JEE Mathematics - Three Dimensional Geometry Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
A line make angles \(\alpha ,\beta ,\gamma ,\delta \) with the diagonals of a cube, then \({ cos }^{ 2 }\alpha +{ cos }^{ 2 }\beta +{ cos }^{ 2 }\gamma +{ cos }^{ 2 }\delta \), is
- (a)
\(\frac { 1 }{ 3 } \)
- (b)
\(\frac { 2}{ 3 } \)
- (c)
1
- (d)
\(\frac { 4}{ 3 } \)
The plane \(\frac { x }{ a } +\frac { y }{ b } +\frac { z }{ c } =1\), meets the coordinates axes in A,B,C; then coordinates of the centroid of the \(\triangle \)ABC, are
- (a)
(a,b,c)
- (b)
\(\left( \frac { a }{ 3 } ,\frac { b }{ 3 } ,\frac { c }{ 3 } \right) \)
- (c)
\(\left( \frac { 1 }{ a } ,\frac { 1 }{ b } ,\frac { 1 }{ c } \right) \)
- (d)
NONE OF THESE
A tetrahedron has vertices 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
- (a)
\({ cos }^{ -1 }\left( \frac { 19 }{ 35 } \right) \)
- (b)
\({ cos }^{ -1 }\left( \frac { 17 }{ 31 } \right) \)
- (c)
300
- (d)
900
A variable planes passes through the fixed point (a,b,c) and meets the coordinates axes in A,B,C. Then locus of the point common to the planes through A,B,C and parallel to the coordinate planes is
- (a)
\(\frac { x }{ a } +\frac { y }{ b } +\frac { z }{ c } =1\)
- (b)
\(\frac { a }{ x } +\frac { b }{ y } +\frac { c }{ z } =1\)
- (c)
\(\frac { a }{ x } +\frac { b }{ y } +\frac { c }{ z } =2\)
- (d)
\(\frac { x }{ a } +\frac { y }{ b } +\frac { z }{ c } =2\)
The distance between the planes 2x+y+2z=8 and 4x+2y+4z+5=0, is
- (a)
\(\frac { 1 }{ 2 } \)
- (b)
\(\frac { 3 }{ 2 } \)
- (c)
\(\frac { 5 }{ 2 } \)
- (d)
\(\frac { 7 }{ 2 } \)
The length of the perpendicular from O(0,0,0) to te plane 3x+4y+12z=26, is
- (a)
2
- (b)
3
- (c)
4
- (d)
6
The angle between the lines
\(\frac { x-1 }{ 5 } =\frac { y-2 }{ -12 } =\frac { z+4 }{ 13 } \)
and \(\frac { x+1 }{ -3 } =\frac { y+2 }{ 4 } =\frac { z-5 }{ 5 } \), is
- (a)
\({ cos }^{ -1 }\left( \frac { 2 }{ 65 } \right) \)
- (b)
\({ cos }^{ -1 }\left( \frac { 1 }{ 65 } \right) \)
- (c)
\({ cos }^{ -1 }\left( \frac { 3 }{ 65 } \right) \)
- (d)
NONE OF THESE
The value of \(\lambda \) for which the lines
\(\frac { x-1 }{ 2 } =\frac { y }{ 3 } =\frac { z+1 }{ 4 } \)
and \(\frac { x+3 }{ 3 } =\frac { y-2 }{ \lambda } =\frac { z-5 }{ 3 } \)
are perpendicular, is
- (a)
6
- (b)
\(\frac { 1 }{ 6 } \)
- (c)
-6
- (d)
\(-\frac { 1 }{ 6 } \)
If the line
\(\frac { x-4 }{ 1 } =\frac { y-2 }{ 1 } =\frac { z-k }{ 2 } \)
lies in the plane
2x-4y+z=7,
then the value of k is
- (a)
1
- (b)
-7
- (c)
7
- (d)
8
The equation of the sphere which passes through the points (1,0,0), (0,1,0), (0,0,1), and has the centre on the plane x+y+z=6, is
- (a)
x2+y2+z2-4x-4y-4z+3=0
- (b)
x2+y2+z2-4x-4y-4z-3=0
- (c)
x2+y2+z2+4x-4y+4z+3=0
- (d)
x2+y2+z2+4x-4y+4z-3=0
For what value of a will the two points(1,a,1) and (-3,0,a) lie on opposite sides of the plane 3x+4y-12z+13=0?
