0,4,0),(0,0,5) is
A. ( frac{sqrt{3+4+5}}{4} )
B. ( frac{sqrt{3+4+5}}{3} )
c. ( frac{sqrt{3^{2}+4^{2}+5^{2}}}{16} )
D. ( frac{sqrt{3^{2}+4^{2}+5^{2}}}{4} )
12
368 The value(s) of ( lambda ), for which the triangle with vertices ( (mathbf{6}, mathbf{1 0}, mathbf{1 0}),(mathbf{1}, mathbf{0},-mathbf{5}) ) and
( (6,-10, lambda) ) will be a right angled
triangle is/ are
A . 1
в. ( frac{70}{3}, 0 )
c. 35
D. ( 0,-frac{70}{3} )
12
369 Find the ratio in which the yz-plane
divides the join of the points (-2,4,7)
and (3,-5,8) and also find the ( c o )
ordinates of the point of intersection of
this line with the ( y z ) – plane.
A ( cdot lambda=frac{2}{3} ) and ( left(0, frac{2}{5}, frac{37}{5}right) )
B. ( lambda=frac{1}{3} ) and ( left(frac{-3}{4}, frac{7}{4}, frac{29}{4}right) )
c. ( lambda=frac{2}{3} ) and ( left(frac{-3}{4}, frac{7}{4}, frac{29}{4}right) )
D ( lambda=frac{1}{3} ) and ( left(0, frac{2}{5}, frac{37}{5}right) )
12
370 The direction cosines of the normal to
the plane ( 5 y+4=0 ) are?
A ( cdot 0, frac{-4}{5}, 0 )
в. 0,1,0
c. 0,-1,0
D. None of these
12
371 The equation of the plane that passes through the points (1,0,2),(-1,1,2),(5,0,3) is
A. ( x+2 y-4 z+7=0 )
B. ( x+2 y-3 z+7=0 )
c. ( x-2 y+4 z+7=0 )
D. ( 2 y-4 z-7+x=0 )
12
372 The extremities of a diagonal of a rectangular parallelopiped whose faces are parallel to the reference planes are (-2,4,6) and ( (3,16,6) . ) The length of the base diagonal is
( A cdot 7 )
B. 10
c. 11
D. 13
12
373 The equation of a line is ( 5 x-3= )
( mathbf{1 5 y}+mathbf{7}=mathbf{3}-mathbf{1 0 z} . ) Write the direction
( operatorname{cosines} ) of the line.
12
374 Find the direction cosines of a line that
pass through the point ( boldsymbol{P}(mathbf{1}, mathbf{4}, boldsymbol{6}) ) and
( Q(5,1,11) ) and is so directed that it
make an acute angle with the positive direction of ( boldsymbol{y}-boldsymbol{a} boldsymbol{x} boldsymbol{i} boldsymbol{s} )
12
375 The distance of point ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}, boldsymbol{z}_{1}right) ) to
the plane ( a x+b y+c z+D=0 ) is
given by
A ( cdot frac{left|a x_{1}+b y_{1}+c z_{1}-dright|}{sqrt{a+b+c}} )
B. ( frac{left|a x_{1}+b y_{1}+c z_{1}right|}{sqrt{a^{2}+b^{2}+c^{2}}} )
c. ( frac{left|a x_{1}+b y_{1}+c z_{1}-dright|}{sqrt{a^{2}+b^{2}+c^{2}}} )
D. None of these
12
376 A plane passes through (1,-2,1) and is
perpendicular to two planes ( 2 x-2 y+ )
( z=0 ) and ( x-y+2 z=4 . ) The distance
of the plane from the point is (1,2,2)
( A cdot 0 )
B.
( c cdot sqrt{2} )
D. ( 2 sqrt{2} )
12
377 intb, z = c.ytd
2003
14. The two lines x=ay+b,z=cy+d and x=a’y+
will be perpendicular, if and only if
(a) aa’+cc’+1 = 0
(b) aa’ + bb’ + cc’ + 1 = 0
(c) aa’+bb’ +cc’ = 0
(d) (a+a’)(b+b) +c+c’)=0.
12
378 A line passes through the points (6,-7,-1) and ( (2,-3,1) . ) The direction cosines of the line so directed that the
angle made by it with the positive direction of x-axis is acute, is?
A ( cdot frac{2}{3},-frac{2}{3},-frac{1}{3} )
B. ( -frac{2}{3}, frac{2}{3}, frac{1}{3} )
c. ( frac{2}{3},-frac{2}{3}, frac{1}{3} )
D. ( frac{2}{3}, frac{2}{3}, frac{1}{3} )
12
379 x – 4
The va
The value of k such that
y-2
z-k
22.
lies in the
plane 2x – 4y +z=7, is
(2003)
(a)
7
(b) -7
(d) 4
(c) no real value
.
12
380 A plane makes intercept 1,2,3 on the co-ordinate axes. If the distance from
origin is ( p ) then find the value of ( p )
12
381 The equation of plane containing the lines ( vec{gamma}=vec{alpha}+overrightarrow{lambda beta} ) and ( vec{gamma}=vec{beta}+mu vec{alpha} )
( mathbf{A} cdotleft[begin{array}{lll}vec{gamma} & vec{alpha} & vec{beta}end{array}right]=vec{alpha} cdot vec{beta} )
B ( cdotleft[begin{array}{lll}vec{gamma} & vec{alpha} & vec{beta}end{array}right]=0 )
( mathbf{c} cdotleft[begin{array}{lll}vec{alpha} & vec{beta} & vec{gamma}end{array}right]=vec{alpha} cdot vec{beta} )
D. None of these
12
382 is a
sa
mu on
19. Consider a pyramid OPQRS located in the first octant (x20.
y=0,220) with O as origin, and OP and OR along the x-axis
and the y-axis, respectively. The base OPQR of the pyramid
is a square with OP=3. The point S is directly above the
mid-point, T of diagonal OQ such that TS=3. Then
(JEE Adv. 2016)
(a) the acute angle between OQ and OS is
(b)
the equation of the plane containing the triangle OOS
is x-y=0
the length of the perpendicular from P to the plane
(c)
containing the triangle OQS is 5
the perpendicular distance from 0 to the straight lin
(d)
containing RS is
12
383 Show that the points ( boldsymbol{A}(-mathbf{3} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+mathbf{5} hat{boldsymbol{k}}), boldsymbol{B}(hat{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}+mathbf{3} hat{boldsymbol{k}}), boldsymbol{C} )
are collinear.
12
384 The points ( (boldsymbol{k}-mathbf{1}, boldsymbol{k}+mathbf{2}),(boldsymbol{k}, boldsymbol{k}+ )
1) ( ,(k+1, k) ) are collinear for
A. any value of ( k )
B. ( k=-frac{1}{2} ) only
c. no value of ( k )
D. integral values of ( k ) only
12
385 If ( boldsymbol{A} ) is ( (2,4,5), ) and ( B ) is (-7,-2,8) then which of the following is collinear with ( A ) and ( B ) is
A ( .(1,2,6) )
В. (2,-1,6)
c. (-1,2,6)
D. (2,6,-1)
12
386 A given unit vector is orthogonal to ( mathbf{5} hat{mathbf{i}}+mathbf{2} hat{mathbf{j}}+mathbf{6} hat{boldsymbol{k}} ) and coplanar with ( hat{mathbf{i}}-hat{boldsymbol{j}}+ )
( hat{k} ) and ( 2 hat{i}+hat{j}+hat{k} ) then the vector is?
A ( cdot frac{3 hat{j}-hat{k}}{sqrt{10}} )
( ^{text {В } cdot frac{6 hat{i}-5 hat{k}}{sqrt{61}}} )
c. ( frac{2 hat{i}-5 hat{k}}{sqrt{29}} )
( frac{2 hat{i}+hat{j}-2 hat{k}}{3} )
12
387 to the plane
68. A equation of a plane parallel to the
x-2y + 2z-5 = 0 and at a unit distance from the origin
2012
(a) x-2y+ 2z – 3 =0
(c) x2y + 2z – 1 =0
(b) x-2y + 2z+1=0
(d) x-2y + 2z+5=0
12
388 The direction ratios of the joining ( A(1,2,1) ) and (2,1,2) are
A .3,3,3
B. -1,1,-1
c. 3,1,3
D. ( frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}} )
12
389 ( boldsymbol{A}=(mathbf{2}, mathbf{4}, mathbf{5}) ) and ( boldsymbol{B}=(mathbf{3}, mathbf{5},-mathbf{4}) ) are two
points. If the ( X Y ) -plane, ( Y Z ) -plane divide ( A B ) in the ratio ( a: b ) and ( p: q )
respectively, then ( frac{a}{b}+frac{p}{q}= )
A ( cdot frac{23}{12} )
B. ( frac{-7}{12} )
c. ( frac{7}{12} )
D. ( frac{-22}{15} )
12
390 The pairs ( bar{a}, bar{b} ) and ( bar{c}, bar{d} ) each determine a plane. Then the planes are parallel if ( (overline{boldsymbol{a}} times overline{boldsymbol{c}}) times(overline{boldsymbol{b}} times overline{boldsymbol{d}})= )
( mathbf{0} 2)(overline{boldsymbol{a}} times overline{boldsymbol{c}}) cdot(overline{boldsymbol{b}} times overline{boldsymbol{d}})=mathbf{0} )
( (bar{a} times bar{b}) times(bar{c} times bar{d})= )
( mathbf{0} 4)(bar{a} times bar{b}) cdot(bar{c} times bar{d})=0 )
12
391 If a ray makes angles ( alpha, beta, gamma ) and ( delta ) with the four diagonals of a cube and
( mathbf{A}: cos ^{2} boldsymbol{alpha}+cos ^{2} boldsymbol{beta}+cos ^{2} boldsymbol{gamma}+cos ^{2} boldsymbol{delta} )
( mathbf{B}: sin ^{2} boldsymbol{alpha}+sin ^{2} boldsymbol{beta}+sin ^{2} boldsymbol{gamma}+sin ^{2} boldsymbol{delta} )
( mathbf{C}: cos 2 boldsymbol{alpha}+cos 2 boldsymbol{beta}+cos 2 gamma+cos 2 boldsymbol{delta} )
Arrange ( A, B, C ) in descending order
A. ( B, A, C )
в. ( A, B, C )
c. ( C, A, B )
D. ( B, C, A )
12
392 If ( R ) divides the line segment joining ( P(2, )
3, 4) and ( Q(4,5,6) ) in the ratio -3: 2 then the value of the parameter which
represents R is
в. ( =(10,9,8) )
( mathbf{c} .=(10,8,9) )
( mathbf{D} .=(9,10,8) )
12
393 Find the coordinates of a point on ( y ) -axis which are at a distance of ( 5 sqrt{2} ) from the
point ( boldsymbol{P}(boldsymbol{3},-boldsymbol{2}, boldsymbol{5}) )
12
394 Find the equation of the set of points ( P ) the sum of whose distances from
( boldsymbol{A}(mathbf{4}, mathbf{0}, mathbf{0}) ) and ( boldsymbol{B}(-mathbf{4}, mathbf{0}, mathbf{0}) ) is equal to
10
12
395 Distance between the points (12,4,7) and (10,5,3) is
A ( cdot sqrt{21} )
B. ( sqrt{5} )
c. ( sqrt{17} )
D. none of these
12
396 The distances of the point ( boldsymbol{P}(mathbf{1}, mathbf{2}, mathbf{3}) )
from the coordinates axes are:
A ( cdot sqrt{13}, sqrt{10}, sqrt{5} )
в. ( sqrt{11}, sqrt{10}, sqrt{5} )
c. ( sqrt{13}, sqrt{20}, sqrt{15} )
D. ( sqrt{23}, sqrt{10}, sqrt{5} )
12
397 If a line makes an angle of ( pi / 4 ) with the positive direction of each of ( x ) -axis and
( boldsymbol{y} ) -axis, then the angle that the line makes with the positive direction of the ( z ) -axis is
A ( cdot frac{pi}{6} )
в.
c.
D.
12
398 Find the vector equation of the line passing through the point (3,1,2) and perpendicular to the plane ( vec{r} cdot(2 hat{i}-hat{j}+widehat{k})=8 ) Also find the point
of intersection of line and plane.
12
399 Which of the following is true for a
plane?
This question has multiple correct options
A. A locus is called a plane if the line joining any two arbitrary points on the locus is also a part of the locus.
B. Value of ( y ) in a ( z x ) plane is non-zero.
C. Value of ( z ) in a ( x y ) plane is zero.
D. None of the above
12
400 The point which divides the line joining the points (1,3,4) and (4,3,1) internally in the ratio ( 2: 1, ) is
A. (2,-3,3)
в. (2,3,3)
c. ( left(frac{5}{2}, 3, frac{5}{2}right) )
D. (-3,3,2)
E ( .(3,3,2) )
12
401 If ( G ) is centroid of ( triangle A B C, ) then
A ( cdot vec{G}=vec{a}+vec{b}+vec{c} )
B. ( vec{G}=frac{vec{a}+vec{b}+vec{c}}{2} )
c. ( 3 vec{G}=vec{a}+vec{b}+vec{c} )
” ( quad 3 vec{G}=frac{vec{a}+vec{b}+vec{c}}{2} )
12
402 A line OP where ( 0=(0,0,0) ) makes
equal angles with ox, oy, oz. The point on OP, which is at a distance of 6 units
from 0 is:
A. ( left(frac{6}{sqrt{3}}, frac{6}{sqrt{3}}, frac{6}{sqrt{3}}right) )
B . ( (2 sqrt{3},-2 sqrt{3}, 2 sqrt{3}) )
( c cdot-(2 sqrt{3}, 2 sqrt{3}, 2 sqrt{3}) )
12
403 Show that the points ( boldsymbol{O}(mathbf{0}, mathbf{0}), boldsymbol{A}(mathbf{2},-mathbf{3}, mathbf{3}), boldsymbol{B}(-mathbf{2}, mathbf{3},-mathbf{3}) ) are
collinear. Find the ratio in which each point divides the segment joining the other two.
12
404 Find the equation of the line passing through the points ( A(3,2,-1) ) and
( boldsymbol{B}(mathbf{4},-mathbf{1}, mathbf{3}) )
12
405 A line makes the same angle ( theta ) with
each of the ( X ) and ( Z ) -axes. 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
A ( cdot frac{2}{5} )
в. ( frac{1}{5} )
( c cdot frac{3}{5} )
D. ( frac{2}{3} )
12
406 The foot of the perpendicular from the
point ( boldsymbol{A}(mathbf{7}, mathbf{1 4}, mathbf{5}) ) to the plane ( mathbf{2} boldsymbol{x}+mathbf{4} boldsymbol{y}- )
( z=2 ) is?
A. (3,1,8)
в. (1,2,8)
c. (3,-3,5)
D. (5,-3,-4)
12
407 The ratio in which the line segment
joining the points whose position vectors are ( 2 hat{i}-4 hat{j}-7 hat{k} ) and ( -3 hat{i}+ ) ( mathbf{5} hat{boldsymbol{j}}-boldsymbol{8} hat{boldsymbol{k}} ) is divided by the plane whose
equation is ( hat{r} cdot(hat{i}-2 hat{j}+3 hat{k})=13 ) is-
A. 13: 12 internally
B. 12: 25 externally
c. 13: 25 internally
D. 37: 25 internally
12
408 Distance between plane ( 3 x+4 y- )
( mathbf{2 0}=mathbf{0} ) and point ( (mathbf{0}, mathbf{0},-mathbf{7}) ) is
A. 4 units
B. 3 units
c. 2 units
D. 1 units
12
409 The scalar product and the magnitude of vector products of two vectors are ( 48 sqrt{3} ) and 144 respectively. Then the angle between the two vectors is
A. 54.7
B. 60
c. 90
D. 120
12
410 If a line has the direction ratios
4,-12,18 then find its direction
cosines.
A ( cdot-frac{2}{11},-frac{6}{11},-frac{9}{11} )
B. ( -frac{2}{11}, frac{6}{11},-frac{9}{11} )
c. ( frac{2}{11},-frac{6}{11}, frac{9}{11} )
D. ( frac{2}{11}, frac{6}{11}, frac{9}{11} )
12
411 The locus of a point ( P ) which moves such
that ( P A^{2}-P B^{2}=2 k^{2} ) where ( A ) and ( B )
( operatorname{are}(3,4,5) ) and (-1,3,-7) respectively
is
A. ( 8 x+2 y+24 z-9+2 k^{2}=0 )
B. ( 8 x+2 y+24 z-2 k^{2}=0 )
c. ( 8 x+2 y+24 z+9+2 k^{2}=0 )
D. ( 8 x-2 y+24 z-2 k^{2}=0 )
12
412 Assertion
In each of the three planes determined by two of the lines ( O A, O B, O C ) ( ( O ) being the origin), a straight line is drawn through ( O ) perpendicular to the third line.
The three lines so determined are
coplanar.
Reason
( (a times b) times c+(b times c) times a+(c times a) times )
( b=0, ) where ( O A=a, O B=b ) and
( boldsymbol{O} boldsymbol{C}=boldsymbol{c} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
413 Prove that
( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{0}), boldsymbol{B}(mathbf{3}, mathbf{1}, mathbf{1}), boldsymbol{C}(mathbf{7},-mathbf{1}, mathbf{3}) ) are
collinear.
12
414 The d.r. of normal to the plane through (1, 0, 0), (0, 1.0
which makes an angle /4 with plane x+y=3 are [20021
(a) 1,12,1
(b) 1,1, 2
(c) 1,1,2
(d) √2,1,1
12
415 If the projections of the line segment ( A B ) on the coordinate axes are 2,3,6
then the square of the sine of the angle made by ( boldsymbol{A B} ) with ( boldsymbol{x}=mathbf{0}, ) is
A ( cdot frac{3}{7} )
B. ( frac{3}{49} )
( c cdot frac{4}{7} )
D. ( frac{40}{49} )
12
416 The angle between any two faces is
( A cdot cos ^{-1}(1 / 3) )
B. ( cos ^{-1}(1 / 4) )
c. ( pi / 3 )
( mathbf{D} cdot cos ^{-1}(1 / 2) )
12
417 The point equidistant from the point ( boldsymbol{O}(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{A}(boldsymbol{a}, mathbf{0}, mathbf{0}), boldsymbol{B}(mathbf{0}, boldsymbol{b}, mathbf{0}) ) and
( C(0,0, c) ) has the coordinates
( mathbf{A} cdot(a, b, c) )
B. ( (a / 2, b / 2, c / 2) )
c. ( (a / 3, b / 3, c / 3) )
D. ( (a / 4, b / 4, c / 4) )
12
418 The distance between (5,1,3) and the
line ( x=3, y=7+t, z=1+t ) is
A . 4
B. 2
( c cdot 6 )
D. 8
12
419 The point of intersection of the lines ( frac{x-5}{3}=frac{y-7}{-1}=frac{z+2}{1} ) and ( frac{x+3}{-36}= )
( frac{y-3}{2}=frac{z-6}{4} ) is
A ( cdotleft(21, frac{5}{3}, frac{10}{3}right) )
в. (2,10,4)
c. (-3,3,6)
D. (5,7,-2)
12
420 Assertion(A): If centroid and
circumcentre of a triangle are known its orthocentre can be found.

Reason (R) : Centriod, orthocentre and
circumcentre of a triangle are collinear
A. Both A and R are individually true and R is the correct explanation of
B. Both A and R individually true but R is not the correct explanation of ( A )
c. ( A ) is true but ( R ) is false
D. A is false but R is true

12
421 ( mathbf{I f A}=(mathbf{1}, mathbf{2}, mathbf{3}), mathbf{B}=(mathbf{2}, mathbf{3}, mathbf{4}) ) and ( mathbf{C} ) is a
point of trisection of AB such that ( mathbf{C}_{mathbf{x}}+ ) ( mathbf{C}_{mathbf{y}}=frac{mathbf{1 3}}{mathbf{3}} ) then ( mathbf{C}_{mathbf{z}}= )
A ( cdot frac{10}{3} )
B. ( frac{11}{3} )
c. ( frac{11}{2} )
D. 11
12
422 +1
33.
x
If the angel e between the line
– 1 -2
22
– and
the plane 2x-y+ Váz+4 = 0 is such that sin 0=
then
the value of 1 is
[2005]
12
423 Find the distance between the points ( R(-3,0), Sleft(0, frac{5}{2}right) ) 12
424 Find the angle between the two lines
whose direction cosines are given by
equations ( l+m+n=0 ) and ( l^{2}+ )
( boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} )
A ( cdot frac{pi}{6} )
в.
( c cdot frac{pi}{2} )
D.