- (a)
\(a,-1< or \ a>{1\over3}\)
- (b)
-2<a<2
- (c)
0<a<1
- (d)
-2<a<1
If the straight line \({x-\alpha\over l}={y-\beta\over m}={z-r\over n}\) intersect the curve ax2+bx2=1, z=0 then the value of a \((\alpha n-\gamma l)^2+b(\beta n-\gamma m)^2\) is
- (a)
n2
- (b)
m2
- (c)
l2
- (d)
0
The coordinate of the point, where the line through the points A(3,4,1) and B(5,1,6) crosses the XY-plane is
- (a)
\(({13\over5},{23\over5},0)\)
- (b)
\(({-13\over5},{23\over5},0)\)
- (c)
\(({13\over5},{-23\over5},0)\)
- (d)
\(({13\over5},{22\over5},0)\)
The two lines x=ay+b, z=cy+d and x=a'y+b', z=c'y+d' are perpendicular to each other, if
- (a)
aa'+cc'=1
- (b)
\({a\over a'}+{c\over c'}=-1\)
- (c)
\({a\over a'}+{c\over c'}=1\)
- (d)
aa'+cc'=-1
The angle between the lines 2x=3y=-z and 6x=-y=-4z is
- (a)
30o
- (b)
45o
- (c)
90o
- (d)
0o
The four lines drawn from the vertices of any tetrahedron to the centroid of the opposite faces meet in a point whose distance from each vertex is k times the distance from each vertex to the opposite face, where k is
- (a)
1/3
- (b)
1/2
- (c)
3/4
- (d)
5/4
The equation of the plane through the point (0, - 4, - 6) and (-2, 9, 3) and perpendicular to the plane x - 4y - 2z = 8 is
- (a)
3x + 3y - 2z = 0
- (b)
x - 2y + z = 2
- (c)
2x + y - z = 2
- (d)
5x - 3y + 2z = 0
The equation of the plane passing through the points (3, 2,- 1), (3, 4, 2)and (7, 0,6) is 5x + 3y - 2z =\(\lambda,\) where \(\lambda\) is
- (a)
23
- (b)
21
- (c)
19
- (d)
27
The point of the plane \(\frac { x-1 }{ 1 } =\frac { y+3 }{ -2 } =\frac { z+5 }{ -2 } \) at a distance of 6 from the point ( 2, -3, -5 ) is
- (a)
( 3, -5, -3 )
- (b)
( 4, -7, -9 )
- (c)
(0, 2, -1)
- (d)
( -3, 5, 3 )
The lines which intersect the skew lines y = mx, z = c ; y = - mx, z = -c and the x-axis lie on the surface
- (a)
cz = mxy
- (b)
cy = mxz
- (c)
xy = cmz
- (d)
none of these
The equation of the line passing through the point (1, 1, -1 ) and perpendicular to the plane x - 2y - 3z = 7 is
- (a)
\(\frac { x-1 }{ -1 } =\frac { y-1 }{ 2 } =\frac { z+1 }{ 3 } \)
- (b)
\(\frac { x-1 }{ -1 } =\frac { y-1 }{ -2 } =\frac { z+1 }{ 3 } \)
- (c)
\(\frac { x-1 }{ 1 } =\frac { y-1 }{ -2 } =\frac { z+1 }{ -3 } \)
- (d)
none of these
A plane meets the coordinate axes in A, B, C such that the centroid of the triangle ABC is the point (a, a, a). Then the equation of the plane is x + y + z = p, where p is
- (a)
a
- (b)
3 / a
- (c)
a / 3
- (d)
3a
The straight lines whose direction cosines are given by al + bm + cn = 0, fmn + gnl + hlm = 0 are perpendicular if
- (a)
\(\frac { f }{ a } +\frac { g }{ b } +\frac { h }{ c } =0\)
- (b)
\(\frac { { a }^{ 2 } }{ f } +\frac { { b }^{ 2 } }{ g } +\frac { { c }^{ 2 } }{ h } =0\)
- (c)
a2(g+h)+b2(h+f)+c2(f+g)=0
- (d)
none of these
The plane containing the two lines \(\frac { x-3 }{ 1 } =\frac { y-2 }{ 4 } =\frac { z-1 }{ 5 } \) and \(\frac { x-2 }{ 1 } =\frac { y+3 }{ -4 } =\frac { z+1 }{ 5 } \)i is 11x = my + nz = 28, where
- (a)
m = -1, n = 3
- (b)
m = 1, n = - 3
- (c)
m = -1, n = - 3
- (d)
m = 1, n = 3
The area of the triangle whose vertices are at the points (2,1,1), (3,1,2), (-4,0,1) is
- (a)
\(\sqrt{19}\)
- (b)
\(\frac{1}{2}\sqrt{19}\)
- (c)
\(\frac{1}{2}\sqrt{38}\)
- (d)
\(\frac {1}{2}\sqrt{5}\)
If p1 p2 p3 denote the distances of the plane 2x - 3y 4z + 2 = 0 from the planes 2x - 3y + 4z + 6 = 0, 4x - 6y + 8z + 3 = 0 and 2x - 3y + 4z - 6 = 0 respectively. Then
- (a)
p1 + 8p2 -p3 = 0
- (b)
p3 = 16p2
- (c)
8p2 = 91
- (d)
p1 + 2p2 + 3p3 = \(\sqrt{29}\)
The extremities of a diameter of a sphere lie on positive y and positive z-axes at distances 2 and 4 from the origin, respectively, then
- (a)
sphere passes through the origin
- (b)
centre of the sphere is (0, 1, 2)
- (c)
radius of the sphere is \(\sqrt{5}\)
- (d)
equation of a diameter is \(\frac{x}{0}=\frac{y-2}{1}=\frac{z-4}{-2}\)
If -2, 2,1 are direction of a line, then its direction cosines are
- (a)
\(-\frac{2}{3},\frac{2}{3},\frac{1}{3}\)
- (b)
\(\frac{2}{3},-\frac{2}{3},\frac{1}{3}\)
- (c)
\(\frac{2}{3},-\frac{2}{3}-,\frac{1}{3}\)
- (d)
\(-\frac{2}{3},\frac{2}{3},-\frac{1}{3}\)
Suppose direction cosines of two lines are given by ul + vm + wn = 0 and al2 + bm2 + cn2 = 0, where u, v, w, a, b, c are arbitrary constants and I, m, n are direction cosines of the lines.
The given lines will be parallel if
- (a)
\(\sum { { u }^{ 2 } } \left( b+c \right) =0\)
- (b)
\(\sum { \frac { { a }^{ 2 } }{ u } } =0\)
- (c)
\(\sum { \frac { { u }^{ 2 } }{ a } } =0\)
- (d)
\(\sum { \frac { \left( b+c \right) }{ { u }^{ 2 } } } =0\)
The equation of motion of a point in space is x = 2t, Y = - 4t, z = 4t; where 't' measured in seconds and coordinates of moving point in kilometers.
The equation of plane which is parallel to tangents plane in Q3 and touches the sphere X2 + y2 + Z2 - 2x - 4y + 2z -3 = 0 is
- (a)
x - 2y + z + 2 = 0
- (b)
x - 2y + z + 3 = 0
- (c)
x - 2y + z + 4 = 0
- (d)
none of the above
Let two planes P1 : 2x - y + z = 2 and P2 : x + 2y - z = 3 are given
The equation of the bisector of angle of the planes P1 and P2 which not containing origin is
- (a)
x - 3y + 2z + 1 = 0
- (b)
x + 3y = 5
- (c)
x + 3y + 2z + 2 = 0
- (d)
3x + y = 5
Let two planes P1 : 2x - y + z = 2 and P2 : x + 2y - z = 3 are given
The image of plane P1 in the plane mirror P2 is
- (a)
x + 7y - 4x + 5 = 0
- (b)
3x + 4y - 5z + 9 = 0
- (c)
7x - y + 4z - 9 = 0
- (d)
none of the above
Direction cosines of the line that makes equal angles with the three axes in space are
- (a)
\(\pm \frac { 1 }{ \sqrt { 3 } } ,\pm \frac { 1 }{ 3 } ,\pm \frac { 1 }{ 3 } \)