12
425 The projection of the line segment joining (0,0,0) and (5,2,4) on the line whose direction ratios are 2,-3,6 is
A . 28
B. 4
c. ( frac{40}{7} )
D. ( sqrt{45} )
12
426 Find the co-ordinates of the points of trisection of the line joining the points (-3,0) and (6,6) 12
427 If the lines ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and
( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect, then the
value of ( k ) is
A ( cdot frac{3}{2} )
B. ( frac{9}{2} )
( c cdot-frac{2}{9} )
D. ( -frac{3}{2} )
12
428 If the distance of the point ( P(4,3,5) ) from the Y-axis is ( lambda ), then the value of
( 7 lambda^{2} ) is
A .287
B. ( 7 sqrt{41} )
c. 63
D. 21
12
429 The shortest distance between z-axis
and the line
( boldsymbol{x}+boldsymbol{y}+mathbf{2} boldsymbol{z}-mathbf{3}=mathbf{0}=mathbf{2} boldsymbol{x}+mathbf{3} boldsymbol{y}+mathbf{4} boldsymbol{z} )
( 4, ) is
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 4 )
D. 3
12
430 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
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot-1 )
D. 0
12
431 17.
The radius of the circle in which the sphere
x2 + y2 + z2 + 2x – 2y – 42 – 19=0 is cut by the plane
x +2y + 2z +7= 0 is
[2003]
(a) 4 (6) (c) 2 (d) 3
12
432 A line with direction cosines
proportional to 2,1,2 meets each of the line ( boldsymbol{x}=boldsymbol{y}+boldsymbol{a}=boldsymbol{z} ) and ( boldsymbol{x}+boldsymbol{a}=boldsymbol{2} boldsymbol{y}= )
2 ( z ). The co-ordinates of each of the
points of intersection are given by:
A. ( (3 a, 3 a, 3 a),(a, a, a) )
в. ( (3 a, 2 a, 3 a),(a, a, a) )
c. ( (3 a, 2 a, 3 a),(a, a, 2 a) )
D. ( (2 a, 3 a, 3 a),(2 a, a, a) )
12
433 If ( vec{A} times vec{B}=vec{C}, ) which of the following
options is wrong?
( A cdot vec{C} ) is ( perp ) to ( vec{A} )
B cdot ( vec{C} ) is ( perp ) to ( vec{B} )
c. ( vec{C} ) is ( perp(vec{A}+vec{B}) )
D ( cdot vec{C} ) is ( perp ) to ( (vec{A} times vec{B}) )
12
434 Find the shortest distance between the
following pair of lines. ( overline{boldsymbol{r}}=(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}+overline{boldsymbol{k}})+boldsymbol{lambda}(mathbf{2} overline{boldsymbol{i}}-overline{boldsymbol{j}}+mathbf{3} overline{boldsymbol{k}}) & )
( overline{boldsymbol{r}}=(overline{boldsymbol{i}}-mathbf{3} overline{boldsymbol{j}}-overline{boldsymbol{k}})+boldsymbol{mu}(mathbf{3} overline{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}-mathbf{5} overline{boldsymbol{k}}) )
12
435 Show that the points (2,-1,3),(4,3,1) and (3,1,2) are collinear. 12
436 f a point is in the ( X Z ) -plane. What can
you say about its ( y ) -coordinate?
12
437 The plane ( x+2 y-z=4 ) cuts the
sphere ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{z}^{2}-boldsymbol{x}+boldsymbol{z}-boldsymbol{2}=mathbf{0} ) in
a circle of radius
A . 1
B. 3
( c cdot sqrt{2} )
D.
12
438 To find the vector and the Cartesian
equation in symmetric form of line passing through the points, (2,0,-3) and ( (mathbf{7}, mathbf{3},-mathbf{1 0}) )
12
439 If ( theta ) is the angle between two lines
whose d.c.s are ( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} )
then the d.cs of one of the angular bisectors of the two lines are
( ^{text {A }} cdot frac{l_{1}+l_{2}}{2}, frac{m_{1}+m_{2}}{2}, frac{n_{1}+n_{2}}{2} )
B. ( frac{l_{1}+l_{2}}{2 cos left(frac{theta}{2}right)}, frac{m_{1}+m_{2}}{2 cos left(frac{theta}{2}right)}, frac{n_{1}+n_{2}}{2 cos left(frac{theta}{2}right)} )
c. ( frac{l_{1}+l_{2}}{cos left(frac{theta}{2}right)}, frac{m_{1}+m_{2}}{cos left(frac{theta}{2}right)}, frac{n_{1}+n_{2}}{cos left(frac{theta}{2}right)} )
D. ( frac{l_{1}+l_{2}}{2 sin left(frac{theta}{2}right)} frac{m_{1}+m_{2}}{2 sin left(frac{theta}{2}right)} frac{n_{1}+n_{2}}{2 sin left(frac{theta}{2}right)} )
12
440 The image of the point with position vector ( hat{i}+3 hat{k} ) in the plane ( r cdot(hat{i}+hat{j}+ )
( hat{boldsymbol{k}})=mathbf{1} ) is
( mathbf{A} cdot hat{i}+2 hat{j}+hat{k} )
B ( cdot hat{i}-2 hat{j}+hat{k} )
c. ( -hat{i}-2 hat{j}+hat{k} )
D. ( hat{i}+2 hat{j}-hat{k} )
12
441 The vector form of the equation of the
line passing through points (3,4,7) and (5,1,6) is
( mathbf{A} cdot vec{r}=(3 hat{i}+4 hat{j}-7 hat{k})+lambda(2 hat{i}-3 hat{j}+13 hat{k}) )
B ( cdot vec{r}=(3 hat{i}+4 hat{j}-7 hat{k})+lambda(8 hat{i}+5 hat{j}-hat{k}) )
C ( . vec{r}=(3 hat{i}+4 hat{j}+7 hat{k})+lambda(2 hat{i}-3 hat{j}-hat{k}) )
D ( cdot vec{r}=(3 hat{i}+4 hat{j}-7 hat{k})+lambda(2 hat{i}-3 hat{j}-13 hat{k}) )
12
442 The angle between any two diagonals of
cube are:
A ( cdot cos ^{-1}left(frac{1}{2}right) )
B. ( cos ^{-1}left(frac{1}{3}right) )
( ^{mathbf{C}} cdot cos ^{-1}left(frac{1}{sqrt{3}}right) )
( mathrm{D} cdot cos ^{-1}left(frac{1}{sqrt{2}}right) )
12
443 Find the point of intersection of the following pair of lines, assuming that the vectors ( vec{a} ) and ( vec{b} ) are not parallel. ( vec{r}=gamma(vec{b}+vec{a}), vec{r}=mu(vec{b}-vec{a}) )
A . origin
в. ( vec{b}+vec{a} )
( c cdot 2 b )
D. no intersection point
12
444 Find the equation of the line in vector and in Cartesian form that passes through the point with position vector ( 2 hat{i}-hat{j}+4 hat{k} ) and is in the direction ( hat{i}+ ) ( mathbf{2} hat{boldsymbol{j}}-hat{boldsymbol{k}} ) 12
445 Find the equation of the plane passing through the points (0,0,0) and (3,-1,2) are parallel to the line ( frac{boldsymbol{x}-mathbf{4}}{mathbf{1}}=frac{boldsymbol{y}+mathbf{3}}{mathbf{- 4}}=frac{boldsymbol{z}+mathbf{1}}{mathbf{7}} ) 12
446 If a line has direction ratio 2,-1,-2
determine its direction cosines.
12
447 10
80
8
12
448 The angle between the lines whose
direction cosines satisfy the equations ( l+m+n=0 ) and ( l^{2}+m^{2}+n^{2} ) is
A ( cdot frac{pi}{2} )
в.
c.
D. ( frac{pi}{6} )
12
449 Find the vector and Cartesian equations
of the plane passing through the points with position vectors ( 3 vec{i}+4 vec{jmath}+ ) ( 2 vec{k}, 2 vec{i}-2 vec{j}-vec{k} ) and ( 7 vec{i}+vec{k} )
12
450 A straight line ( L ) on the ( x y ) -plane
bisects the angle between ( O X ) and ( O Y )
What are the direction cosines of ( L ? )
A ( cdot(1 / sqrt{2}, 1 / sqrt{2}, 0) )
в. ( (1 / 2, sqrt{3} / 2,0) )
c. (0,0,1)
D. ( (2 / 3,2 / 3,1 / 3) )
12
451 ( mathbf{A}=(mathbf{1}, mathbf{1}, mathbf{4}) ) and ( mathbf{B}=(mathbf{5},-mathbf{3}, mathbf{4}) ) are two
points. If the points ( P, Q ) are on the line
( A B ) such that ( A P=P Q=Q B ) then ( P Q= )
A ( cdot 2 sqrt{2} )
в. 4
c. ( sqrt{frac{32}{9}} )
D. ( sqrt{2} )
12
452 The angle between the lines, whose direction ratios are 1,1,2 and ( sqrt{3}- ) ( 1,-sqrt{3}-1,4, ) is
A . ( 45^{circ} )
B. ( 30^{circ} )
( c cdot 60^{circ} )
D. ( 90^{circ} )
12
453 Show that the points
( boldsymbol{A}(mathbf{1}, mathbf{1}, mathbf{1}), boldsymbol{B}(mathbf{1}, mathbf{2}, mathbf{3}) ) and ( boldsymbol{C}(mathbf{2},-mathbf{1}, mathbf{1}) ) are
vertices of an isosceles triangle.
12
454 Find the vector and Cartesian equation of the line that passes through the points (3,-2,-5) and (3,-2,6) 12
455 If two vertices of an equilateral triangle ( operatorname{are}(2,1,5) ) and ( (3,2,3), ) then its third vertex is:
A. (1,2,4)
в. (4,0,4)
c. (0,-4,4)
a 5
D. (4,4,1)
12
456 Given that ( boldsymbol{P}(boldsymbol{3}, boldsymbol{2},-boldsymbol{4}), boldsymbol{Q}(boldsymbol{5}, boldsymbol{4},-boldsymbol{6}) ) and
( R(9,8,-10) ) are collinear. Find the ratio in which ( Q ) divides ( P R )
12
457 If the lines ( frac{x-0}{1}=frac{y+1}{2}=frac{z-1}{-1} ) and ( frac{x+1}{k}= )
( frac{y-3}{-2}=frac{z-2}{1} ) are at right angles, then the
value of k is
( mathbf{A} cdot mathbf{5} )
B.
( c cdot 3 )
D. –
12
458 2.
The equation x +2y + 2z = 1 and 2x + 4y + 4z=9 have
(a) Only one solution
ono (1979)
(b) Only two solutions
(c) Infinite number of solutions i n the
(d) None of these.
12
459 Find the angles at which the normal
vector to the plane ( 4 x+8 y+z=5 ) is
inclined to the coordinate axes.
12
460 Find ( a ), if the distance between the
points ( P(11,-2) ) and ( Q(a, 1) ) is 5 units.
12
461 In what ratio, the line joining (-1,1) and (5,7) is divided by the line ( x+y= )
( 4 ? )
12
462 If the origin is the centroid of the triangle whose vertices are ( boldsymbol{A}(mathbf{2}, boldsymbol{p},-mathbf{3}), boldsymbol{B}(boldsymbol{q},-mathbf{2}, mathbf{5}) ) and ( boldsymbol{R}(-mathbf{5}, mathbf{1}, boldsymbol{r}) )
then find the values of ( boldsymbol{p}, boldsymbol{q}, boldsymbol{r} )
12
463 The equation of the plane passing through the point (1,1,1) and perpendicular to the planes ( 2 x+y- )
( 2 z=5 ) and ( 3 x-6 y-2 z=7 ) is?
A. ( 14 x+2 y-15 z=1 )
B. ( -14 x+2 y+15 z=3 )
c. ( 14 x-2 y+15 z=27 )
D. ( 14 x+2 y+15 z=31 )
12
464 f points (1,2),(3,5) and ( (0, b) ) are collinear the value of b is
A ( cdot frac{1}{2} )
в. ( frac{7}{2} )
( c cdot 2 )
D. –
12
465 Two distinct lines in a plane
A. always intersect
B. always either intersect or are parallel
C . always have two common points
D. none of these
12
466 h the origin.
ant distance
P.: 2x-y+z
17. In R’, let L be a straight line passing through the ori
Suppose that all the points on L are at a constant dista
from the two planes P, :x+2y-z+1=0 and P, : 2x –
-1= 0. Let M be the locus of the feet of the perpendicula,
drawn from the points on L to the plane P,. Which of th
following points lie (s) on M?
(JEE Adv. 2015)
(b)
12
467 ( boldsymbol{L} ) and ( boldsymbol{M} ) are two points with position vectors ( 2 bar{a}-bar{b} ) and ( a+2 bar{b} ) respectively.
The position vector of the point ( N ) which
divides the line segment ( L M ) in the ratio 2: 1 externally is
( mathbf{A} cdot 3 bar{b} )
в. ( 4 bar{b} )
( c .5 bar{b} )
D. ( 3 bar{a}+4 bar{b} bar{hline} bar{b}+4+4 bar{a} bar{a} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} )
12
468 A non-zero vector ( vec{a} ) is parallel to the
line of intersection of plane ( boldsymbol{p}_{mathbf{1}} )
determined by ( hat{i}+hat{j} ) and ( hat{i} ) and plane ( P_{2} ) determined by vectors ( hat{mathbf{i}}-hat{mathbf{j}} ) and ( hat{mathbf{i}}+hat{boldsymbol{k}} )
then angle between a and vector ( hat{mathbf{i}}- ) ( 2 hat{j}+2 hat{k} ) is
This question has multiple correct options
A ( cdot frac{pi}{4} )
B. ( frac{pi}{2} )
c. ( frac{pi}{3} )
D. ( frac{3 pi}{4} )
12
469 Find the coordinates of the points of trisection of the line segment joining the points ( A(-4,3) ) and ( B(2,-1) ) 12
470 The position vectors of three points are ( mathbf{2} overrightarrow{boldsymbol{a}}-overrightarrow{boldsymbol{b}}+mathbf{3} overrightarrow{boldsymbol{c}}, overrightarrow{boldsymbol{a}}-mathbf{2} overrightarrow{boldsymbol{b}}+boldsymbol{lambda} overrightarrow{boldsymbol{c}} ) and ( boldsymbol{mu} overrightarrow{boldsymbol{a}}-mathbf{5} overrightarrow{mathbf{b}} )
where ( vec{a}, vec{b}, vec{c} ) are non coplanar vectors then the points are collinear when
A ( cdot lambda=-2, mu=frac{9}{4} )
B. ( lambda=-frac{9}{4}, mu=2 )
c. ( lambda=frac{9}{4}, mu=-2 )
D. None of these
12
471 If the projections ofthe line segment ( A B )
on the ( y z ) -plane, ( z x ) -plane, ( x y ) -plane are ( sqrt{160}, sqrt{153}, 5 ) respectively, then the projection of ( A B ) on the ( z ) -axis is
A ( cdot sqrt{12} )
B. ( sqrt{13} )
c. 12
D. 144
12
472 Find the distance of the point (-1,-5,-10) from the point of intersection of the line
( boldsymbol{r}=mathbf{2} overline{boldsymbol{i}}-overline{boldsymbol{j}}+mathbf{2} overline{boldsymbol{k}}+overline{boldsymbol{lambda}}(mathbf{3} overline{boldsymbol{i}}+mathbf{4} overline{boldsymbol{j}}+mathbf{2} overline{boldsymbol{k}}) ) and
the plane ( bar{r} .(bar{i}-bar{j}+bar{k})=5 )
12
473 The coordinates of a point which divides the line joining the points ( P(2,3,1) ) and ( Q(5,0,4) ) in the ratio 1: 2 are
( ^{mathbf{A}} cdotleft(frac{7}{3}, 1, frac{5}{3}right) )
в. (4,1,3)
c. (3,2,2)
D. (1,-1,1)
12
474 Find the value of ( k ) if the pts ( A equiv )
( (1,2,-1), B equiv(4,-2,4) ) and ( C equiv )
( (0,0, k) ) form a triangle right angled at
( boldsymbol{C} )
12
475 If ( P(3,2,-4), Q(5,4,-6) ) and ( R )
(9,8,-10) are collinear, then ( R ) divides PQ in the ratio
A. 3: 2 internally
B. 3: 2 externally
c. 2: 1 internally
D. 2: 1 externally
12
476 Are the points (1,1),(2,3) and (8,11) collinear?
A. collinear
B. Non collinear
c. coplaner
D. None of above
12
477 The distance between the parallel planes given by the equations, ( vec{r} .(2 hat{i}- ) ( mathbf{2} hat{mathbf{j}}+hat{boldsymbol{k}})+mathbf{3}=mathbf{0} ) and ( vec{r} cdot(mathbf{4} hat{mathbf{i}}-mathbf{4} hat{mathbf{j}}+ )
( mathbf{2} hat{boldsymbol{k}})+mathbf{5}=mathbf{0} ) is –
A ( cdot 1 / 2 )
в. ( 1 / 3 )
c. ( 1 / 4 )
D. ( 1 / 6 )
12
478 The equation of a plane which passes through the point of intersection of lines ( frac{x-1}{3}=frac{y-2}{1}=frac{z-3}{2}, ) and ( frac{x-3}{1}= )
( frac{boldsymbol{y}-mathbf{1}}{mathbf{2}}=frac{boldsymbol{z}-mathbf{2}}{mathbf{3}} ) and at greatest
distance from point (0,0,0) is-
A. ( 4 x+3 y+5 z=25 )
B. ( 4 x+3 y+5 z=50 )
c. ( 3 x+4 y+5 z=49 )
D. ( x+7 y-5 z=2 )
12
479 ( boldsymbol{A}=(mathbf{1},-mathbf{1}, mathbf{2}) ) and ( boldsymbol{B}=(mathbf{2}, mathbf{3}, mathbf{7}) ) are two
points. If ( boldsymbol{P}, boldsymbol{O} ) divide ( boldsymbol{A B} ) in the ratios
2: 3,-2: 3 respectively then ( P_{x}+ )
( boldsymbol{Q}_{boldsymbol{y}}= )
A ( cdot frac{-38}{5} )
в. ( frac{38}{5} )
c. ( frac{-2}{5} )
D. ( frac{-47}{6} )
12
480 A line makes the same angle ( theta ) with
each of the ( x ) and ( z ) -axes. If the angle ( beta )
which it makes with ( y ) -axis, is such that ( sin ^{2} beta=3 sin ^{2} theta, ) then ( cos ^{2} theta ) is equal to
A ( cdot frac{2}{3} )
B. ( frac{1}{5} )
( c cdot frac{3}{5} )
D.
12
481 If
( boldsymbol{A}(1,2,-1), B(4,0,-3), C(1,2,-1), D( )
find the distance between ( A B ) and ( C D )
12
482 90.
The equation of the line passing through (4,3,1), parallel
to the plane x + 2y – Z-5 = 0 and intersecting the line
x +1 y-3 2-2
[JEE M 2019-9 Jan (M)
-3 2

1.
is:
x

4
y
+3
z+1
2
1
12
483 The image of the point ( P(1,2,3) ) in the
plane ( 2 x-y+z+3=0 ) is
A ( cdot(-3,4,1) )
в. (3,5,2)
c. (-3,5,2)
D. (3,-5,2)
12
484 If ( A ) and ( B ) be the points (3,4,5) and (-1,3,-7) respectively. Find the equation of the set of points ( boldsymbol{P} ) such
that ( boldsymbol{P} boldsymbol{A}^{2}+boldsymbol{P} boldsymbol{B}^{2}=boldsymbol{K}^{2}, ) where ( boldsymbol{K} ) is a
constant
12
485 If sum of the perpendicular distances of a variable point ( boldsymbol{P}(boldsymbol{x}, boldsymbol{y}) ) from the lines
( boldsymbol{x}+boldsymbol{y}-mathbf{5}=mathbf{0} ) and ( mathbf{3} boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{7}=mathbf{0} ) is
always ( 10 . ) Show that ( P ) must move on a
line.
12
486 14. The image of the line *<!-Y-3-2-4 in the plane
3
2x – y +2+3= 0 is the line:
[JEE M 2014)
x-3
y + 5
z-2
X-3 y +5 2-2
-3 –
15
x +3
15
Z
-2
x+3y-5z +2
-3 -1
3
-5
12
487 Two planes intersect each other to form
( mathbf{a}: )
A. plane
B. point
c. straight line
D. angle
12
488 Find the direction cosines of vector ( vec{r} )
which is equally inclined to ( O X, O Y )
and ( O Z ). Find total number of such
vectors.
A ( cdot frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}} ; 6 )
B. ( frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}, frac{1}{sqrt{3}} ; 8 )
( ^{mathbf{C}} pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}} ; 8 )
D. None of these
12
489 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
begin{tabular}{l}
A ( 3 sqrt{2} ) \
hline
end{tabular}
B. ( 5 sqrt{2} )
c. ( 6 sqrt{2} )
D. None of these
12
490 Show that the points with position vectors ( vec{a}+vec{b}, vec{a}-vec{b} ) and ( vec{a}+k vec{b} ) are
collinear for all values of ( k )
12
491 The equation of the plane through the points (2,3,1) and (4,-5,3) and parallel to ( x ) -axis is
A. ( x-z-1=0 )
в. ( 4 x+y-11=0 )
c. ( y+4 z-7=0 )
D. None of these
12
492 If the d.rs of ( O A ) and ( O B ) are 1,-1,-1
and ( 2,-1,1, ) then the d.cs of the line perpendicular to both ( boldsymbol{O} boldsymbol{A} ) and ( boldsymbol{O B} ) are
в. -2,-3,1
c. ( frac{-2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{1}{sqrt{14}} )
D. ( frac{2}{sqrt{41}}, frac{3}{sqrt{41}}, frac{-1}{sqrt{41}} )
12
493 Point ( D ) has coordinates as (3,4,5)
Find the coordinates of point ( G )
( mathbf{A} cdot(0,3,5) )
B. (3,0,4)
( mathbf{C} cdot(3,5,4) )
( mathbf{D} cdot(3,0,5) )
12
494 The number of straight lines that are equally inclined to the threedimensional coordinate axes, is
A .2
B. 4
( c .6 )
D. 8
12
495 ( frac{x-2}{1}=frac{y-3}{1}=frac{z-4}{-1} & frac{x-1}{k}= )
( frac{boldsymbol{y}-boldsymbol{4}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{5}}{boldsymbol{2}} ) are coplanar then ( mathbf{k}=? )
A. any value
B. exactly one value
c. exactly 2 values
D. exactly 3 values
12
496 Determine if the points (1,5)(2,3) and (-2,-11) are collinear.
A. True
B. False
12
497 The direction cosines of a line whose
equations are ( frac{x-1}{2}=frac{y+3}{4}=frac{z-2}{-3} )
A ( cdot frac{1}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{2}{sqrt{14}} )
В ( cdot frac{2}{sqrt{29}}, frac{4}{sqrt{29}}, frac{-3}{sqrt{29}} )
c. ( frac{1}{sqrt{29}}, frac{-3}{sqrt{29}}, frac{2}{sqrt{29}} )
D. 2,4,-3
12
498 Solve the following differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}-mathbf{1} )
A ( cdot y=x^{2}+x )
B . ( y=x^{2} )
c. ( y=x^{2}-x )
D. None of the above
12
499 If the foot of the perpendicular from
(0,0,0) to a plane is ( P(1,2,2) . ) Then, the equation of the plane is
A. ( -x+2 y+8 z-9=0 )
B. ( x+2 y+2 z-9=0 )
c. ( x+y+z-5=0 )
D. ( x+2 y-3 z+3=0 )
12
500 The ratio in which ( x y- ) plane divides the line joining the points (1,0,-3) and (1,-5,7) is given by
A . 7: 3
в. 3: 7
( c .3: 4 )
D. 4: 7
12
501 ( vec{A} cdot(vec{A} times vec{B}) )
( A cdot vec{A} cdot vec{B} )
( mathbf{B} cdot underset{A}{longrightarrow} times underset{B}{longrightarrow} )
( c cdot 0 )
D. 1
12
502 The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
parallel to the line. ( frac{x}{2}=frac{y}{3}=frac{z}{-6}, quad ) is :
( A )
B. 6/7
( c cdot 7 / 6 )
D. ( 1 / 6 )
12
503 Find the vector equation of the line through ( boldsymbol{A}(boldsymbol{3}, boldsymbol{4},-boldsymbol{7}) ) and ( boldsymbol{B}(boldsymbol{6},-1,1) )
Also find the cartesian form.
12
504 One of the rectangular components of a
force of ( 40 N ) is ( 20 N . ) Find the angle it
makes with this component and
magnitude of other component.
12
505 Find the distance of the point (1,2,-1) from the plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}-mathbf{1 0}= )
( mathbf{0} )
12
506 Consider three vectors ( overrightarrow{boldsymbol{p}}=boldsymbol{i}+boldsymbol{j}+ )
( boldsymbol{k}, overrightarrow{boldsymbol{q}}=2 boldsymbol{i}+boldsymbol{4} boldsymbol{j}-boldsymbol{k} ) and ( overrightarrow{boldsymbol{r}}=boldsymbol{i}+boldsymbol{j}+boldsymbol{3} boldsymbol{k} . )
( p, q ) and ( r ) denotes the position vector of three non-collinear points, then the equation of the plane containing these points is
A. ( 2 x-3 y+1=0 )
B. ( x-3 y+2 z=0 )
c. ( 3 x-y+z-3=0 )
D. ( 3 x-y-2=0 )
12
507 A line makes equal angles with the coordinate axis. The direction cosines of this line are
A ( cdotleft(frac{1}{3}, frac{1}{3}, frac{1}{3}right) )
B ( cdotleft(frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}}right) )
( ^{mathbf{c}} cdotleft(frac{1}{sqrt{3}}, frac{1}{3}, frac{1}{3}right) )
D ( cdotleft(frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, frac{1}{sqrt{2}}right) )
12
508 [2004]
25.
If the straight lines
1+t, z=2-t,
x=1+s, y=-3-as,z=1+as and x
with parameters s and t respectively, are co-planar, then a
equals.
(a) 0 (6) 1 (c) — 1 (d) -2
1.
1
.
12
509 The expression in the vector form for the
point ( vec{r}_{1} ) of intersection of the plane ( vec{r}_{1} cdot vec{n}=d ) and the perpendicular line
( vec{r}=vec{r}_{0}+t vec{n} ) where ( t ) is a parameter
given by
A ( cdot overrightarrow{r_{1}}=overrightarrow{r_{0}}+left(frac{d-overrightarrow{r_{0}} cdot vec{n}}{vec{n}^{2}}right) vec{n} )
B ( cdot overrightarrow{r_{1}}=overrightarrow{r_{0}}+left(frac{overrightarrow{r_{0}} cdot vec{n}}{vec{n}^{2}}right) vec{n} )
C・ ( _{overrightarrow{r_{1}}}=overrightarrow{r_{0}}+left(frac{overrightarrow{r_{0}} cdot vec{n}-d}{|vec{n}|}right) vec{n} )
D ( overrightarrow{r_{1}}=overrightarrow{r_{0}}+left(frac{overrightarrow{r_{0}} cdot vec{n}}{|vec{n}|}right) vec{n} )
12
510 what is the distance of the point (p,q,r) from the ( x ) -axis ? 12
511 ( A B C D ) is a parallelogram. ( L ) is a point on ( B C ) which divides ( B C ) in the ratio
( mathbf{1}: mathbf{2} . boldsymbol{A} boldsymbol{L} ) intersects ( boldsymbol{B} boldsymbol{D} ) at ( boldsymbol{P} . boldsymbol{M} ) is a
point on ( D C ) which divides ( D C ) in the
ratio 1: 2 and ( A M ) intersects ( B D ) in ( Q )
Point ( P ) divides ( A L ) in the ratio
A . 1: 2
B. 1: 3
c. 3: 1
D. 2: 1
12
512 For waht value of ( lambda ), the three numbers
( 2 lambda-1, frac{1}{4}, lambda-frac{1}{2} ) can be the direction
cosines of a straight line?
A ( cdot frac{1}{2} pm frac{sqrt{3}}{4} )
B. ( frac{3}{4} )
( c cdot pm frac{3}{4} )
D. ( frac{sqrt{3}}{2} pm frac{1}{4} )
12
513 If the sum of the squares of the distance of a point from the three
coordinate axes be ( 36, ) then its distance
from the origin is
A. 6 units
B. ( 3 sqrt{2} ) units
c. ( 2 sqrt{3} ) units
D. none of these
12
514 If (1,-1,0),(-2,1,8) and (-1,2,7) are three consecutive vertices of a
parallelogram then the fourth vertex is
A. (2,0,-1)
В. (1,0,-1)
c. (1,-2,0)
a 5
D. (0,-2,1)
12
515 In ( triangle A B C ) the mid points of the sides
( A B, B C ) and ( C A ) are respectively
( (l, 0,0),(0, m, 0) ) and ( (0,0, n) . ) Then
( frac{A B^{2}+B C^{2}+C A^{2}}{l^{2}+m^{2}+n^{2}} ) is equal to
( A cdot 2 )
B. 4
c. 8
D. 16
12
516 A point at which all the three perpendicular coordinate axes meets is known as
A. Meeting point
B. Origin
c. Triple point
D. None of these
12
517 Find the coordinate of the point ( boldsymbol{P} ) where the line through ( A(3,-4,-5) ) and ( B(2,-3,1) ) crosses the plane passing through three points ( boldsymbol{L}(mathbf{2}, mathbf{2}, mathbf{1}), boldsymbol{M}(mathbf{3}, mathbf{0}, mathbf{1}) ) and ( boldsymbol{N}(mathbf{4},-mathbf{1}, mathbf{0}) )
Also, find the ratio in which ( P ) divides
the line segment ( boldsymbol{A B} )
12
518 35.
If the plane 2ax – 3ay + 4az + 6 = 0 passes through the
midpoint of the line joining the centres of the spheres
x2 + y2 +z2 + 6x – 8y – 2z = 13 and
x2 + y2 +22 – 10x + 4y – 2z = 8 then a equals [2005]
(a) -1 (b) 1 (c) – (d) 2
12
519 The ratio in which the plane ( vec{r} .(hat{i}-2 hat{j}+ ) ( mathbf{3} hat{k})=17 ) divides the line joining the points ( (-2 hat{i}+4 hat{j}+7 hat{k}) ) and ( (3 hat{i}-5 hat{j}+ )
( mathbf{8} hat{boldsymbol{k}}) ) is
A .1: 5
B. 1: 10
( c .3: 5 )
D. 3: 10
12
520 Find the equation of line of intersection of planes ( vec{r} cdot(3 vec{i}-vec{j}+vec{k})=1 ) and ( vec{r} )
( (3 vec{i}+4 vec{j}-2 vec{k})=2 )
12
521 Find he equation of the line passing through (1,2,3) and perpendicular to the two lines ( frac{x}{1}=frac{y}{2}=frac{z}{-1} ) and ( frac{x-1}{3}=frac{y}{2}=frac{z}{6} ) 12
522 Plane passing through the points ( A(2,1 )
3), ( mathrm{B}(-1,2,4) ) and ( mathrm{C}(0,2,1) ). Determine its
point of intersection with the line ( r= )
( boldsymbol{j}+boldsymbol{k}+boldsymbol{t}(boldsymbol{2} boldsymbol{i}+boldsymbol{k}) )
A ( cdot(7,+1,4) )
B. (9,+1,-2)
c. (7,-1,4)
D. (9,-1,2)
12
523 If ( theta ) is the angle between two lines
whose d.cs are ( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} )
then
( frac{Sigmaleft(l_{1}+l_{2}right)^{2}}{4 cos ^{2}left(frac{theta}{2}right)}+frac{Sigmaleft(l_{mathrm{I}}-l_{2}right)^{2}}{4 sin ^{2}left(frac{theta}{2}right)}= )
( A )
B.
c. -1
( D )
12
524 ( mathbf{2} boldsymbol{x}-boldsymbol{y}+mathbf{3} boldsymbol{z}-mathbf{1}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}-boldsymbol{y}+ )
( mathbf{3} boldsymbol{z}+mathbf{3}=mathbf{0} )
12
525 Find the direction cosines of a line
which is perpendicular to the lines whose direction ratios are (1,-1,2) and (2,1,-1)
12
526 The line passes through the points ( (5,1, a) &(3, b, 1) ) crosses the ( y z ) plane at the point ( left(0, frac{17}{2},-frac{13}{2}right), ) then
A ( . a=4, b=6 )
B. ( a=6, b=4 )
c. ( a=8, b=2 )
D. ( a=2, b=8 )
8
12
527 The point in the ( x y- ) plane which is
equidistant from (2,0,3),(0,3,2) and (0,0,1) is
A ( .(1,2,3) )
В. (-3,2,0)
c. (3,-2.0)
D. (3,2,0)
E ( .(3,2,1) )
12
528 The equation of the plane, which bisects the line joining the points (1,2,3) and (3,4,5) at right angles is?
A. ( x+y+z=0 )
B. ( x+y-z=9 )
c. ( x+y+z=9 )
D. ( x+y-z+9=0 )
12
529 Find vector equation of line passing through the point whose position vector
is ( mathbf{3} hat{mathbf{i}}-mathbf{4} hat{mathbf{j}}+hat{boldsymbol{k}} ) and parallel to the vector
( 2 hat{i}+hat{j}-3 hat{k} . ) Also write the equation in
Cartesian form.
12
530 Find the coordinates of the point ( boldsymbol{P} ) where the line through ( boldsymbol{A}(mathbf{3},-mathbf{4},-mathbf{5}) )
and ( B(2,-3,1) ) crosses the plane
passing through three points ( boldsymbol{L}(mathbf{2}, mathbf{2}, mathbf{1}), boldsymbol{M}(mathbf{3}, mathbf{0}, mathbf{1}) ) and ( boldsymbol{N}(mathbf{4},-mathbf{1}, mathbf{0}) )
Also, find the ratio in which ( P ) divides
the line segment ( boldsymbol{A B} )
12
531 The distance between two points (1,1) and ( left(frac{2 t^{2}}{1+t^{2}}, frac{(1-t)^{2}}{1+t^{2}}right) ) is
( A cdot 4 t )
B. 3t
( c )
D. none of these
12
532 The coordinates of the foot of the
perpendicular drawn from the point ( A(1,0,3) ) to the join of the points
( B(4,7,1) ) and ( C(3,5,3) ) are
A ( cdot(5,7,17) )
B ( cdotleft(frac{-5}{7}, frac{7}{3}, frac{-17}{3}right) )
( ^{mathbf{c}} cdotleft(frac{5}{7}, frac{-7}{3}, frac{17}{3}right) )
D. ( left(frac{5}{7}, frac{7}{3}, frac{17}{3}right) )
12
533 1. The angles which a vector i + j + 2 k makes with X, Y
and Z axes respectively are
(a) 60°, 60°, 60° (b) 45°, 45°, 45°
(c) 60°, 60°, 45°
(d) 45°, 45°, 60°
12
534 The angle between the line ( 2 x=3 y= )
( -z ) and ( 6 x=-y=-4 z ) is
A ( .90^{circ} )
B . ( 0^{circ} )
( c cdot 30^{0} )
D. ( 45^{circ} )
12
535 The equation of a line passing through
(2,-3) and inclined at an angle of ( 135^{circ} )
with the positive direction of ( x ) -axis is.
A. ( x+y-1=0 )
в. ( x+y+1=0 )
c. ( x-y-1=0 )
D. ( -x+y+1=0 )
12
536 A ray makes angles ( frac{pi}{3}, frac{pi}{3} ) with ( overrightarrow{O X} ) and ( boldsymbol{O Y} ) respectively. Find the angle made by it with ( overrightarrow{O Z} ) 12
537 49.
If (2,3,5) is one end of a diameter of the sphere x2 + y2 +22
– 6x – 12y – 2z + 20 = 0, then the cooordinates of the other
end of the diameter are
[2007]
(a) (4,3,5)
(b) (4,3,-3)
(c) (4,9, -3)
(d) (4, -3,3).
12
538 If ( boldsymbol{A} times boldsymbol{B}=boldsymbol{B} times boldsymbol{A}, ) then the angle
between ( A ) and ( B ) is
A. ( pi )
в. ( pi / 3 )
c. ( pi / 2 )
D. ( pi / 4 )
12
539 The chord of contact of tangents from a
point ( boldsymbol{P} ) to a circle passes through ( boldsymbol{q} ). If
( l_{1} ) and ( l_{2} ) are the lengths of the tangents
from ( boldsymbol{P} ) and ( boldsymbol{Q} ) to the circle, then ( boldsymbol{P} boldsymbol{Q} ) is
equal to
A. ( frac{l_{1}+l_{2}}{2} )
в. ( frac{l_{1}-l_{2}}{2} )
C ( . sqrt{left|l_{1}^{2}-l_{2}^{2}right|} )
D. ( sqrt[2]{l_{1}^{2}+l_{2}^{2}} )
12
540 In what ratio does the plane ( 2 x+y- )
( z=3 ) divide line segment joining the
point ( boldsymbol{a}=(mathbf{2}, mathbf{1}, mathbf{3}), boldsymbol{b}=(mathbf{9},-mathbf{2}, mathbf{5}) )
12
541 If (1,-2,-2) and (0,2,1) are direction ratios of two lines, then the direction
cosines of a perpendicular to both the
lines are
( ^{mathrm{A}} cdotleft(frac{1}{3},-frac{1}{3}, frac{2}{3}right) )
в. ( left(frac{2}{3},-frac{1}{3}, frac{2}{3}right) )
( ^{c} cdotleft(-frac{2}{3},-frac{1}{3}, frac{2}{3}right) )
D. ( left(frac{2}{sqrt{14}},-frac{1}{sqrt{14}}, frac{3}{sqrt{14}}right) )
12
542 If the vectors ( 3 bar{p}+bar{q} ; 5 bar{p}-3 bar{q} ) and ( 2 bar{p}+ )
( bar{q} ; 4 bar{p}-2 bar{q} ) are pairs of mutually
perpendicular vectors then ( sin (theta) ) is ( (theta ) is the angle between ( overline{boldsymbol{p}} ) and ( overline{boldsymbol{q}} )
A ( cdot sqrt{55} / 4 )
B . ( sqrt{55} / 8 )
c. ( 3 / 16 )
D. ( sqrt{247} / 16 )
12
543 A hall has dimensions ( 24 m times 8 m times )
6 ( m ). The length of the longest pole which
can be accommodated in the hall is
A. 26
B. 28 m
( c cdot 30 m )
D. 36 m
12
544 Let ( vec{a}=x^{2} hat{i}+2 hat{j}-2 hat{k}, vec{b}=hat{i}-hat{j}+hat{k} )
and ( vec{c}=x^{2} hat{i}+5 hat{j}-4 hat{k} ) be three vectors.
Find the values of ( x ) for which the angle between ( vec{a} ) and ( vec{b} ) is acute and the angle between ( vec{b} ) and ( vec{c} ) is obtuse.
A ( cdot(-3,-2) cup(2,3) )
B . (-3,-1)( cup(1,3) )
c. (-3,-1)( cap(1,3) )
D. (-3,-2)( cap(2,3) )
12
545 91.
The plane through the intersection of the planes
x+y+z=1 and 2x+3y-z+4= 0 and parallel to y-axis also
passes through the point:
JEEM 2019-9 Jan (MI
(a) (-3,0,-1)
(b) (-3,1,1)
(c) (3,3,-1)
(d) (3, 2, 1)
12
546 Find ( x ) so that the point (6,5,-3) is at a distance of 13 from the point ( (x,-7,0) ) 12
547 Find the locus of the point, the sum of the squares of whose distances from
the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{0}, boldsymbol{x}-boldsymbol{y}=mathbf{0} )
( x+y-2 z=0 ) is 7
12
548 Find the equation of line passing through ( (5,0,5) &(2,1,3) . ) Also show that ( (5,0,5),(2,1,3) &(-4,3,-1) ) are
collinear.
12
549 Let the equation of the plane through
the points (-1,1,1) and (1,-1,1) and
perpendicular to the plane ( boldsymbol{x}+mathbf{2} boldsymbol{y}+ )
( mathbf{2} z=mathbf{7} ) be ( boldsymbol{k} boldsymbol{x}+boldsymbol{m} boldsymbol{y}-boldsymbol{n} boldsymbol{z}+boldsymbol{p}=mathbf{0} . ) Find
( boldsymbol{k}+boldsymbol{m}+boldsymbol{n}+boldsymbol{p} ? )
12
550 ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{1} ) 12
551 The distance of the point (1,-5,9) from the planer. ( (hat{mathbf{i}}-hat{mathbf{j}}+hat{boldsymbol{k}})=mathbf{5} )
measured long the line ( r=hat{i}+hat{j}+hat{k} ) is
A ( 3 sqrt{5} )
5
B. ( 10 sqrt{3} )
( c cdot 5 sqrt{3} )
D. ( 3 sqrt{10} )
12
552 Find the direction cosines of the unit
vector perpendicular to the plane ( vec{r} cdot(6 hat{i}-3 hat{j}-2 hat{k})+1=0 )
12
553 The coordinates of any point, which lies in ( boldsymbol{y} boldsymbol{z} ) plane, are
This question has multiple correct options
A ( .(x, y, y) )
в. ( (0, y, y) )
c. ( (0, y, x) )
D. ( (x, y, z) )
12
554 34.