- (b)
\(\pm \frac { 6 }{ 7 } ,\pm \frac { 2 }{ 7 } ,\pm \frac { 3 }{ 7 } \)
- (c)
\(\pm \frac { 1 }{ \sqrt { 3 } } ,\pm \frac { 1 }{ \sqrt { 3 } } ,\pm \frac { 1 }{ \sqrt { 3 } \quad } \)
- (d)
\(\sqrt { \frac { 1 }{ 7 } } ,\pm \sqrt { \frac { 3 }{ 14 } } ,\frac { 1 }{ \sqrt { 14 } } \)
If a line makes an angle θ1,θ2,θ3 with the axces respectively, then cos 2θ1+cos 2θ2+cos2θ3=
- (a)
-4
- (b)
-2
- (c)
-3
- (d)
-1
If A(3,5,-4) B(-1,1,2) and C (-5,-5,-2) are the vertices of a ∆ABC, then
- (a)
direction cosines of AB are \(\frac { -2 }{ \sqrt { 17 } } ,\frac { -2 }{ \sqrt { 17 } } ,\frac { 3 }{ \sqrt { 17 } } \)
- (b)
direction cosines of AC are\(\frac { 4 }{ \sqrt { 42 } } ,\frac { 5 }{ \sqrt { 42 } } ,\frac { -1 }{ \sqrt { 42 } } \)
- (c)
direction cosines of BC are\(\frac { -2 }{ \sqrt { 17 } } ,\frac { -3 }{ \sqrt { 17 } } ,\frac { -2 }{ \sqrt { 17 } } \)
- (d)
All of these
The coordinates of a point on the line \(\frac { x-2 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-3 }{ 2 } \) at a distance of \(\frac{6}{\sqrt2}\) from the points (1,2,3) is
- (a)
(56,43,111)
- (b)
\(\left( \frac { 56 }{ 17 } ,\frac { 43 }{ 17 } ,\frac { 111 }{ 17 } \right) \)
- (c)
(2,1,3)
- (d)
(-2,-1,-3)
The equation of the line joining the points (-3,4,11) and (1,-2,7) is
- (a)
\(\frac { x+3 }{ 2 } =\frac { y-4 }{ 3 } =\frac { z-11 }{ 4 } \)
- (b)
\(\frac { x+3 }{ -2 } =\frac { y-4 }{ 3 } =\frac { z-11 }{ 4 } \)
- (c)
\(\frac { x+3 }{ -2 } =\frac { y+4 }{ 3 } =\frac { z+11 }{ 4 } \)
- (d)
\(\frac { x+3 }{ 2 } =\frac { y+4 }{ -3 } =\frac { z+11 }{ 4 } \)
Vector equation of the line 6x-3=3y+4=2z-2
- (a)
\(\vec { r } =\hat { i } -\hat { j } +\hat { k } +\lambda (6\hat { i } +\hat { j } +\hat { k } )\)
- (b)
\(\vec { r } =6\hat { i } +3\hat { j } +2\hat { k } +\lambda (3\hat { i } +4\hat { j } -2\hat { k } )\)
- (c)
\(\vec { r } =\frac { 1 }{ 2 } \hat { i } -\frac { 4 }{ 3 } \hat { j } +\hat { k } +\lambda (\frac { 1 }{ 6 } \hat { i } +\frac { 1 }{ 3 } \hat { j } +\frac { 1 }{ 2 } \hat { k } )\)
- (d)
None of these
Equation of line passing through (1,2,-3) and parallel to the line .\(\frac { x-2 }{ 1 } =\frac { y+1 }{ 3 } =\frac { z-1 }{ 4 } \)is
- (a)
\(\frac { x-1 }{ 1 } =\frac { y-2 }{ 3 } =\frac { z+3 }{ 4 } \)
- (b)
\(\frac { x-2 }{ 1 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ -3 } \)
- (c)
\(\frac { x-1 }{ 1 } =\frac { y-3 }{ 2 } =\frac { z-1 }{ -3 } \)
- (d)
None of these
Shortest distance between the two lines \(\vec { r } =(8+3\lambda )\hat { i } -(9+16\lambda )\hat { j } +(10+7\lambda )\hat { k } and\vec { r } =15\hat { i } +29\hat { j } +5\hat { k } +\mu (3\hat { i+8\hat { j } -5\hat { k } ) } \)
- (a)
84
- (b)
14
- (c)
21
- (d)
16
Find the shortest distence between the lines