The line passing through the points (5, 1, a) and (3, b, 1)
crosses the yz-plane at the point (0,7 -13). Then
(a) a=2, b=8
(b) a=4, b=6
(c) a=6, b=4
(d) a=8, b=2
12
555 A line passes through the point (6,-7,-1) and ( (2,-3,1) . ) if the angle ( alpha ) which the line makes with the positive
direction of ( x ) -axis is acute, the
direction cosines of the line are,
A ( .2 / 3,-2 / 3,-1 / 3 )
в. ( 2 / 3,2 / 3,-1 / 3 )
c. ( 2 / 3,-2 / 3,1 / 3 )
D. ( 2 / 3,2 / 3,1 / 3 )
12
556 Find the equations to the straight lines which are conjugate to the coordinate
axes with respect to the conic ( A x^{2}+ )
( 2 H x y+B y^{2}=1 )
Find the condition that they may
coincide, and interpret the result.
12
557 80.
Z+4
3
lies in the plane, kx +my-z=9,
x-3 y +2
If the line,
2 -1
then 12 + m2 is equal to :
(a) 5 (b) 2
(c) 26
[JEE M 2016]
(d) 18
12
558 x

1
V
-4
z
5
1
1
-k
15. The lines *-2 =973 – 3-4 and **=
coplanar if
(a) k=3 or-2
b) k=0 or -1
(c) k=1 or-1
1) k=0 or-3
=-7 are
[2003]
12
559 Let ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}) boldsymbol{B}= )
( (-1,-2,-1) C=(2,3,2) ) and ( D= )
( (4,7,6) . ) Then ( A B C D ) is a
A. rectangle
B. square
c. parallelogram
D. none of these
12
560 If the projections of the line segment ( A B ) on the coordinate axes are ( 12,3, k )
and ( A B=13, ) then ( k^{2}-2 k+3 ) is equal
to:
This question has multiple correct options
A . 0
B.
c. 11
D. 27
12
561 Find the coordinate of the points which trisect the line segment joining the points ( boldsymbol{A}(mathbf{2}, mathbf{1},-mathbf{3}) ) and ( boldsymbol{B}(mathbf{5},-mathbf{8}, mathbf{3}) ) 12
562 If ( mathbf{A}=(-mathbf{1}, mathbf{6}, mathbf{6}), mathbf{B}=(-mathbf{4}, mathbf{9}, mathbf{6}), mathbf{G}= )
( frac{1}{3}(-5,22,22) ) and ( G ) is the centroid of
the ( Delta A B C ) then the name of the
triangle ( mathbf{A B C} ) is
A. an isosceles triangle
B. a right angled triangle
c. an equilateral triangle
D. a right-angled isosceles triangle
12
563 Find ( x, ) if ( triangle A B C ) is right-angled at ( A )
where ( boldsymbol{A} equiv(mathbf{4}, mathbf{2}, mathbf{3}), boldsymbol{B} equiv(mathbf{3}, mathbf{1}, mathbf{8}), boldsymbol{C} equiv )
( (x,-1,2) )
12
564 A plane ( pi ) makes intercepts 3 and 4
respectively on ( z- ) axis. If ( pi ) is parallel to ( boldsymbol{y}- ) axis, then its equation is ?
A. ( 3 x+4 z=12 )
B. ( 3 z+4 x=12 )
c. ( 3 y+4 z=12 )
D. ( 3 z+4 y=12 )
12
565 Determine whether the points are
collinear.
( boldsymbol{P}(-mathbf{2}, mathbf{3}), boldsymbol{B}(mathbf{1}, mathbf{2}), boldsymbol{C}(mathbf{4}, mathbf{1}) )
12
566 If ( P(x, y, z) ) is a point on the line segment joining ( boldsymbol{A}(mathbf{2}, mathbf{2}, mathbf{4}) ) and
( B(3,5,6) ) such that projection of ( overrightarrow{O P} ) on axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} ) respectively, then ( P ) divide AB in the ratio
A .3: 2
B. 2: 3
c. 1: 2
D. 1: 3
12
567 Find the direction cosines of the
line passing through the two points (-2,4,-5) and (1,2,3)
12
568 Which one of the following is best condition for the plane ( a x+b y+c z+ )
( d=0 ) to intersect the ( x ) and ( y ) axes at
equal angle
A ( cdot|a|=|b| )
( b mid )
B . ( a=-b )
c. ( a=b )
D. ( a^{2}+b^{2}=1 )
12
569 The point of intersection of the line
joining the points (-3,4,-8) and (5,-6,4) with the ( X Y ) -plane is
( mathbf{A} cdotleft(frac{7}{3},-frac{8}{3}, 0right) )
в. ( left(-frac{7}{3},-frac{8}{3}, 0right) )
( ^{mathbf{c}} cdotleft(-frac{7}{3}, frac{8}{3}, 0right) )
D. ( left(frac{7}{3}, frac{8}{3}, 0right) )
12
570 If ( P(x, y, x) ) is a point on the line
segment joining ( Q(2,2,4) ) and ( R(3,5,6) ) such that the projection of ( O P ) on the axis are ( frac{13}{5}, frac{19}{5}, frac{26}{5} )
respectively, then ( P ) divides ( Q R ) in the
ratio
A . 1: 2
B. 3: 2
( c cdot 2: 3 )
D. 1: 3
12
571 The xy-plane divides the line joining the points (-1,3,4) and (2,-5,6)
A. internally in the ratio 2: 3
B. externally in the ratio 2: 3
c. internally in the ratio 3: 2
D. externally in the ratio 3: 2
12
572 Point ( (boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma}) ) lies on the plane ( boldsymbol{x}+boldsymbol{y}+ )
( z=2 . ) Let ( vec{a}=alpha hat{i}+beta hat{j}+gamma hat{k} ) and ( hat{k} times )
( (hat{k} times vec{a})=0 ) then ( gamma= )
A .
B. 1
c. 2
D.
12
573 Write the abscissa of the following
point
(0,5)
12
574 The ratio in which the line joining
(2,-4,3) and (-4,5,-6) is divided by the plane ( 3 x+2 y+z-4=0 ) is
A .2: 1
B . 4: 3
( c cdot-1: 4 )
D. 2: 3
12
575 If the point ( P(a, b, c), ) with reference to Eq. ( (i) ) lies on the plane ( 2 x+y+z=1 )
then the value of ( 7 a+b+c ) is
( mathbf{A} cdot mathbf{0} )
B. 12
( c cdot 7 )
D. 6
12
576 The image of the point ( 3 hat{i}-2 hat{j}+hat{k} ) in the plane ( bar{r} .(3 hat{i}-hat{j}+4 hat{k})=2 )
( mathbf{A} cdot-hat{j}+3 hat{k} )
B . ( hat{j}-3 hat{k} )
( mathbf{c} .-hat{j}-3 hat{k} )
D ( .-2 hat{j}-3 hat{k} )
12
577 Given planes are ( boldsymbol{P}_{1}: boldsymbol{c} boldsymbol{y}+boldsymbol{b} boldsymbol{z}=boldsymbol{x} quad boldsymbol{P}_{2}: boldsymbol{a} boldsymbol{z}+boldsymbol{c} boldsymbol{x}=boldsymbol{y} quad boldsymbol{P}_{3} )
( P_{1}, P_{2} ) and ( P_{3} ) pass through one line, if
A ( cdot a^{2}+b^{2}+c^{2}=a b+b c+c a )
B . ( a^{2}+b^{2}+c^{2}+2 a b c=1 )
( mathbf{c} cdot a^{2}+b^{2}+c^{2}=1 )
D. ( a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a+2 a b c=1 )
12
578 The plane which bisects the line
segment joining the points (-3,-3,4) and (3,7,6) at right angles, passes through which one of the following points?
A ( .(4,1,7) )
В. (4,1,-2)
c. (2,3,5)
D. (2,1,3)
12
579 The point (3,0,-4) lies on the
A. Y-axis
B. z-axis
c. XY-plane
D. xz-plane
E. YZ-plane
12
580 The point of intersection of the line ( frac{x-1}{3}=frac{y+2}{4}=frac{z-3}{-2} ) and plane
( 2 x-y+3 z-1=0 ) is.
B . (10,10,-3)
c. (-10,10,3)
D. None of these
12
581 Show that the points ( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{3}) )
( boldsymbol{B}(-1,-2,-3), C(2,3,2) ) and ( D(4,7,3) )
are the vertices of a parallelogram.
12
582 The distance of the point (1,-2,3)
from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} )
measured parallel to the line whose direction cosines are proportional to
2,3,-5 is
A ( cdot frac{9}{7} )
B. ( frac{11}{7} )
c. ( frac{15}{7} )
D. None of these
12
583 Find the distance between the following pairs of points:
(i) (2,3,5) and (4,3,1)
( (i i)(-3,7,2) ) and ((2,4,-1)
(iii) (-1,3,-4) and (1,-3,4)
(iv) (2,-1,3) and (-2,1,3)
12
584 In the ( Delta A B C, ) if ( A B=sqrt{2} ; A C= )
( sqrt{mathbf{2 0}}, boldsymbol{B}=(mathbf{3}, mathbf{2}, mathbf{0}) ) and ( boldsymbol{C}=(mathbf{0}, mathbf{1}, mathbf{4}) )
then the length of the median passing
through ( boldsymbol{A} ) is
A ( cdot frac{3}{2} )
B. ( frac{9}{2} )
c. ( frac{3}{sqrt{2}} )
D. ( frac{sqrt{3}}{2} )
12
585 A point on the line ( frac{boldsymbol{x}+mathbf{2}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{3}}{-mathbf{4}}= )
( frac{z-1}{2 sqrt{2}} ) at a distance 6 from the point ( (2, )
3, 1) is
A ( cdot(4-21,1+12 sqrt{2}) )
В. ( left(frac{-4}{5}, frac{-9}{5}, 1right) )
C ( cdotleft(frac{-16}{5}, frac{39}{5}, frac{5-12 sqrt{2}}{5}right) )
D. ( left(frac{-16}{5},-21,1+12 sqrt{2}right) )
12
586 The coordinates of the point where the line segment joining ( boldsymbol{A}(mathbf{5}, mathbf{1}, boldsymbol{6}) ) and ( B(3,4,1) ) crosses the yz plane are
A ( cdotleft(0, frac{17}{2}, frac{13}{2}right) )
в. ( left(0,-frac{17}{2}, frac{13}{2}right) )
c. ( left(0, frac{17}{2},-frac{13}{2}right) )
D. ( left(0,-frac{17}{2},-frac{13}{2}right) )
12
587 Show that the points ( boldsymbol{A}(boldsymbol{3}, boldsymbol{2},-boldsymbol{4}), boldsymbol{B}(boldsymbol{5}, boldsymbol{4},-boldsymbol{6}) ) and
( C(9,8,-10) ) are collinear, find the ratio in which ( B ) divides ( overline{A C} ).
12
588 Find the direction cosines of the sides
of the triangle whose vertices are (3,5,-4),(-1,1,2) and (-5,-5,-2)
12
589 The distance between the X-axis and
the point (3,12,5) is
A. 3
B. 13
( c cdot 14 )
D. 12
E. 5
12
590 The line ( frac{x-2}{3}=frac{y+1}{2}=frac{z-1}{-1} )
intersects the curve ( x y=c^{2}, z=0 ) if ( c ) is
equal to:
( A cdot pm 1 )
B. ( pm frac{1}{3} )
( mathrm{c} cdot pm sqrt{5} )
D. None of these
12
591 If the points ( (h, 3,-4),(0,-7,10) ) and ( (1, k, 3) ) are collinear, then ( h+k ) is
( mathbf{A} cdot mathbf{4} )
B.
c. -4
D. 14
12
592 The direction cosines of the line
passing through ( mathbf{P}(mathbf{2}, mathbf{3},-mathbf{1}) ) and the origin are
A ( cdot frac{2}{sqrt{14}}, frac{3}{sqrt{14}}, frac{1}{sqrt{14}} )
B. ( frac{2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{1}{sqrt{14}} )
c. ( frac{-2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{1}{sqrt{14}} )
D. ( frac{2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{-1}{sqrt{14}} )
12
593 The vertices of a triangle are 2,3,5)( ,(-1,3,2),(3,5,-2), ) then the
angles are
( mathbf{A} cdot 30^{circ}, 30^{circ}, 30^{circ} )
( ^{mathrm{B}} cos ^{-1}left(frac{1}{sqrt{5}}right), 90^{circ}, cos ^{-1}left(frac{sqrt{5}}{sqrt{3}}right) )
c. ( 30^{circ}, 60^{circ}, 90^{circ} )
D ( cdot cos ^{-1}left(frac{1}{sqrt{3}}right), 90^{circ}, cos ^{-1}left(frac{sqrt{2}}{sqrt{3}}right) )
12
594 Determine the three planes through the intersection of the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{1} )
and ( 2 x+3 y-z+4=0 ) which are
parallel to the three co – ordinate axes.
Also find the equation of the plane
perpendicular to the ( y z- ) plane and
passing through the point (2,3,1) and
(4,-5,3)
A ( cdot y-3 z-6=0 ; x-4 z=7 ) and ( 3 x-4 y-3=0 )
( y+4 z+7=0 )
B. ( y-3 z+6=0 ; x+4 z=7 ) and ( 3 x+4 y+3=0 )
( y-4 z+7=0 )
c. ( y+3 z+6=0 ; x+4 z=7 ) and ( 3 x+4 y+3=0 )
( y-4 z-7=0 )
D. ( y-3 z+6=0 ; x+4 z=7 ) and ( 3 x+4 y+3=0 )
( y+4 z-7=0 )
12
595 Find the angle between the planes
whose vector equations are ( vec{r} cdot(2 hat{i}+2 hat{j}-3 hat{k})=5 ) and
( vec{r} cdot(3 hat{i}-3 hat{j}+5 hat{k})=3 )
12
596 If a line makes an angle ( theta_{1}, theta_{2}, theta_{3} ) which
the axis respectively, then ( cos 2 theta_{1}+ )
( cos 2 theta_{2}+cos 2 theta_{3}=? )
A . -4
B . 2
( c .3 )
D. –
12
597 The projection of the join of the two points (1,4,5),(6,7,2) on the line whose d.s’s are (4,5,6) is
A ( cdot frac{17}{sqrt{77}} )
B. ( frac{7}{6} )
c. 21
D. ( frac{7}{9} )
12
598 ( boldsymbol{A}(boldsymbol{3}, boldsymbol{2}, boldsymbol{0}), boldsymbol{B}(boldsymbol{5}, boldsymbol{3}, boldsymbol{2}), boldsymbol{C}(-boldsymbol{9}, boldsymbol{6},-boldsymbol{3}) ) are
three points forming a triangle. If ( boldsymbol{A} boldsymbol{D} )
the bisector of ( angle B A C ) meets ( B C ) in ( D )
then coordinates of ( D ) are
( ^{mathrm{A}} cdotleft(-frac{19}{8}, frac{57}{16}, frac{17}{16}right) )
В ( cdotleft(frac{19}{8},-frac{57}{16}, frac{17}{16}right) )
( ^{mathrm{C}} cdotleft(frac{19}{8}, frac{57}{16}, frac{17}{16}right) )
D. None of these
12
599 Let ( P(3,2,6) ) be point in space and ( Q ) be appoint on the line ( vec{r}=(hat{i}-hat{j}+2 hat{k})+ )
( mu(-3 hat{i}+hat{j}+5 hat{k}) . ) Then the value of ( mu ) for which the vector ( overrightarrow{P Q} ) is parallel to the
plane ( boldsymbol{x}-mathbf{4} boldsymbol{y}+mathbf{3} boldsymbol{z}=mathbf{1} ) is
A. ( 1 / 4 )
B. – 1/4
c. ( 1 / 8 )
D. ( -1 / 8 )
12
600 Find the direction cosines of the vector
( hat{mathbf{i}}+mathbf{2} hat{mathbf{j}}+mathbf{3} hat{boldsymbol{k}} )
12
601 If the planes ( boldsymbol{x}-boldsymbol{c} boldsymbol{y}-boldsymbol{b} boldsymbol{z}=boldsymbol{0}, boldsymbol{c} boldsymbol{x}-boldsymbol{y}+ )
( boldsymbol{a} boldsymbol{z}=mathbf{0} ) and ( boldsymbol{b} boldsymbol{x}+boldsymbol{a} boldsymbol{y}-boldsymbol{z}=mathbf{0} ) pass
through a stright line,then the value of ( a^{2}+b^{2}+c^{2}+2 a b c ) is:
A . 1
B. 2
( c .3 )
D. none of these
12
602 The equations of two planes are ( P_{1} )
( 2 x-y+z=2, ) and ( P_{2}: x+2 y-z= )
3. The equation of the plane which
passes through the point (-1,3,2) and is perpendicular to each of the planes
( P_{1} ) and ( P_{2} ) is
A ( cdot x+3 y-5 z+2=0 )
B. ( x+3 y+5 z-18=0 )
c. ( x-3 y-5 z+20=0 )
D. ( x-3 y+5 z=0 )
12
603 Let the equation of the plane through the intersection of the planes ( x+2 y+ )
( mathbf{3} boldsymbol{z}-mathbf{4}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}+boldsymbol{y}-boldsymbol{z}+mathbf{5}=mathbf{0} ) and
perpendicular to the plane ( 5 x+3 y+ )
( mathbf{6} z+mathbf{8}=mathbf{0} ) be ( boldsymbol{k} boldsymbol{x}+mathbf{1 5} boldsymbol{y}+boldsymbol{m} boldsymbol{z}+mathbf{1 7 3}= )
0. Find ( k+m )
12
604 The ( x ) -coordinate of a point on the line joining the points ( P(2,2,1) ) and ( Q(5,1,-2) ) is ( 4 . ) Find its z-coordinate.
A . -1
B. -2
( c .1 )
D.
12
605 The Cartesian equation of the plane ( overrightarrow{boldsymbol{r}}=(1+boldsymbol{lambda}-boldsymbol{mu}) hat{boldsymbol{i}}+(boldsymbol{2}-boldsymbol{lambda}) hat{boldsymbol{j}}+(boldsymbol{3}- )
( 2 lambda+2 mu) hat{k} ) is-
( mathbf{A} cdot 2 x+y=5 )
В . ( 2 x-y=5 )
c. ( 2 x+z=5 )
D. ( 2 x-z=5 )
12
606 The projection of the join of the points (3,4,2),(5,1,8) on the line whose d.c’s ( operatorname{are}left(frac{2}{7}, frac{3}{7}, frac{6}{7}right) ) is
A. 7
B. ( frac{31}{71} )
( mathbf{c} cdot frac{42}{13} )
( D cdot frac{38}{138} )
12
607 Find the value of ( p ) for which the points ( (-5,1),(1, p) ) and (4,-2) are collinear
( mathbf{A} cdot mathbf{1} )
B.
( c .-1 )
D. 2
12
608 Find the shortest distance between the ( operatorname{lines} bar{r}=4 bar{i}-bar{j}+lambda(bar{i}+2 bar{j}-5 bar{k}) ) and
( overline{boldsymbol{r}}=overline{boldsymbol{i}}-overline{boldsymbol{j}}+2 overline{boldsymbol{k}}+boldsymbol{mu}(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}-mathbf{5} overline{boldsymbol{k}}) )
A. ( sqrt{220} )
B. ( frac{sqrt{221}}{sqrt{30}} )
c. 432
D. ( sqrt{33} )
12
609 For what value of ( boldsymbol{m}, ) the points ( (boldsymbol{3}, mathbf{5}) ) ( (m, 6) ) and ( left(frac{1}{2}, frac{15}{2}right) ) are collinear?
( A cdot 9 )
B. 5
( c .3 )
D.
12
610 The plane ( 2 x+3 y+k z-7=0 ) is
parallel to the line whose direction
ratios are (2,-3,1) then ( k= )
( mathbf{A} cdot mathbf{5} )
B. 8
c. 1
( D )
12
611 Image of point ( mathrm{P}(1,2,3) ) with respect to plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{1 2}, ) is
A. (5,4,3)
)
B. (9,6,3)
c. (5,6,7)
D. (3,4,5)
12
612 The point of intersection of the lines ( overrightarrow{boldsymbol{r}}=(-overrightarrow{boldsymbol{i}}+mathbf{2} overrightarrow{boldsymbol{j}}+boldsymbol{3} overrightarrow{boldsymbol{k}})+boldsymbol{t}(-boldsymbol{2} overrightarrow{boldsymbol{i}}+overrightarrow{boldsymbol{j}}+overrightarrow{boldsymbol{k}}) )
and ( vec{r}=(2 vec{i}+3 vec{j}+5 vec{k})+s(vec{i}+2 vec{j}+ )
( mathbf{3} overrightarrow{boldsymbol{k}}) ) is:
A ( .(1,1,2) )
в. (2,1,1)
c. (1,1,1)
D. (1,2,1)
12
613 The equation of the plane which is parallel to the ( x y- ) plane is
A. ( x=y )
B. ( z=c )
( mathbf{c} cdot y=c )
D. ( z=x y )
12
614 The sum of the intercepts on the coordinate axes of the plane passing
through the point (-2,-2,2) and
containing the line joining the points
(1,-1,2) and ( (1,1,1), ) is
A . 4
B. -4
c. 12
D. -8
12
615 Determine if the points (5,-1,1),(7,-4,7),(1,-8,10) and
(-1,-3,4) are the vertices of a rhombus or a square
12
616 Assertion ( (A) ). The direction ratios of the line joining origin and point ( (x, y, z) ) must be ( x, y, z )

Reason (R): If ( P(x, y, z) ) is a point in space and ( |O P|=r, ) then the direction cosines of ( O P ) are ( frac{x}{r}, frac{y}{r}, frac{z}{r} )
A. Both A and R are individually true and R is the correct explanation of
B. Both A and R individually true but R is not the correct explanation of A
c. A is true but R is false
D. A is false but R is true

12
617 A line separates a plane into three parts namely the two half-planes and the line itself
A. True
B. False
12
618 If the line passing through the origin
makes angles ( theta_{1}, theta_{2}, theta_{3} ) with the planes
( X O Y, X O Z ) and ( Z O X ) respectively
then prove that ( cos ^{2} theta_{1}+cos ^{2} theta_{2}+ )
( cos ^{2} theta_{3}=2 )
12
619 A triangle ( A B C ) is placed so that the
mid-points of the sides are on the ( x, y, z )
axes. Lengths of the intercepts made by the plane containing the triangle on
these axes are respectively ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} )
Coordinates of the centroid of the
triangle ( boldsymbol{A B C} ) are
A ( cdot(-alpha / 3, beta / 3, gamma / 3) )
B. ( (alpha / 3,-beta / 3, gamma / 3) )
c. ( (alpha / 3, beta / 3,-gamma / 3) )
D. ( (alpha / 3, beta / 3, gamma / 3) )
12
620 The point on the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}=frac{boldsymbol{y}+mathbf{3}}{-mathbf{2}}= )
( frac{z+5}{-2} ) at a distance of 6 from the point
(1,-3,-5) is
в. (3,-7,-9)
D. (-3,5,3)
12
621 Find the direction cosines of the vector
joining the points ( A(1,2,-3) ) and ( B(-1,-2,1) ) directed from ( A ) to ( B )
12
622 The ordinate of the point which divides the lines joining the origin and the point (1,2) externally in the ratio of 3: 2 is
A . -2
B. ( frac{3}{5} )
( c cdot frac{2}{5} )
D. 6
12
623 A tangent to the curve ( y=f(x) ) at
( boldsymbol{p}(boldsymbol{x}, boldsymbol{y}) ) meets ( boldsymbol{x}-boldsymbol{a} boldsymbol{x} boldsymbol{i} boldsymbol{s} ) at ( boldsymbol{A} ) and ( boldsymbol{y}- )
axis at ( B . ) If ( overline{A P}: overline{B P}=1: 3 ) and
( f(1)=1 ) then the curve also passes
through the point.
( ^{mathbf{A}} cdotleft(frac{1}{2}, 4right) )
B ( cdotleft(frac{1}{3}, 24right) )
c. ( left(2, frac{1}{8}right) )
D. ( left(3, frac{1}{28}right) )
12
624 The equation of the plane passing through ( (a, b, c) ) and parallel to the plane ( r cdot(hat{i}+hat{j}+hat{k})=2 ) is
A. ( x+y+z=1 )
B . ( a x+b y+c z=1 )
c. ( x+y+z=a+b+c )
D. None of these
12
625 If a unit vector ( vec{a} ) makes angles ( frac{pi}{3} ) with ( hat{i} )
( frac{pi}{4} ) with ( hat{j} ) and an acute angle ( theta ) with ( hat{k} )
then find ( theta ) and hence, the components of ( overrightarrow{boldsymbol{a}} )
12
626 The three planes divides the space into
A. four parts
B. six parts
c. eight parts
D. sixteen parts
12
627 Assertion
The points (1,1)( &(-1,-1) ) lie on the
same side of the line ( boldsymbol{x}-boldsymbol{y}+mathbf{1}=mathbf{0} )
Reason
The algebraic perpendicular distances from the given points to the line have
same sign
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
628 4.
A vector Ă has components A,, A2, A, in a right-handed
rectangular Cartesian coordinate system oxyz. The
coordinate system is rotated about the x-axis through an
T
angle =. Find the components of A in the new coordinate
system, in terms of A,, A2, Az.
(1983 – 2 Marks)
12
629 The three point ( boldsymbol{A}(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{B}(mathbf{2},-mathbf{3}, mathbf{3}), boldsymbol{C}(-mathbf{2}, mathbf{3},-mathbf{3}) ) are
collinear. Find in what ratio each point
divides the segment joining other two
12
630 The ratio in which the join of (1,-2,4)
and (4,2,-1) divided by the ( X-Y )
plane is
A . 1: 3
B. 3: 1
c. 4: 1
D. 1: 4
12
631 The graph of the equation ( y^{2}+z^{2}=0 )
in three dimensional space is
A. x- axis
B. y- axis
c. z- axis
D. yz-plane
12
632 If ( frac{1}{2}, frac{1}{2}, n(n<0) ) are the dos of a line,
then the angle made by that line with ( boldsymbol{O} boldsymbol{Z} ) where ( boldsymbol{O}=(mathbf{0}, mathbf{0}, mathbf{0}) ) is
A ( cdot frac{-1}{sqrt{2}} )
B . ( 45^{circ} )
( c cdot 60^{circ} )
D. ( 135^{circ} )
12
633 Find the co-ordinates of the points on the join of (-3,7,-13) and (-6,1,-10) which is nearest to the intersection of the planes: ( 3 x-y- )
( mathbf{3} z+mathbf{3 2}=mathbf{0} ) and ( mathbf{3} boldsymbol{x}+mathbf{2} boldsymbol{y}-mathbf{1 5 z}-mathbf{8}= )
0
12
634 The equation of plane passing through (-1,0,-1) parallel to ( x z ) plane is
В. ( y=0 )
c. ( -x-z=0 )
D. None of the above
12
635 The condition that the line ( frac{x-alpha^{prime}}{l}= ) ( frac{boldsymbol{y}-boldsymbol{beta}^{prime}}{boldsymbol{m}}=frac{boldsymbol{z}-boldsymbol{gamma}^{prime}}{boldsymbol{n}} ) in the plane ( boldsymbol{A} boldsymbol{x}+ )
( boldsymbol{B} boldsymbol{y}+boldsymbol{C} boldsymbol{z}+boldsymbol{D}=mathbf{0} ) is
( mathbf{A} cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}+D=0 ) and ( A l+B m+C n neq 0 )
B ( cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}+D neq 0 ) and ( A l+B m+C n=0 )
C ( cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}+D=0 ) and ( A l+B m+C n=0 )
D ( cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}=0 ) and ( A l+B m+C n=0 )
12
636 A rectangular parallelopiped is formed by drawing planes through the point (-1,2,5) and (1,-1,-1) and paralle to the coordinates planes. The length of the diagonal of the parallelopiped is 12
637 82.
The distance of the point (1, -5,9) from the plane x-y+z=5
measured along the line x=y=zis: [JEE M 2016]
10
a)
TT
(b)
(c) 3/10
(d) 103
12
638 If the points (-1,3,2),(-4,2,-2) and ( (5,5, lambda) ) are collinear, then ( lambda ) is equal to
A . -10
B. 5
( c .-5 )
D. 10
12
639 The equations of the line of intersection
of the planes ( x+y+z=2 ) and ( 3 x- )
( boldsymbol{y}+mathbf{2} z=mathbf{5} ) in symmetric form are
A ( frac{x-frac{7}{4}}{4}=frac{y-frac{1}{4}}{-1}=frac{z}{-3} )
B. ( quad frac{x}{3}=frac{y+frac{1}{3}}{1}=frac{z-frac{7}{4}}{-4} )
c. ( frac{x}{1}=frac{3 y+1}{1}=frac{3 z-7}{-4} )
D. none of these
12
640 23.
(1)
Find
Find the equation of the plane passing through the
points (2,1,0), (5,0, 1) and (4,1,1).
If P is the point (2,1, 6) then find the point Q such that
PQ is perpendicular to the plane in (i) and the mid point
of PQ lies on it.
(2003 – 4 Marks)
12
641 Find the angle between the following pair of lines:
(i) ( vec{r}=2 hat{i}-5 hat{j}+hat{k}+lambda(3 hat{i}-2 hat{j}+6 hat{k}) )
and ( vec{r}=mathbf{7} hat{boldsymbol{i}}-boldsymbol{6} hat{boldsymbol{k}}+boldsymbol{mu}(hat{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}+mathbf{2} hat{boldsymbol{k}}) )
(ii) ( vec{r}=3 hat{i}+hat{j}-2 hat{k}+lambda(hat{i}-hat{j}-2 hat{k}) )
and ( vec{r}=2 hat{i}-hat{j}-56 hat{k}+ )
( mu(hat{mathbf{3}} mathbf{i}-mathbf{5} hat{mathbf{j}}-mathbf{4} hat{boldsymbol{k}}) )
12
642 2.
Let : 2x + y – z = 3 and P2 : x +2y +z = 2 be two planes.
Then, which of the following statement(s) is (are) TRUE?
(JEE Adv. 2018)
(a) The line of intersection of P, and P, has direction
ratios 1, 2, -1
3x -4 1-3yZ
(b) The line . -= =
is perpendicular to the line of intersection of P, and
P2
c)
(d)
The acute angle between P, and P, is 60°.
If P, is the plane passing through the point (4, 2, -2)
and perpendicular to the line of intersection of P, and
P, then the distance of the point (2, 1, 1) from the
plane Pz is 7
12
643 14
The equation of a plane passing through the line of
intersection of the planes x + 2y + 3z=2 and x-y+z=3 and
at a distance
from the point (3,1,-1) is
(2012)
(a) 5x-1ly+z=17
(c) x+y+z= 13
(b) V2x+y = 3/2 – 1
(d) x-V2y =1-2
12
644 Find the vector equation of the line joining (1,2,3) and (-3,4,3) and show pependicular to the z-axis 12
645 In the given figure, co-ordinates of the
midpoint of ( boldsymbol{A B} ) are
A . (0,2)
в. (0,3)
c. (1,2)
D. (3,1)
12
646 The centroid of triangle ( boldsymbol{A}(mathbf{3}, mathbf{4}, mathbf{5}) ; boldsymbol{B}(mathbf{6}, mathbf{7}, mathbf{2}) ; boldsymbol{C}(mathbf{0},-mathbf{5}, mathbf{2}) ) is
A ( .(3,2,3) )
в. (5,2,1)
c. (2,5,1)
D. (3,4,1)
12
647 Find the coordinates of point which
divides the line joining the points (3,4) and (6,1) in the ratio of 1: 2
12
648 ( 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?
A . -1
B. 0
c. 1
D. 2
12
649 If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with
coordinate axes, find
( cos ^{2} alpha+cos ^{2} beta+cos ^{2} gamma+1 )
12
650 The straight lines ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{2}}{mathbf{2}}= )
( frac{z-3}{3} ) and ( frac{x-1}{1}=frac{y-2}{2}=frac{z-3}{3} ) are
A. Parallel lines
B. Intersecting at ( 60^{circ} )
c. skew lines
D. Intersecting at right angle
12
651 Find the angles between the lines,
whose direction cosines are give by the equation ( l^{2}-m^{2}+n^{2}=0, l+m+ )
( boldsymbol{n}=mathbf{0} )
A. 0
в. ( frac{pi}{6} )
( c cdot frac{pi}{4} )
D.
12
652 Find the equation of the line passing through (1,2,-4) and perpendicular to both the lines ( frac{x-1}{2}=frac{y+2}{-3}=frac{z-4}{4} )
and ( frac{boldsymbol{x}-mathbf{3}}{mathbf{5}}=frac{boldsymbol{y}+mathbf{6}}{mathbf{1}}=frac{boldsymbol{z}+mathbf{1 0}}{mathbf{2}} )
12
653 If the ( z x ) -plane divides the line segment joining (1,-1,5) and (2,3,4) in the ratio ( p: 1, ) then ( p+1= )
A ( cdot frac{1}{3} )
B. 1: 3
( c cdot frac{3}{4} )
D. ( frac{4}{3} )
12
654 The image of the point (-1,3,4) in the
plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}=mathbf{0} ) is
( ^{mathbf{A}} cdotleft(-frac{17}{3},-frac{19}{3}, 4right) )
В. (15,11,4)
( ^{mathbf{C}} cdotleft(-frac{17}{3},-frac{19}{3}, 1right) )
D. ( left(frac{9}{5},-frac{13}{5}, 4right) )
12
655 75. The angle between the lines whose direction cosines satisfy
the equations 1+m+n= 0 and 12 = m² +nis
[JEE M 2014
wa
12
656 f a plane passes through the point (1,1,1) and is perpendicular to the line ( frac{x-1}{3}=frac{y-1}{0}=frac{z-1}{4} ) then its
perpendicular distance from the origin is
A ( cdot frac{3}{4} )
B. ( frac{4}{3} )
( c cdot frac{7}{5} )
D.
12
657 If the lines ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and
( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect, then ( boldsymbol{k}= )
A. 0
B. 3
( c cdot frac{7}{2} )
D. –
12
658 If the vertices of a triangle are (-1,6,-4),(2,1,1) and (5,-1,0) then
the centroid of the triangle is
В. (2,2,-1)
( ^{mathbf{c}} cdotleft(3,3,-frac{3}{2}right) )
D. none of these
12
659 An equation of sphere with centre at
origin and radius ( r ) can be represented
as
A ( cdot x^{2}+y^{2}+z^{2}=r )
B . ( x^{2}+y^{2}+z^{2}=r^{2} )
c. ( x^{2}+y^{2}+z^{2}=2 r^{2} )
D. None of the above
12
660 Obtain the equation of the line passing through (1,1,2) and (2,1,2) in the vector form. 12
661 The reflection of the plane ( 2 x+3 y+ )
( 4 z-3=0 ) in the plane ( x-y+z- )
( mathbf{3}=mathbf{0} ) is the plane
A. ( 4 x-3 y+2 z-15=0 )
B. ( x-3 y+2 z-15=0 )
c. ( 4 x+3 y-2 z+15=0 )
D. none of these
12
662 A normal to the plane ( x=2 ) is…
A ( .(0,1,1) )
в. (2,0,2)
c. (1,0,0)
D. (0,1,0)
12
663 An ordered triplet corresponds to in three dimensional
space.
A. three points
B. a unique point
c. a point in each octant
D. infinite number of points
12
664 A point at a distance of ( sqrt{6} ) from the origin which lies on the straight line ( frac{x-1}{1}=frac{y-2}{2}=frac{z+1}{3} ) will be
( mathbf{A} cdot(1,-1,2) )
В. (1,2,-1)
C ( cdotleft(frac{5}{7}, frac{10}{7}, frac{-13}{7}right) )
D. ( left(frac{5}{7}, frac{2}{7}, frac{-6}{7}right) )
12
665 If ( (3, lambda, mu) ) is a point on the line then
( 2 x+y+z=0=x-2 y+z-1 ) then
A ( cdot lambda=frac{-8}{3}, mu=-frac{1}{3} )
B. ( lambda=frac{-1}{3}, mu=-frac{8}{3} )
c. ( lambda=frac{-4}{3} mu=frac{-14}{3} )
D. ( lambda=-5, mu=-1 )
12
666 Find the point of intersection of the plane ( bar{r} .(1,1,1)=2 ) and the line ( bar{r}= ) ( (4,5,3)+k(2,2,1), k in R ) 12
667 The equation of the plane which is equidistant from the two parallel planes ( 2 x-2 y+z+3=0 ) and ( 4 x-4 y+ )
( mathbf{2} z+mathbf{9}=mathbf{0} ) is :
A. ( 8 x-8 y+2 z+15=0 )
B. ( 8 x-8 y+4 z+15=0 )
c. ( 8 x-8 y+4 z+3=0 )
D. ( 8 x-8 y+4 z-3=0 )
E ( .8 x-8 y+4 z+4=0 )
12
668 The distance between the line ( r=2 hat{i}- ) ( 2 hat{j}+3 hat{k}+lambda(hat{i}-hat{j}+4 hat{k}) ) and the plane
( r cdot(hat{i}+5 hat{j}+hat{k})=5 ) is
A ( cdot frac{10}{9} )
в. ( frac{10}{3 sqrt{3}} )
c. ( frac{10}{3} )
D. None of these
12
669 If the distance between a point ( P ) and
the point (1,1,1) on the line ( frac{x-1}{3}= )
( frac{y-1}{4}=frac{z-1}{12} ) is ( 13, ) then the coordinates
of ( P ) are
A. (3,4,12)
В. ( left(frac{3}{13}, frac{4}{13}, frac{12}{13}right) )
c. (4,5,12)
D. (40, 53, 157)
12
670 The equation of the plane through the
intersection of ( p_{1} & p_{2} ) containing the
point (1,1,2) is
A. ( 5 x-6 y+4 z=4 )
в. ( 5 x+6 y-4 z=3 )
c. ( 3 x-2 y+4 z=9 )
D. Nonoe of these
12
671 If a plane passes through a fixed point
(2,3,4) and meets the axes of reference
in ( A, B ) and ( C, ) the point of intersection
of the planes through ( A, B, C ) parallel to the coordinate planes can be
This question has multiple correct options
A ( cdot(6,9,12) )
B ( cdot(4,12,16) )
( mathbf{c} cdot(1,1,-1) )
D. (2,3,-4)
12
672 ( boldsymbol{L}_{1}: frac{boldsymbol{x}-mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{4}} )
( L_{2}: frac{x-2}{3}=frac{y-4}{2}=frac{z-5}{5} ) be two
given lines, point P lies on ( L_{1} ) and Q lies
on ( L_{2} ) then distance between ( P ) and ( Q )
can be
This question has multiple correct options
A ( cdot frac{1}{3} )
B.
c. 15
D. 30
12
673 Let ( X ) and ( Y ) be two related variables.
The two regression lines are given by ( boldsymbol{x}-boldsymbol{y}+mathbf{1}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}-boldsymbol{y}+mathbf{4}=mathbf{0} . ) The
two regression lines pass through the point:
A ( cdot(-4,-3) )
в. (-6,-5)
c. (3,-2)
D. (-3,-2)
12
674 The values of a for which
( (8,-7, a),(5,2,4) ) and (6,-1,2) are
collinear, is given by?
A .2
B. -2
c. -1
D.
12
675 If ( z_{1} ) and ( z_{2} ) are ( z ) co-ordinates of the
points of trisection of the segment joining the points ( boldsymbol{A}(mathbf{2}, mathbf{1}, mathbf{4}), boldsymbol{B}(-mathbf{1}, mathbf{3}, mathbf{6}) )
then ( z_{1}+z_{2}= )
A . 1
B. 4
( c .5 )
D. 10
12
676 If ( boldsymbol{P}=(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{Q}=(mathbf{3}, mathbf{6}, mathbf{9}) ) and ( boldsymbol{R} ) is a
point of trisection of ( boldsymbol{P Q}, ) then ( boldsymbol{R}_{boldsymbol{y}}= )
( A cdot frac{4}{3} )
B . 2
( c .3 )
D.
12
677 Find the image of (1,5,1) in the plane
( boldsymbol{x}-mathbf{2} boldsymbol{y}+boldsymbol{z}+mathbf{5}=mathbf{0} )
12
678 If the centroid of tetrahedron ( O A B C )
where ( A, B, C ) are given by ( (a, 2,3),(1, b, 2) ) and ( (2,1, c) )
respectively is ( (1,2,-2), ) then distance of ( boldsymbol{P}(boldsymbol{a}, boldsymbol{b}, boldsymbol{c}) ) from origin is
( mathbf{A} cdot sqrt{195} )
в. ( sqrt{14} )
c. ( sqrt{frac{107}{14}} )
D. ( sqrt{13} )
12
679 1.
From a point O inside a triangle ABC, perpendiculars OD,
OE, OF are drawn to the sides BC, CA, AB respectively.
Prove that the perpendiculars from A, B, C to the sides EF,
FD, DE are concurrent.
(1978)
hone
12
680 A parallelepiped is formed by planes
drawn through the point ( P(6,8,10) ) and
( Q(3,4,8) ) parallel to the coordinate
planes. Find the length of edges and edges and diagonals of the
parallelepiped.
12
681 The angle between the lines whose
direction cosines satisfy the equations ( l+m+n=0 ) and ( l^{2}+m^{2}+n^{2} ) is
A ( cdot frac{pi}{2} )
в.
c.
D. ( frac{pi}{6} )
12
682 The line passing through the points ( 10 hat{i}+3 hat{j}, 12 hat{i}+5 hat{j} ) also passes through
the point ( a hat{i}+11 hat{j}, ) then ( a= )
A . -8
B. 4
c. 18
D. 12
12
683 Direction ratio of line given by ( frac{x-1}{3}= ) ( frac{6-2 y}{10}=frac{1-z}{-7} ) are:
( A cdot )
в. ( )
c. ( )
D. ( )
12
684 Point ( D ) has coordinates as (3,4,5)
Referring to the given figure, find the
coordinates of point ( boldsymbol{B} )
A ( cdot(3,0,4) )
( mathbf{B} cdot(4,3,0) )
C. (4,0,3)
D. (3,4,0)
12
685 Which of the following are equations for
the plane passing through the points
( P(1,1,-1), Q(3,0,2) ) and ( R(-2,1,0) ? )
A ( cdot(2 hat{i}-3 hat{j}+3 hat{k}) cdot((x+2) hat{i}+(y-1) hat{j}+z hat{k})=0 )
B . ( x=3-t, y=-11 t, z=2-3 t )
( mathbf{c} cdot(x+2)+11(y-1)=3 x )
D. ( (2 hat{i}-hat{j}+3 hat{k}) times(-3 hat{i}+hat{k}) cdot((x+2) hat{i}+(y-1) hat{j}+z hat{k})= )
0
12
686 The direction ratios of a vector are
( 2,-3,4 . ) Find its direction cosines
12
687 If ( A=5 ) units, ( B=6 ) units and ( |vec{A} times vec{B}|= ) 15 units, then the angle between ( vec{A} ) and ( vec{B} )
is:
This question has multiple correct options
( A cdot 30^{circ} )
B. ( 60^{circ} )
c. ( 90^{circ} )
D. ( 150^{circ} )
12
688 Find the shortest distance between the
skew lines:
( l_{1}: frac{x-1}{2}=frac{y+1}{1}=frac{z-2}{4} )
( l_{2}: frac{x+2}{4}=frac{y-0}{-3}=frac{z+1}{1} )
12
689 Consider the plane ( (boldsymbol{x}, boldsymbol{y}, boldsymbol{z})= )
( (0,1,1)+lambda(1,-1,1)+mu(2,-1,0) . ) The
distance of this plane from the origin is:
A ( cdot frac{1}{3} )
B. ( frac{sqrt{3}}{2} )
( c cdot sqrt{frac{3}{2}} )
D. ( frac{2}{sqrt{3}} )
12
690 71. Distance between two parallel planes 2x + y + 2z = 8 and
4x + 2y + 4z +5=0 is
[JEE M 2013]
12
691 Vector equation of line ( frac{mathbf{3}-boldsymbol{x}}{mathbf{3}}= )
( frac{2 y-3}{5}=frac{z}{2} ) is ( _{–}——-k in R )
A ( cdot bar{r}=(3,5,2)+k(3,3,0) )
B ( cdot quad bar{r}=left(3, frac{3}{2}, 0right)+k(-6,5,4) )
c. ( bar{r}=(3,3,0)+k(3,5,2) )
D ( cdot_{bar{r}}=(-6,5,4)+kleft(3, frac{3}{2}, 0right) )
12
692 If ( (1,1, a) ) is the centroid of the triangle formed by the points ( (1,2,-3),(mathrm{b}, 0,1) ) and (-1,1,-4) then ( a-b= )
A . -5
B. -7
( c .5 )
( D )
12
693 Show that the lines ( frac{x-1}{2}=frac{y-2}{3}= ) ( frac{z-3}{4} ) and ( 4 x-3 y+1=0=5 x- )
( 3 z+2 ) are interesting lines. Also find
point of intersection.
12
694 88. If L, is the line of intersection of the planes
2x-2y+3z-2=0, x-y+z+1=0 and L, is the line of
intersection of the planes x+2y-z-3=0,
3x-y+2z-1=0, then the distance of the origin from the
plane, containing the lines L, and L,, is: [JEEM 2018]
(a) ZNZ
(6) 212
12
695 What is the direction cosine of angle which the vector ( sqrt{2 hat{i}}+hat{j}+ )
( hat{k} ) makes with ( y- ) axis ( ? )
12
696 2. If a vector P making angles a, b, and yrespectively with
the X, Y and Z axes respectively.
Then sin? a + sin2 B + sin2 y=
(a) (b) 1 (c) 2 (d) 3
12
697 Point ( D ) has coordinates as (3,4,5)
Find the coordinates of the point ( boldsymbol{F} )
( mathbf{A} cdot(0,4,0) )
B. (0,0,4)
( mathbf{C} cdot(0,0,5) )
( mathbf{D} cdot(0,5,0) )
12
698 If ( overline{O A}=3 bar{i}+bar{j}-bar{k},|overline{A B}|=2 sqrt{6} ) and ( A B )
has the direction ratios 1,-1,2 then
( |boldsymbol{O} boldsymbol{B}|= )
A . ( sqrt{35} )
B. ( sqrt{41} )
c. ( sqrt{26} )
D. ( sqrt{55} )
12
699 Find the equation of the plane through the points ( boldsymbol{A}(mathbf{2}, mathbf{2}-mathbf{1}), boldsymbol{B}(mathbf{3}, mathbf{4}, mathbf{2}) ) and
( boldsymbol{C}(boldsymbol{7}, boldsymbol{0}, boldsymbol{6}) )
A. ( 5 x+2 y-3 z=17 )
B. ( 5 x+2 y+3 z=17 )
c. ( 5 x+y-3 z=7 )
D. ( 5 x+y+3 z=7 )
12
700 Show that the three lines with direction
( operatorname{cosines} )
( frac{12}{13}, frac{-3}{13}, frac{-4}{13}: frac{4}{13}, frac{12}{13}, frac{3}{13} ; frac{-4}{13}, frac{12}{13} ) are mutually
perpendicular
12
701 If the projection of point ( boldsymbol{P}(overrightarrow{boldsymbol{p}}) ) on the
plane ( vec{r} cdot vec{n}=q ) is the point ( S(vec{s}), ) then?
A ( cdot vec{s}=frac{(q-vec{p} cdot vec{n}) vec{n}}{|vec{n}|^{2}} )
В ( cdot vec{s}=vec{p}+frac{(vec{p} cdot vec{n}-q)}{|vec{n}|^{2}} vec{n} )
C・ ( _{vec{s}}=vec{p}-frac{(vec{p} cdot vec{n}) vec{n}}{|vec{n}|^{2}} )
D ( vec{s}=vec{p}-frac{(q-vec{p} cdot vec{n}) vec{n}}{|vec{n}|^{2}} )
12
702 Two systems of rectangular axes have
the same origin. If a plane cuts them at
distance ( a, b, c ) and ( d, b^{prime}, c^{prime} ) from the
origin, then
A ( cdot frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}-frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}=0 )
B ( cdot frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}-frac{1}{a^{2}}-frac{1}{b^{2}}+frac{1}{c^{2}}=0 )
c. ( frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}-frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}=0 )
D ( frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}+frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}=0 )
12
703 Find the equation of the line joining the points (-1,3) and (4,-2) 12
704 If the position vectors of the points ( boldsymbol{A}, boldsymbol{B} ) and ( C ) be ( i+j, i-j ) and ( a i+b j+c k )
respective;y, then the points ( A, B ) and
( C ) are collinear if:
A ( . a=b=c=1 )
B. ( a=1, b ) and ( c ) are arbitrary scalars
C. ( a=b=c=0 )
0
12
705 A point ( R ) with ( x ) -coordinate 4 lies on the line segment joining the points
( P(2,-3,4) ) and ( Q(8,0,10) . ) Find the
coordinates of the point ( boldsymbol{R} )
12
706 Fill in the blanks:
(i) The ( x ) -axis and ( y ) -axis taken together
determine a plane known as
(ii) The coordinates of points in the ( boldsymbol{X} boldsymbol{Y} )
-plane are of the form
(iii) Coordinate planes divide the space
into octants
12
707 The Cartesian equation of a line is
( frac{boldsymbol{x}-mathbf{5}}{mathbf{3}}=frac{boldsymbol{y}+mathbf{4}}{mathbf{7}}=frac{boldsymbol{z}-mathbf{6}}{mathbf{2}} . ) Write its vector
form.
12
708 The direction cosines of the lines
bisecting the internal angle ( theta ) between the lines whose direction cosines are
( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} ) are
( mathbf{A} cdot )
B. ( )
c. ( )
D. none of these
12
709 Find the distance of the point (-6,0,0)
from the plane ( 2 x-3 y+6 z=2 )
12
710 Let ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{2}), boldsymbol{B}=(mathbf{2}, mathbf{3}, mathbf{6}) ) and ( boldsymbol{C}= )
( (3,4,12) . ) The direction cosines of a line
equally inclined with ( O A, O B ) and ( O C )
where ( boldsymbol{O} ) is the origin, are
A ( cdot frac{1}{sqrt{2}}, frac{-1}{sqrt{2}}, 0 )
B. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, 0 )
c. ( frac{1}{sqrt{3}}, frac{-1}{sqrt{3}}, frac{1}{sqrt{3}} )
D. ( frac{1}{sqrt{3}}, frac{-1}{sqrt{3}}, frac{-1}{sqrt{3}} )
12
711 Find the coordinates of the point on the ( x ) -axis that is equidistant from
( boldsymbol{P}(mathbf{4}, mathbf{3}, mathbf{1}) ) and ( boldsymbol{Q}(-mathbf{2},-mathbf{6},-mathbf{2}) )
A ( cdotleft(frac{3}{2}, 0,0right) )
B ( cdotleft(-frac{3}{2}, 0,0right) )
c. ( left(0,-frac{3}{2}, 0right) )
D. ( left(0, frac{3}{2}, 0right) )
12
712 In the triangle with vertices ( boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{5},-mathbf{6}, mathbf{2}) ) and ( boldsymbol{C}(mathbf{1}, mathbf{3},-mathbf{1}) )
find the altitude ( n=|B D| )
( A cdot 5 )
B. 10
c. ( 5 sqrt{2} )
D. ( frac{10}{sqrt{2}} )
12
713 If the origin is the centroid of the
triangle ( P Q R ) with vertices ( boldsymbol{P}(mathbf{2} boldsymbol{a}, mathbf{2}, mathbf{6}), boldsymbol{Q}(-mathbf{4}, mathbf{3} boldsymbol{b},-mathbf{1 0}) ) and
( boldsymbol{R}(mathbf{8}, mathbf{1 4}, mathbf{2 c}), ) then find the values of ( boldsymbol{a}, boldsymbol{b} )
and ( c )
12
714 The points with the co-ordinates
( (2 a, 3 a),(3 b, 2 b) &(c, c) ) are collinear.
12
715 Using section formula show that the
points ( boldsymbol{A}(mathbf{2},-mathbf{3}, mathbf{4}), boldsymbol{B}(-mathbf{1}, mathbf{2}, mathbf{1}) ) and
( Cleft(0, frac{1}{3}, 2right) ) are collinear.
12
716 If the line joining ( boldsymbol{A}(mathbf{1}, mathbf{3}, mathbf{4}) ) and ( boldsymbol{B} ) is divided by the point (-2,3,5) in the ratio ( 1: 3, ) then ( B ) is
A. (-11,3,8)
(年) (-1,3,8)
в. (-11,3,-8)
D. (13,6,-13)
12
717 16. In R}, consider the planes P, :y=0 and P2: x+z=1. Let P
be the plane, different from P, and P2, which passes through
the intersection of P, and P,. If the distance of the point
(0, 1, 0) from P, is 1 and the distance of a point (a, b, y) from
Pz is 2, then which of the following relations is (are) true
(a) 2a+3+2y+2=0
(c) 2a+B-2y-10=0
(JEE Adv. 2015)
(b) 20-3+2y+4=0
(d) 2a-B+2y-8=0
12
718 Find the distance between the points
(3,4,-2),(1,0,7)
12
719 W
J
84. If the image of the point P(1, -2, 3) in the plane,
X
Y
Z
2x + 3y– 4z + 22=0 measured parallel to line, = = is
Q, then PQ is equal to :
JJEE M 2017]
(a) 615
(b) 315
(c) 2742
(d) 142
12
720 The points ( boldsymbol{A}(mathbf{1}, mathbf{2},-mathbf{1}), boldsymbol{B}(mathbf{2}, mathbf{5},-mathbf{2}), boldsymbol{C}(mathbf{4}, mathbf{4},-mathbf{3}) )
and ( D(3,1,-2) ) are
A. collinear
B. vertices of a rectangle
c. vertices of a square
D. vertices of a rhombus
12
721 Find the directions cosines of ( x, y ) and ( z )
axis.
12
722 The perpendicular distance from the
point (3,1,1) on the plane passing through the point (1,2,3) and containing the line, ( vec{r}=hat{i}+hat{j}+ ) ( lambda(2 hat{i}+hat{j}+4 hat{k}), ) is:
( ^{A} cdot frac{1}{sqrt{11}} )
в. ( frac{4}{sqrt{41}} )
( c .0 )
D. ( frac{3}{sqrt{11}} )
12
723 If ( (0, b, 0) ) is the centroid of the triangle
formed by the points (4,2,-3) ( (a,-5,1) ) and ( (2,-6,2) . ) If ( a, b ) are the
roots of the quadratic equation ( x^{2}+ )
( boldsymbol{p} boldsymbol{x}+boldsymbol{q}=mathbf{0}, ) then ( boldsymbol{p}, boldsymbol{q} ) are
A .9,18
в. -9,-18
c. 3,-18
D. -3,18
12
724 For two vectors ( overrightarrow{boldsymbol{A}} ) and ( overrightarrow{boldsymbol{B}}, overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}}=overrightarrow{boldsymbol{C}} )
and ( |vec{A}|+|vec{B}|=|vec{C}| . ) The angle between
two vectors is:
A .
в. ( pi / 3 )
c. ( pi / 2 )
D.
12
725 A plane meets the co-ordinate axes in A,B,C such that the centroid of the
triangle ( A B C ) is the point ( (p, q, r) . ) The
equation of the plane is
A ( cdot frac{x}{p}+frac{y}{q}+frac{z}{r}=0 )
В ( cdot frac{x}{p}+frac{y}{q}+frac{z}{r}=1 )
c. ( frac{x}{p}+frac{y}{q}+frac{z}{r}=2 )
D. none of these
12
726 What is the sum of the squares of direction cosines of the line joining the points (1,2,-3) and (-2,3,1)( ? )
( A cdot O )
B.
( c cdot 3 )
D. ( frac{2}{sqrt{26}} )
12
727 The image of: (-4,0,0) in the ( x z ) – plane is (4,0,0)
A. True
B. False
12
728 The point which is equidistant from the points ( (boldsymbol{a}, boldsymbol{0}, boldsymbol{0}),(boldsymbol{0}, boldsymbol{b}, boldsymbol{0}),(boldsymbol{0}, boldsymbol{0}, boldsymbol{c}) ) and
(0,0,0) is:
( mathbf{A} cdot(a, b, c) )
B . ( (sqrt{a}, sqrt{b}, sqrt{c}) )
c. ( (2 a, 2 b, 2 c) )
D. ( left(frac{a}{2}, frac{b}{2}, frac{c}{2}right) )
12
729 Find distance of a point (3,4) from the origin. 12
730 42.
The plane x +2y-z=4 cuts the sphere x + y< +22-x+,
– 2 = 0 in a circle of radius
(a) 3 (6) 1 (c) 2 (d) &
[2005]
12
731 34.
The angle between the lines 2x = 3y = – z and
6x=-y=-4z is
[2005]
(a) O
). (b) 90°
(c) 45°
(d) 30°
12
732 The number of lines which are equally
inclined to the axes is
( A cdot 2 )
B. 4
( c cdot 6 )
D. 8
12
733 VEU
43.
point P is the intersection of the straight line joining
points (2,3,5) and R(1,-1, 4) with the plane 5x – 4y
1. If S is the foot of the perpendicular drawn from the
point T(2, 1, 4) to QR, then the length of the line segment PS
(2012)
7
(b) √2
(c) 2
(2) 252
12
734 ( operatorname{lines} frac{x-1}{2}=frac{y-1}{2}=frac{z-2}{3} ) and
( frac{x-1}{2}=frac{y-2}{2}=frac{z-3}{-2} )
Check whether the lines are parallel, mutually perpendicular or intersecting in acute angle)
12
735 A line is perpendicular to the plane ( x+ )
( 2 y+2 z=0 ) and passes through
( (0,1,0) . ) The perpendicular distance of this line from the origin is
A ( frac{sqrt{5}}{3} )
B. ( frac{sqrt{7}}{3} )
( c cdot frac{2}{3} )
D. 3
12
736 The name of the figure formed by the points (0,0,0),(1,0,1) and (0,1,1) is
A . a straight line
B. an isosceles triangle
c. an equilateral triangle
D. a scalene triangle
12
737 The cartesian equation of the plane ( overline{boldsymbol{r}}=(mathbf{1}+boldsymbol{s}-boldsymbol{t}) hat{boldsymbol{i}}+(boldsymbol{2}-boldsymbol{s}) hat{boldsymbol{j}}+ )
( (3-2 s+2 t) hat{k} )
A. ( 2 x-y=5 )
B. ( 2 x+z=5 )
c. ( 2 x+y=5 )
D. ( 2 x-z=5 )
12
738 27.
A variable plane at a distance of the one unit from the
origin cuts the coordinates axes at A, B and C. If the
centroid D (x, y, z) of triangle ABC satisfies the relation
2
then the value k is
(2005)
y
(a) 3
(b) 1
(c)
(d) 9
12
739 21. Let ✓ = 2i +1 -k and W = i +3k . If Ū is a unit vector,
then the maximum value of the scalar triple product |ŪVW | is
(a) -1
(b) V10 + V6 (2002)
(c) 159
(d) 160
12
740 Two opposite vertices of a square are (2,-3,4) and ( (4,1,-2) . ) The length of the side of the square is
A ( cdot sqrt{58} )
B. ( 2 sqrt{7} )
c. ( sqrt{14} )
D. ( sqrt{7} )
12
741 If the angle between the planes ( boldsymbol{r} cdot(boldsymbol{m} hat{boldsymbol{i}}-hat{boldsymbol{j}}+boldsymbol{2} hat{boldsymbol{k}})+boldsymbol{3}=boldsymbol{0} ) and ( overline{boldsymbol{r}} cdot(boldsymbol{2} hat{boldsymbol{i}}- )
( m hat{j}-hat{k})-5=0 ) is ( frac{pi}{3} ) then ( m= )
( A cdot 2 )
B. ±3
( c cdot 3 )
D. – –
12
742 Find the ratio in which (the plane) ( 2 x+ ) ( 3 y+5 z=1 ) divides the line joining the
points (1,0,-3) and (1,-5,7)
A .1: 2
B . 2: 3
( c .3: 1 )
D. None of these
12
743 Let ( O ) be the origin and ( P ) be the point at a distance 3 units from origin. If d.x.s’ of OP are ( 1,-2,-2, ) then coordinates of ( P ) is
given by
A. 1,-2,-2
в. 3,-6,-6
c. ( frac{1}{3},-frac{2}{3},-frac{2}{3} )
D. ( frac{1}{9},-frac{2}{9},-frac{2}{9} )
12
744 Write the direction cosines of ( x ) -axis 12
745 The product of the d.cs of the line which makes equal angles with ( o x, o y, o z ) is
( mathbf{A} cdot mathbf{1} )
B. ( sqrt{3} )
c. ( frac{1}{3 sqrt{3}} )
D. ( frac{1}{sqrt{3}} )
12
746 If a line makes ( theta_{1}, theta_{2}, theta_{3} ) angles with the
co-ordinates axes, then prove that
( cos 2 theta_{1}+cos 2 theta_{2}+cos 2 theta_{3}+1=0 )
12
747 Verify the following:
(0,7,-10),(1,6,-6) and (4,9,-6) are
the vertices of an isosceles triangle.
12
748 If ( vec{A} times vec{B}=vec{B} times vec{A}, ) then the angle between ( vec{A} ) and ( vec{B} ) is
A . ( pi )
в.
( c cdot frac{pi}{2} )
D.
12
749 Show that the lines whose d.c.s are
given by ( l+m+n=0,2 m n+3 l n- )
( 5 l m=0 ) are perpendicular to each
other.
12
750 Show that the following set of point are collinear?
(2,3,-4),(-1,0,5),(3,4,-7)
12
751 Equation of a plane making X-intercept
4, Y-intercept ( (-6), mathrm{Z} ) -intercept 3 is
A. ( 3 x-4 y+6 z=12 )
B. ( 3 x-2 y+4 z=12 )
c. ( 4 x-6 y+3 z=1 )
D. ( 4 x-3 y+2 z=12 )
12
752 Find the distance between (12,3,4) and (4,5,2)
A ( cdot sqrt{72} )
B. ( sqrt{62} )
( c cdot sqrt{64} )
D. None of these
12
753 The general equation of plane which is
parallel to x-axis is
( mathbf{A} cdot a x+b y+c z+d=0, a neq 0, b neq 0, c neq 0 )
B. ( b y+a x+d=0, a neq 0, b neq 0 )
c. ( a x+c z+d=0, a neq 0 . c neq 0 )
D. ( b y+c z+d=0, b neq 0, c neq 0 )
12
754 If from the point ( boldsymbol{P}(boldsymbol{f}, boldsymbol{g}, boldsymbol{h}) )
perpendiculars ( P L, P M ) be drawn to ( y z )
and ( z x ) planes, then the equation to the
plane ( boldsymbol{O} boldsymbol{L} boldsymbol{M} ) is
A ( cdot frac{x}{f}+frac{y}{g}-frac{z}{h}=0 )
В ( cdot frac{x}{f}+frac{y}{g}+frac{z}{h}=0 )
c. ( frac{x}{f}-frac{y}{g}+frac{z}{h}=0 )
D. ( -frac{x}{f}+frac{y}{g}+frac{z}{h}=0 )
12
755 The position vectors of point ( A ) and ( B ) ( operatorname{are} hat{boldsymbol{i}}-hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}} ) and ( boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}} )
respectively. The equation of a plane is
( r cdot(5 hat{i}+2 hat{j}-7 hat{k})+9=0 . ) The point ( A )
and ( B )
A. lie on the plane
B. are on the same side of the plane
c. are on the opposite side of the plane
D. None of these
12
756 Find the distance between the following
pairs of points
(-5,7) and (-1,3)
12
757 The equation of the plane which is parallel to ( x y ) plane and cuts intercept of length 3 from the z-axis
( mathbf{A} cdot x=3 )
B. ( y=3 )
( mathbf{c} cdot z=3 )
D. ( x+y+z=3 )
12
758 The coordinate of any point, which lies
in ( boldsymbol{x} boldsymbol{y} ) plane, is
( mathbf{A} cdot(x, 0, y) )
в. ( (x, x, 0) )
c. ( (x, 0, x) )
D. ( (y, 0, x) )
12
759 What is the angle between the lines
( frac{boldsymbol{x}-mathbf{2}}{mathbf{1}}=frac{boldsymbol{y}+mathbf{1}}{-mathbf{2}}=frac{boldsymbol{z}+mathbf{2}}{mathbf{1}} ) and ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}= )
( frac{2 y+3}{3}=frac{z+5}{2}=? )
( ^{A} cdot frac{pi}{2} )
в.
c. ( frac{pi}{6} )
D. None of the above
12
760 The distance between the origin and the centroid of the tetrahedron whose vertices are ( (mathbf{0}, mathbf{0}, mathbf{0}) )
( (a, 0,0),(0, b, 0),(0,0, c) ) is?
A ( cdot sqrt{a^{2}+b^{2}+c^{2}} )
B. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{2} )
c. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{4} )
D. ( 4 sqrt{a^{2}+b^{2}+c^{2}} )
12
761 A parallelepiped is formed by planes
drawn through the points ( boldsymbol{P}(boldsymbol{6}, boldsymbol{8}, boldsymbol{1} boldsymbol{0}) )
and ( Q(3,4,8) ) parallel to the coordinate
planes. Find the length of edges and diagonals of the parallelepiped.
12
762 Find the equation of the plane passing through the points ( A=(2.3,-1), B=(4,5, )
2), ( C=(3,6,5) )
12
763 The cartesian equation of plane ( bar{r} cdot(2,-3,4)=5 ) is
A. ( 3 y-2 x-4 z+5=0 )
B. ( 2 x-3 y+4 z=0 )
c. ( 2 x-3 y+4 z+5=0 )
D. ( frac{x-1}{2}=frac{y-1}{-3}=frac{z-1}{4} )
12
764 A plane which passes through the point (3,2,0) and the line ( frac{x-3}{1}=frac{y-7}{5}= )
( frac{z-4}{4} ) is?
A. ( x-y+z=1 )
в. ( x+y+z=5 )
c. ( x+2 y-z=1 )
D. ( 2 x-y+z=5 )
12
765 55. If the straight lines *;?v=2=27
and
k
2
=intersect at a point, then the integer k
2
=
3 k
is equal to
(a) 5
[2008]
(b) 5
(c) 2
(d) -2
12
766 The line ( frac{x-3}{2}=frac{y-4}{5}=frac{z-6}{7} )
A . lies in ( 3 x+2 y+4 z-6=0 )
B. is parallel to ( 2 x-5 y+3 z=0 )
( mathbf{c} cdot ) is ( perp ) to ( 2 x-5 y+3 z=0 )
D. passing through (1,2,3)
12
767 Direction cosines of ray from
( boldsymbol{P}(mathbf{1},-mathbf{2}, mathbf{4}) ) to ( boldsymbol{Q}(-mathbf{1}, mathbf{1},-mathbf{2}) ) are
в. 2,-3,6
( mathbf{c} cdot 2,3,6 )
D. ( frac{-2}{7}, frac{3}{7}, frac{-6}{7} )
12
768 If the line, ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and
( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect, then find
the value of ( k )
12
769 Point of intersection of the point
(2,4,5)(3,6,-4)
12
770 Find the position vector of a point ( boldsymbol{R} )
which divides the line joining two
points ( P ) and ( Q ) whose position vectors ( operatorname{are}(2 vec{a}+vec{b}) ) and ( (vec{a}-3 vec{b}) ) externally in
the ratio ( 1: 2 . ) Also, show that ( P ) is the
mid point of the line segment ( boldsymbol{R} boldsymbol{Q} )
12
771 13. Two lines L, : x=5, 3a = – and Ly :x=a, -2-a
are coplanar. Then a can take value(s) (JEE Adv. 2013)
(a) 1 (b) 2 (c) 3 (d) 4
12
772 The distance between the points ( (cos theta, sin theta) ) and ( (sin theta-cos theta) ) is
A. ( sqrt{3} )
B. ( sqrt{2} )
( c cdot 2 )
D.
12
773 Find the coordinates of those points on the line ( frac{boldsymbol{x}+mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}+mathbf{2}}{mathbf{3}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{6}} ) which
is at a distance of 3 units from the point
(1,-2,3)
12
774 Find ( m ) if the point on the ( x ) -axis which
is equidistant from (7,6) and (3,4) is ( left(frac{m}{2}, 0right) )
12
775 93. A plane passing through the points (0,–1, 0) and (0,0,1)
and making an angle with the plane y-z+5 = 0, also
passes through the point: [JEEM 2019-9 April (M)
(a) (-,1,-4) (b) (12,-1,4)
(C) (12,-1,-4) (d) (V2,1,4)
12
776 The distance between the line ( vec{r}=2 vec{i}- ) ( mathbf{2} overrightarrow{boldsymbol{j}}+boldsymbol{3} overrightarrow{boldsymbol{k}}+boldsymbol{lambda}(overrightarrow{boldsymbol{i}}-overrightarrow{boldsymbol{j}}+boldsymbol{4} overrightarrow{boldsymbol{k}}) ) and the plane
( vec{r} cdot(vec{i}+5 vec{j}+vec{k})=5 ) is
A ( cdot frac{10}{3 sqrt{3}} )
в. ( frac{10}{9} )
c. ( frac{10}{3} )
D.
12
777 If ( vec{a}, vec{b} ) and ( vec{c} ) are mutually perpendicular vectors of equal magnitudes, If the angles which the vector ( 2 vec{a}+vec{b}+2 vec{c} ) makes with the vectors ( vec{a} ) is ( cos ^{-1} sqrt{frac{2}{m}} ) Find ( boldsymbol{m} ) 12
778 Find the equation of following planes: 12
779 The plane ( a x+b y+c z+d=0 ) divides
the line joining the points ( left(x_{1}, y_{1}, z_{1}right) )
and ( left(x_{2}, y_{2}, z_{2}right) ) in the ratio
A ( cdot frac{-left(a x_{1}+b y_{1}+c z_{1}+dright)}{left(a x_{2}+b y_{2}+c z_{2}+dright)} )
B. ( frac{left(a x_{1}+b y_{1}+c z_{1}+dright)}{left(a x_{2}+b y_{2}+c z_{2}+dright)} )
c. ( frac{a x_{1} x_{2}+b y_{1} y_{2}+c z_{1} z_{2}}{d^{2}} )
D. None of these
12
780 The vector equation ( boldsymbol{r}=boldsymbol{i}-boldsymbol{2} boldsymbol{j}-boldsymbol{k}+ )
( t(6 j-k) ) represents a straight line
passing through the points:
A. (0,6,-1) and (1,-2,-1)
в. (0,6,-1) and (-1,-4,-2)
c. (1,-2,-1) and (1,4,-2)
D. (1,-2,-1) and (0,-6,1)
12
781 If the distance of a point ( (a, a, a) ) from the origin is ( sqrt{108}, ) then the value of ( a ) is
This question has multiple correct options
( A cdot 9 )
B. 6
( c .-9 )
D. – 6
12
782 Equation of a plane 12
783 In which ratio the plane ( Y Z ) divides the
lines joining the points (2,1,2) and (-6,3,4)
12
784 A straight line passes through (1,-2,3) and perpendicular to the plane ( 2 x+ ) ( 3 y-z=7 . ) Find the direction ratios of
normal to plane
( A cdot )
В. ( )
c. ( )
D. None of the above
12
785 ( X O Z ) plane divides the join of (2,3,1) and (6,7,1) in the ratio
A . 3: 7
B. 2: 7
c. -3: 7
D. -2: 7
12
786 85. The distance of the point (1,3, -7) from the plane passing
through the point (1, -1, -1), having normal perpendicular
to both the lines
Z-4
X-2
y+1
Z+7
,
X-1
1
y +2
– 2
and 1-2
3
2
-1 , is:
2
JEE M 2017]
S
12
787 If ( P(x, y, z) ) is point in the space at a
distance ( r ) from the origin ( O ), then
direction cosines of the line ( O P ) are
12
788 If the extremities of a diagonal of a
square are (1,-2,3) and (2,-3,5)
then area of the square is
( A cdot 6 )
B. 3
( c cdot frac{3}{2} )
D. ( sqrt{3} )
12
789 The equation of the plane passing through the intersection of the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=boldsymbol{6} ) and ( boldsymbol{2} boldsymbol{x}+boldsymbol{3} boldsymbol{y}+boldsymbol{4} boldsymbol{z}+boldsymbol{5}= )
( 0, ) and the point (1,1,1) is
A. ( 20 x+23 y+26 z-69=0 )
B. ( 20 x+23 y+26 z+69=0 )
c. ( 23 x+20 y+26 z-69=0 )
D. None of these
12
790 What are the direction ratios of the line
if it passes through the intersection of the planes ( x=3 z+4 ) and ( y=2 z-3 ? )
A ( .(1,2,3) )
в. (2,1,3)
c. (3,2,1)
D. (1,3,2)
12
791 The ratio in which the joint of
(2,1,5),(3,4,3) is divided by the plane ( 2 x+2 y-2 z-1=0 )
( mathbf{A} cdot 5: 12 )
B. 12: 5
( c .5: 7 )
D. 7: 5
12
792 Three vertices of a tetrahedron are
(0,0,0),(6,-5,-1) and ( (-4,1,3) . )
the centroid of the tetrahedron be
(1,-2,5) then the fourth vertex is
A. (2,-4,18)
в. (1,-4,18)
( ^{c} cdotleft(frac{3}{2}, frac{-3}{2}, frac{7}{4}right) )
D. none of these
12
793 If ( boldsymbol{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 ( boldsymbol{R} ) is
( A cdot 3 )
B . 2
( c . )
D. –
12
794 If ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} in[mathbf{0}, boldsymbol{2} boldsymbol{pi}], ) then the sum of all
possible values of ( alpha, beta, gamma ) if ( sin alpha= ) ( -frac{1}{sqrt{2}}, cos beta=-frac{1}{2}, tan gamma=-sqrt{3}, ) is
A ( cdot frac{22 pi}{3} )
B. ( frac{21 pi}{3} )
c. ( frac{20 pi}{3} )
D. ( 8 pi )
12
795 A plane intersects the co ordinate axes
at ( A, B, C . ) If ( O=(0,0,0) ) and (1,1,1) is
the centroid of the tetrahedron ( O A B C )
then the sum of the reciprocals of the
intercepts of the plane
( mathbf{A} cdot 12 )
B. ( frac{4}{3} )
c. 1
( D cdot frac{3}{4} )
12
796 The plane ( a x+b y+c z+(-3)=0 )
meet the co-ordinate axes in ( A, B, C )
The centroid of the triangle is
B ( cdotleft(frac{3}{a} cdot frac{3}{b}, frac{3}{c}right) )
c. ( left(frac{a}{3} cdot frac{b}{3}, frac{c}{3}right) )
D. ( left(frac{1}{a} cdot frac{1}{b}, frac{1}{c}right) )
12
797 The area of triangle whose vertices are (1,2,3),(2,5,-1) and (-1,1,2) is
A . 150 sq.units
B. 145 sq.units
c. ( sqrt{155} / 2 ) sq.units
D. ( 155 / 2 ) sq.units
12
798 Find the shortest distance between
lines: ( frac{x-1}{1}=frac{y-2}{3}=frac{z-3}{2} ) and
( frac{x-4}{2}=frac{y-5}{3}=frac{z-6}{1} )
A . ( sqrt{6} )
B. ( sqrt{5} )
( c cdot sqrt{3} )
D. 6
12
799 Let the equation of the plane through the points
(-2,-2,2),(1,1,1),(1,-1,2) be ( k x+ )
( boldsymbol{m} boldsymbol{y}+boldsymbol{n} boldsymbol{z}+boldsymbol{p} . ) Find ( boldsymbol{k}+boldsymbol{m}+boldsymbol{n}+boldsymbol{p} )
( A cdot 7 )
B.
( c cdot 4 )
D. 6
12
800 Find the distance between the following pairs of points (-2,-3) and (3,2) 12
801 If ( C_{1}: x^{2}+y^{2}-20 x+64=0 ) and
( C_{2}: x^{2}+y^{2}+30 x+144=0 . ) Then the
length of the shortest line segment ( boldsymbol{P Q} )
which touches ( C_{1} ) at ( P ) and to ( C_{2} ) at ( Q )
is
A . 10
B. 15
( c cdot 22 )
D. 27
12
802 If ( x y- ) plane and ( y z- ) plane divides the line segment joining ( A(2,4,5) ) and ( B(3,5,-4) ) in the ratio ( a: b ) and ( p: q ) respectively then value of ( left(frac{a}{b}+frac{p}{q}right) ) may be
A ( cdot frac{23}{12} )
в. ( frac{7}{5} )
( c cdot frac{7}{12} )
D. ( frac{21}{10} )
12
803 The image of the point (2,-1,1) by the
plane ( 3 x+4 y-5 z=0 ) is
A ( cdot(-2,1,-1) )
в. ( left(frac{2}{3}, frac{-1}{4}, frac{-1}{5}right) )
( ^{mathbf{C}} cdotleft(frac{59}{25}, frac{-13}{25}, frac{2}{5}right) )
D. none of these
12
804 Distance between ( vec{r}=hat{i}+lambda(hat{j}+hat{k}) )
and ( vec{r}=hat{j}+mu(hat{j}+hat{k}) ) is equal to
12
805 ( operatorname{Let} boldsymbol{A}(mathbf{2} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+mathbf{5} hat{boldsymbol{k}}) boldsymbol{B}(-hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+2 hat{boldsymbol{k}}) )
and ( C(lambda hat{i}+5 hat{j}+mu hat{k}) ) are vertices of ( a ) triangle and its median through ( A ) is equally inclined to the positive directions of the axes. The value of ( lambda+ )
( mu ) is equal to
A . -7
B . 2
c. 7
D. 17
12
806 Direction cosines of the line ( frac{x+2}{2}= )
( frac{2 y-5}{3}, z=-1 ) are
A ( cdot frac{4}{5}, frac{3}{5}, 0 )
B. ( frac{3}{5^{prime}}, frac{4}{5^{prime}} frac{1}{5} )
c. ( quad-frac{3}{5}, frac{4}{5}, 0 )
D. ( frac{4}{5},-frac{2}{5}, frac{1}{5} )
12
807 The plane through the intersection of the planes ( x+y+z=1 ) and ( 2 x+ )
( 3 y-z+4=0 ) and parallel to ( y ) -axis
also passes through the point.
A. (-3,0,1)
В. (3,3,-1)
c. (3,2,1)
D. (-3,1,1)
12
808 Find the equation of the plane passing through the points
( (mathbf{1}, mathbf{1}, mathbf{1}),(mathbf{3},-mathbf{1}, mathbf{2}),(-mathbf{3}, mathbf{5},-mathbf{4}) )
12
809 Find the equation of the plane bisecting the line segment joining the points
(-3,-2,1) and (1,6,-5)
perpendicularly.
12
810 is
46. Let P be the image of the point (3,1,7) with respect to the
plane x-y+z=3. Then the equation of the plane passing
x y z .
through P and containing the straight line =
1 z 1
(JEE Adv. 2016)
(a) x+y-3z=0
(b) 3x+z=0
(c) X-4y+z=0
(d) 2x-y=0
hub the point (1 1 1
12
811 The plane ( a x+b y+c z+(-3)=0 )
meet the co-ordinate axes in ( A, B, C . ) Then
centroid
of the triangle is
A. ( (3 a, 3 b, 3 c) )
в. ( left(frac{3}{a} frac{3}{b}, frac{3}{c}right) )
( ^{mathbf{C}} cdotleft(frac{a}{3}, frac{b}{3}, frac{c}{3}right) )
D. ( left(frac{1}{a}, frac{1}{b}, frac{1}{c}right) )
12
812 If the points ( boldsymbol{A}(mathbf{3},-mathbf{2}, mathbf{4}), boldsymbol{B}(mathbf{1}, mathbf{1}, mathbf{1}) ) and
( C(-1,4,-2) ) are collinear, then the ratio in which ( C ) divides ( A B ) is
A .1: 2
B . -2: 1
c. -1: 2
D. 4: 0
12
813 Find the direction cosines (d.cs) of
directed line ( O P ) if coordinates of ( P ) is
( (2,3,7), O ) being the origin.
12
814 Find the co-ordinates of a point lying on the line ( frac{boldsymbol{x}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{y}+mathbf{3}}{mathbf{4}}=frac{boldsymbol{z}-mathbf{1}}{mathbf{7}} ) which
is at a distance 10 units from (2,-3,1)
begin{tabular}{l}
A. (32,37,71) \
hline
end{tabular}
в. (-28,-43,-69)
c. (-32,-37,-71)
D. None of these
12
815 The direction angles of the line ( x= )
( mathbf{4} z+mathbf{3}, boldsymbol{y}=mathbf{2}-mathbf{3} z ) are ( boldsymbol{alpha}, boldsymbol{beta} ) and ( gamma, ) then
( cos alpha+cos beta+cos gamma= )
A ( cdot frac{2}{sqrt{26}} )
B. ( frac{8}{sqrt{26}} )
c. 1
D. 2
12
816 The equation of a plane passing through the point ( A(2,-3,7) ) and making equal intercepts on the axes,
is?
A. ( x+y+z=3 )
в. ( x+y+z=6 )
c. ( x+y+z=9 )
D. ( x+y+z=4 )
12
817 If the dr’s the line are ( (1+lambda, 1-lambda, 2) )
and it makes an angle ( 60^{circ} ) with the ( Y ) –
axis then ( lambda ) is
A ( .1 pm sqrt{3} )
B. ( 4 pm sqrt{5} )
c. ( 2 pm 2 sqrt{3} )
D. ( 2 pm sqrt{5} )
12
818 If direction ratios of the normal of the
plane which contains the lines ( frac{x-2}{3}= ) ( frac{boldsymbol{y}-boldsymbol{4}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{1}}{boldsymbol{1}} & frac{boldsymbol{x}-boldsymbol{6}}{boldsymbol{3}}=frac{boldsymbol{y}+boldsymbol{2}}{boldsymbol{2}}= )
( frac{z-2}{1} ) are ( (a, 1,-26), ) then ( a ) is equal
to
A . 5
B. 6
( c cdot 7 )
D.
12
819 What is the distance of the point
( (p, q, r) ) from the ( x- ) axis.
12
820 What is the angle between ( vec{A} ) and the resultant of ( (overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}}) ) and ( (overrightarrow{boldsymbol{A}}-overrightarrow{boldsymbol{B}}) )
( mathbf{A} cdot 0^{circ} )
B ( cdot tan ^{-1}left(frac{A}{B}right) )
( ^{mathbf{c}} cdot tan ^{-1}left(frac{B}{A}right) )
D. ( tan ^{-1}left(frac{A-B}{A+B}right) )
12
821 Show that angles between any two ( operatorname{diagonals} boldsymbol{theta}=cos ^{-1}left(frac{1}{3}right) ) 12
822 A triangle ( A B C ) is placed so that the
midpoints of its sides are on the ( boldsymbol{x}, boldsymbol{y} )
and ( z ) axes respectively. Lengths of the
intercepts made by the plane containing the triangle on these axes
( operatorname{are} ) respectively ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma}, ) then
the coordinates of the centroid of the
triangle ( A B C ) are
A ( cdotleft(-frac{alpha}{3}, frac{beta}{3}, frac{gamma}{3}right) )
В ( cdotleft(frac{alpha}{3},-frac{beta}{3}, frac{gamma}{3}right) )
( ^{mathbf{C}} cdotleft(frac{alpha}{3}, frac{beta}{3},-frac{gamma}{3}right) )
D ( cdotleft(frac{alpha}{3}, frac{beta}{3}, frac{gamma}{3}right) )
12
823 Using vectors, find the value of ( lambda ) such
that the points ( (boldsymbol{lambda},-mathbf{1 0}, mathbf{3}),(mathbf{1},-mathbf{1}, mathbf{3}) )
and (3,5,3) are collinear.
12
824 Find the equation of the plane through
the intersection of the planes ( 3 x- )
( 4 y+5 z=10 ) and ( 2 x+2 y-3 z=4 )
and parallel to the line ( boldsymbol{x}=mathbf{2} boldsymbol{y}=mathbf{3} boldsymbol{z} )
12
825 Find the direction cosines of the line PQ
joining the points ( P(2,3,4) ) and ( Q(2,1,1) )
12
826 A symmetrical form of the line of intersection of the planes ( boldsymbol{x}=boldsymbol{a} boldsymbol{y}+boldsymbol{b} )
and ( boldsymbol{z}=boldsymbol{c} boldsymbol{y}+boldsymbol{d} ) is :
A. ( frac{x-b}{a}=frac{y-1}{1}=frac{z-d}{c} )
( ^{text {В }} cdot frac{x-b-a}{a}=frac{y-1}{1}=frac{z-d-c}{c} )
c. ( frac{x-a}{b}=frac{y-0}{1}=frac{z-c}{d} )
D. ( frac{x-b-a}{b}=frac{y-1}{0}=frac{z-d-c}{d} )
12
827 Find the distance of the point ( boldsymbol{P}(boldsymbol{3}, boldsymbol{4}, boldsymbol{4},) )
from the point, where the line joining
the points ( boldsymbol{A}(boldsymbol{3},-boldsymbol{4},-boldsymbol{5}) ) and ( boldsymbol{B}(boldsymbol{2},-boldsymbol{3}, boldsymbol{1}) )
intersected the plane ( 2 x+y+z=7 )
12
828 The distance of the point (1,-5,9) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
along the line ( x=y=z ) is:
( A cdot 3 sqrt{10} )
в. ( 10 sqrt{3} )
c. ( frac{10}{sqrt{3}} )
D. ( frac{20}{3} )
12
829 Find the equation of the plane passing through the intresection of the planes ( x-2 y+z=1 ) and ( 2 x+y+z=8 )
and parallel to the line with direction ratio proportional to ( 1,2,1, ) find also the perpendicular distance of (1,1,1) from this plane.
12
830 A rectangular parallelopiped is formed by drawing planes through the points (-1,2,5) and (1,-1,-1) and paralle to the coordinate planes. the length of the diagonal of the parallelopiped is
( A cdot 2 )
B. 3
c. 6
D.
12
831 Derive the equation of the locus of a point twice as far from (-2,3,4) as from
(3,-1,-2)
12
832 Prove that 1,1,1 cannot be direction cosines of a straight line 12
833 The acute angle between two lines such
that the direction cosines ( I, m, n ) of each
of them satisfy the equations ( l+m+ )
( boldsymbol{n}=mathbf{0} ) and ( l^{2}+boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} ) is :
A . 30
B . 45
c. 60
D. 15
12
834 Vector components of the vector with
initial points (2,1) and terminal point (-5,7) are
( mathbf{A} cdot-6 hat{i}+7 widehat{j} )
B. ( -7 hat{i}+6 hat{j} )
c. ( -6 hat{i}-7 widehat{j} )
D. None
12
835 A mirror and a source of light are situated at the origin ( mathrm{O} ) and at a point on ( mathrm{O} X, ) respectively. A ray of light from the source strikes the mirror and is
reflected. If the direction ratios of the
normal to the plane are ( 1,-1,1, ) then find the DCs of the reflected ray.
A ( cdot frac{1}{3}, frac{2}{3}, frac{2}{3} )
в. ( -frac{1}{3}, frac{2}{3}, frac{2}{3} )
( c cdot-frac{1}{3},-frac{2}{3},-frac{2}{3} )
D. ( -frac{1}{3},-frac{2}{3}, frac{2}{3} )
12
836 The points (2,5) and (5,1) are the two opposite vertices of a rectangle. If the other two vertices are points on the straight line ( y=2 x+k, ) then the value
of k is
( A cdot 4 )
B. 3
( c cdot-4 )
( D cdot-3 )
( E )
12
837 Let ( vec{A}, vec{B} ) and ( vec{C} ) be unit vectors. Suppose that ( vec{A} cdot vec{B}=vec{A} cdot vec{C}=0 ) and that the
angle between ( vec{B} ) and ( vec{C} ) is ( frac{pi}{6} ) then ( overrightarrow{boldsymbol{A}}= )
A ( cdot pm 2(vec{B} times vec{C}) )
B ( cdot pm(vec{B} times vec{C}) )
c. ( pm 2(vec{B}+vec{C}) )
D. ( pm(vec{B}+vec{C}) )
12
838 If the lines through the points (4,1,2) and ( (5, k, 0) ) is parallel to the line through the points (2,1,1) and (3,3,1)
find ( k )
12
839 The coordinates of the foot of the
perpendicular from the point (1,-2,1) on the plane containing the lines, ( frac{x+1}{6}=frac{y-1}{7}=frac{z-3}{8} ) and ( frac{x-1}{3}= )
( frac{boldsymbol{y}-boldsymbol{2}}{mathbf{5}}=frac{boldsymbol{z}-boldsymbol{3}}{boldsymbol{7}}, ) is :
A ( cdot(2,-4,2) )
в. (-1,2,-1
c. (0,0,0)
D. (1,1,1)
12
840 If the points ( boldsymbol{A}(1,2,-1), B(2,6,2) ) and
( C(lambda,-2,-4) ) are collinear, then ( lambda ) is
( mathbf{A} cdot mathbf{0} )
B.
( c .-2 )
( D )
12
841 Find the centroid of a triangle, midpoints of whose sides are
(1,2,-3),(3,0,1) and (-1,1,-4)
12
842 If the points ( (a, 1),(1,2) ) and ( (0, b+1) )
are collinear, then show that ( frac{1}{a}+frac{1}{b}=1 )
12
843 The ratio of ( y z ) -plane divide the line
joining the points ( A(3,1,-5), B(1,4,-6) ) is
A . 3: 1
B. -1: 3
c. 1: 3
D ( .-3: 1 )
12
844 ( P(1,1,1) ) and ( Q(lambda, lambda, lambda) ) are two points in the space such that ( P Q=sqrt{27} ), then the value(s) of ( lambda ) can be
A . -4
в. -2,4
( c cdot 2 )
D. 4,3
12
845 The line ( boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}+mathbf{4}=mathbf{0}, boldsymbol{x}+boldsymbol{y}+ )
( z-8=0 ) intersects the plane ( x-y+ )
( mathbf{2} z+mathbf{1}=mathbf{0} ) at the point
A ( cdot(3,2,3) )
B . (5,2,1)
c. (2,5,1)
D . (3,4,1)
12
846 In which ratio does the ( Y Z ) plane divide
the line joining the points (-2,4,7) and (3,-5,8)
12
847 A sphere of constant radius ( 2 k ) passes
through the origin and meets the axes in ( A, B, C . ) The locus of the centroid of
the tetrahedron ( boldsymbol{O} boldsymbol{A} boldsymbol{B} boldsymbol{C} ) is
A ( cdot x^{2}+y^{2}+z^{2}=4 k^{2} )
B cdot ( 9left(x^{2}+y^{2}+z^{2}right)=4 k^{2} )
C . ( x^{2}+y^{2}+z^{2}=k^{2} )
D. None of these
12
848 The point lying on angle bisector of the
planes ( boldsymbol{x}+mathbf{2} boldsymbol{y}+mathbf{2} boldsymbol{z}-mathbf{6}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}- )
( boldsymbol{y}+boldsymbol{4}=mathbf{0} ) is
A. (2,4,0)
B. (-1,3,2)
D. (-2,4,0)
12
849 Find the values of ( a ) and ( b ) so that the
points ( (boldsymbol{a}, boldsymbol{b}, mathbf{3}),(mathbf{2}, mathbf{0},-mathbf{1}) ) and
(1,-1,-3) are collinear.
12
850 If ( |overrightarrow{boldsymbol{A}} times overrightarrow{boldsymbol{B}}|=overrightarrow{boldsymbol{A}} cdot overrightarrow{boldsymbol{B}}, ) then ( |overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}}| ) is:
( mathbf{A} cdot|vec{A}|+|vec{B}| )
B. ( sqrt{|vec{A}|^{2}+|vec{B}|^{2}} )
( ^{mathrm{c}} cdot sqrt{|vec{A}|^{2}+|vec{B}|^{2}+frac{|vec{A}||vec{B}|}{sqrt{2}}} )
D cdot ( sqrt{|vec{A}|^{2}+|vec{B}|^{2}+sqrt{2}|vec{A}||vec{B}|} )
12
851 61. Statement-1: The point A(3, 1, 6) is the mirror image of the
point B(1, 3, 4) in the plane x-y +z=5.
Statement-2: The plane x-y+z=5 bisects the line segment
joining A(3, 1,6) and B(1,3,4).
[2010]
(a) Statement -1 is true, Statement -2 is true; Statement-2
is not a correct explanation for Statement-1.
(6) Statement -1 is true, Statement -2 is false.
(c) Statement-1 is false, Statement -2 is true.
(d) Statement – 1 is true, Statement 2 is true; Statement-2
is a correct explanation for Statement -1.
12
852 Write the vector equation of the line equation ( frac{x-1}{2}=frac{y-2}{3}=frac{z-3}{4} ) 12
853 The name of the figure formed by the points (3,-5,1),(-1,0,8) and
(7,-10,-6) is
A. a triangle
B. a straight line
c. an isosceles triangle
D. an equilateral triangle
12
854 The lines ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and
( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect if ( boldsymbol{K} )
equals
( A cdot frac{3}{2} )
в. ( frac{9}{2} )
c. ( frac{-2}{9} )
D. ( frac{-3}{2} )
12
855 Find unit vector perpendicular to the plane passing through the points (1,2,3),(2,-1,1) and (1,2,-4) 12
856 If ( P ) is a point ( (x, y) ) on the line ( y=-3 x )
such that ( boldsymbol{P} ) and ( boldsymbol{Q}(boldsymbol{3}, boldsymbol{4}) ) are on opposite
side of the line ( 3 x-4 y=8, ) then:
A ( cdot x>frac{8}{5}, yfrac{8}{15}, y<-frac{8}{5} )
c. ( x=frac{8}{15}, y=frac{-8}{5} )
D. ( x=2, y=-2 )
12
857 ( P ) is a variable points which moves
such that ( 3 P A=2 P B . ) If ( A= )
(-2,2,3) and ( B=(13,-3,13) ) prove
that ( P ) satisfies the equation ( x^{2}+y^{2}+ )
( z^{2}+28 x-12 y+10 z-247=0 )
12
858 For what value of ( mathrm{m}, ) the points (3,5)
( (m, 6) ) and ( left(frac{1}{2}, frac{15}{2}right) ) are collinear?
12
859 Find the direction cosines of the line
( frac{boldsymbol{x}+mathbf{2}}{mathbf{2}}=frac{mathbf{2} boldsymbol{y}-mathbf{5}}{mathbf{3}} ; boldsymbol{z}=-mathbf{1} )
12
860 Find the co-ordinates of a point, which is at a distance of 21 units from the
point ( boldsymbol{A}=(mathbf{1},-mathbf{3}, mathbf{4}) ) in the direction of vectors ( 2 hat{i}-3 hat{j}-6 hat{k} )
12
861 72.
22. If the lines *-72 173
If the lines
Z
y-3
1
-4
-ka
and
-1
and

k
z-5
– are coplanar, then k can have
[JEE M 2013]
(a) any value
(c) exactly two values
(b) exactly one value
(d) exactly three values
12
862 If two vertices of a triangle ( A B C ) are ( A(-1,2,4) ) and ( B(2,-3,0), ) and the
centroid is (2,0,2) then the vertex ( C ) has the coordinates
A. (5,1,2)
в. ( left(1,-frac{1}{3}, frac{7}{3}right) )
( ^{mathbf{C}} cdotleft(3,-frac{2}{3}, frac{5}{3}right) )
D. none of these
12
863 Find the distance of the point (2,12,5) from the point of intersection of the line ( overrightarrow{boldsymbol{r}}=2 hat{hat{boldsymbol{i}}}-mathbf{4} hat{boldsymbol{j}}+mathbf{2} hat{boldsymbol{k}}+boldsymbol{lambda}(boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{4} hat{boldsymbol{j}}+boldsymbol{2} hat{boldsymbol{k}}) )
and the plane ( vec{r} cdot(hat{boldsymbol{i}}-boldsymbol{2} hat{boldsymbol{j}}+hat{boldsymbol{k}})=mathbf{0} )
12
864 Find the direction ratio of the line
( frac{x-1}{2}=3 y=frac{2 z+3}{4} )
12
865 The equation of line passing through (1,-2,3) and having ( operatorname{drs}(2,3,1) ) is
( ^{text {A }} cdot frac{x-1}{2}=frac{y+2}{3}=frac{z-3}{1} )
B. ( frac{x+1}{2}=frac{y+2}{3}=frac{z-3}{1} )
c. ( frac{x-1}{2}=frac{y-2}{3}=frac{z-3}{1} )
D. none of these
12
866 If a point ( P ) in the space such that ( overline{O P} )
is inclined to ( O X ) at 45 and ( O Z ) to 60
then ( overline{O P} ) inclined to ( O Y ) is
A ( .75^{circ} )
B . ( 75^{circ} ) or ( 105^{circ} )
C. ( 60^{circ} ) or ( 120^{circ} )
D. None of these
12
867 Find the root of the perpendicular from point (2,3,2) to the line ( frac{4-x}{2}=frac{y}{6}= ) ( frac{1-z}{3} ) also find perpendicular distance
from the point to the line.
12
868 11.
– and
25
x-1 y+1
x +1
Z.
Ifthe straight lines
4+1 Z
1 2 k
5 2 k
are coplanar, then the plane (s) containing these two lines is
(are)
(2012)
(a) y + 2z=-1
(b) y+z=-1
(c) y-z=-1 .
(d) y-2z=-1
12
869 The projection of a directed line segment on the co-ordinate axes are
( 12,4,3, ) then the direction cosines of
the line are
A ( cdot frac{-12}{13}, frac{-4}{13}, frac{-3}{13} )
В. ( frac{12}{13}, frac{4}{13}, frac{3}{13} )
c. ( frac{12}{13}, frac{-4}{13}, frac{3}{13} )
D. ( frac{12}{13}, frac{4}{13}, frac{-3}{13} )
12
870 If the point ( (x, y) ) is equidistant from the points ( (a+b, b-a) ) and ( (a- )
( b, a+b), ) then ( b x=a y )
A. True
B. False
12
871 If the points ( a(1,2,-1), B(2,6,2) ) and ( c(lambda,-2,-4) ) are collinear then ( lambda ) is
( mathbf{A} cdot mathbf{0} )
B . 2
( c .-2 )
D.
12
872 The image of the point (1,2,3) through
the plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{4} boldsymbol{z}=boldsymbol{0} ) is
A ( cdotleft(frac{-2}{3}, frac{1}{3}, frac{-11}{3}right) )
в. ( left(frac{3}{7},-frac{6}{7}, frac{19}{7}right) )
( ^{mathbf{c}} cdotleft(frac{2}{3}, frac{1}{3}, frac{11}{3}right) )
D. ( left(frac{-2}{3}, frac{-1}{3}, frac{11}{3}right) )
12
873 Show that the points ( (3,3),(h, 0) ) and
( (0, k) ) are collinear, if ( frac{1}{n}+frac{1}{k}=frac{1}{3} )
12
874 Angle between lines whose direction
( operatorname{cosine} operatorname{satisfy} l+m+n=0, l^{2}+ )
( boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} )
12
875 a b c a b c
23. Distance between two parallel planes
2x+y+2z=8 and 4x + 2y +4z +5=0 is
[2004]
(a)
2
min
12
876 ( operatorname{can} frac{2}{sqrt{3}}, frac{-2}{sqrt{3}}, frac{-1}{sqrt{3}} ) be the direction ratios
of any directed line? Justify your
answer
12
877 A vector equation of the line of
intersection of the planes ( boldsymbol{r}=boldsymbol{b}+ )
( boldsymbol{lambda}_{1}(boldsymbol{b}-boldsymbol{a})+boldsymbol{mu}_{1}(boldsymbol{a}+boldsymbol{c}) )
( boldsymbol{r}=boldsymbol{c}+boldsymbol{lambda}_{2}(boldsymbol{b}-boldsymbol{c})+boldsymbol{mu}_{1}(boldsymbol{a}+boldsymbol{b}) boldsymbol{a}, boldsymbol{b}, boldsymbol{c} )
being non-coplanar vectors is.
A ( cdot r=a+mu_{1}(b+c) )
B . ( r=b+mu_{1}(a+2 c) )
C . ( r=a+mu_{1}(b+2 c) )
D. ( r=b+mu_{1}(a+c) )
12
878 The length of the perpendicular from the origin to the plane passing through the point ( a ) and containing the line ( r=bar{b}+ )
( lambda bar{c} ) is
A ( cdot frac{[a b c]}{|a times b+b times c+c times a|} )
В. ( frac{[a b c]}{|a times b+b times c|} )
c. ( frac{[a b c]}{|b times c+c times a|} )
D. ( frac{[a b c]}{|a times b+c times a|} )
12
879 Let the equation of the plane which contains the line ( boldsymbol{x}=frac{boldsymbol{y}-boldsymbol{3}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{5}}{boldsymbol{3}} )
and which is perpendicular to the plane ( 2 x+7 y-3 z=1 . ) be ( k x-m y-z+ )
( boldsymbol{p}=mathbf{0} . ) Find ( boldsymbol{p}-boldsymbol{k}-boldsymbol{m} ? )
12
880 Find direction cosine line ( boldsymbol{x}=mathbf{3} boldsymbol{z}+ )
( mathbf{2}, boldsymbol{y}=mathbf{2}-mathbf{5} boldsymbol{z} )
12
881 Find the coordinates of the points which divides the line joining the points (2,-4,3),(-4,5,-6) in the ratio
( (i) 1:-4 )
( (i i) 2: 1 )
12
882 Direction ratio of two lines are ( l_{1}, boldsymbol{m}_{1}, boldsymbol{n}_{1} )
and ( l_{2}, m_{2}, n_{2} ) then direction ratios of the line perpendicular to both the lines
are
A ( . l_{1}-l_{2}, m_{1}-m_{2}, n_{1}-n_{2} )
в. ( l_{1}+l_{2}, m_{1}+m_{2}, n_{1}+n_{2} )
c. ( m_{1} n_{2}-n_{1} m_{2}, n_{1} l_{2}-n_{2} l_{1}, l_{1} m_{2}-m_{1} l_{2} )
D. ( m_{1} n_{2}-n_{1} m_{2}, n_{1} l_{2}-n_{1} l_{1}, l_{1} m_{2}-m_{1} l_{2} )
12
883 The points (3,2,0),(5,3,2) and ( (-9,6,-3), ) are the vertices of a triangle ( A B C . A D ) is the internal
bisector of ( angle B A C ) which meets ( B C ) at
D. Then the co-ordinates of ( D ), are
A ( cdotleft[frac{17}{16}, frac{57}{16}, frac{19}{8}right] )
В. ( left[frac{19}{8}, frac{57}{16}, frac{17}{16}right] )
( ^{mathbf{c}} cdotleft[0,0, frac{17}{16}right] )
D. ( left[frac{17}{16}, 0,0right] )
12

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