\(\vec { r } =\hat { i } -2\hat { j } +-4\hat { k } +\lambda(2\hat { i } +3\hat { j } +6\hat { k })\)and
\(\vec { r } =3\hat { i } +3\hat { j } -5\hat { k } +\mu (-2\hat { i+3\hat { j } +8\hat { k } ) } \)
- (a)
\(\frac{14}{241}\)
- (b)
\(\frac{14}{243}\)
- (c)
\(\frac{14}{\sqrt{243}}\)
- (d)
\(\frac{14}{\sqrt{241}}\)
The distance between lines \(\vec { r } =\vec { { a }_{ 1 } } +t\vec { b } \)and\(\vec { r } =\vec { { a }_{ 1 } } +s\vec { b } \)
- (a)
\( |({ \vec { a } }_{ 2 }-{ \vec { a } }_{ 1 })\times \vec { b } | \)
- (b)
\(\frac { |({ \vec { a } }_{ 2 }-{ \vec { a } }_{ 1 })\times \vec { b } | }{ |\vec { b } | } \)
- (c)
\(\frac { |({ \vec { a } }_{ 2 }-{ \vec { a } }_{ 1 })\times \vec { b } | }{ |{ \vec { a } }_{ 2 }-{ \vec { a } }_{ 1 }|}\)
- (d)
\(\frac { |({ \vec { a } }_{ 2 }-{ \vec { a } }_{ 1 })\times \vec { b } | }{ |{ \vec { a } }_{ 2 }-{ \vec { a } }_{ 1 }|.|\vec { b } | } \)
The equation of the plane passing through three non-collinear points with position vectors \(\vec { a } ,\vec { b } ,\vec { c } \)is
- (a)
\(\vec { r } .(\vec { b } \times \vec { c } +\vec { c } \times \vec { a } +\vec { a } \times \vec { b } )=0\)
- (b)
\(\vec { r } .(\vec { b } \times \vec { c } +\vec { c } \times \vec { a } +\vec { a } \times \vec { b } )=[\vec { a } \vec { b } \vec { c } ]\)
- (c)
\(\vec { r } .(\vec { a } \times (\vec { b } +\vec { c } ))=[\vec { a } \vec { b } \vec { c } ]\)
- (d)
\(\vec { r } .(\vec { a } \times \vec { b } +\vec { c } )=0\)
An equation of the plane passing through the points (3,2,-1),(3,4,2) and (7,0,6) is 5x+3y-2z=λ, where λis
- (a)
23
- (b)
21
- (c)
19
- (d)
27
The line and \(\frac { x-1 }{ 2 } =\frac { y+1 }{ -3 } =\frac { z+10 }{ 8 } and\frac { x-4 }{ 1 } =\frac { y+3 }{ k } =\frac { z+1 }{ 7 } \) are coplanar if K=
- (a)
4
- (b)
-4
- (c)
2
- (d)
-2
The distance of the origin from the plane through the points (1,1,0),(1,2,1)and(-2,2,-1) is
- (a)
\(\frac{3}{\sqrt{11}}\)
- (b)
\(\frac{5}{\sqrt{22}}\)
- (c)
3
- (d)
\(\frac{4}{\sqrt{22}}\)
The angle between the line \(\frac { x+1 }{ 3 } =\frac { y-1 }{ 2 } =\frac { z-2 }{ 4 } \) and the plane 2x+y-3z+4=0 is
- (a)
\({ cos }^{ -1 }\left( \frac { 4 }{ \sqrt { 406 } } \right) \)
- (b)
\({ sin }^{ -1 }\left( \frac { 4 }{ \sqrt { 406 } } \right) \)
- (c)
30o
- (d)
None of these
Consider the lines
\({ L }_{ 1 }:\frac { x+1 }{ 3 } =\frac { y+2 }{ 1 } =\frac { z+1 }{ 2 } ,{ L }_{ 2 }:\frac { x-2 }{ 1 } =\frac { y+2 }{ 2 } =\frac { z-3 }{ 3 } \)
Statement -I: The distance of point (1,1,1) from the plane passing through the point (-1,-2,-1) and whose normal is perpendicular to both the lines L1 and L2 is \(\frac{13}{5\sqrt3}\)
Statement -II: The unit vector perpendicular to both the lines L1 and L2 is \(\frac { -\hat { i } -7\hat { j } +5\hat { k } }{ 5\sqrt { 3 } } \).
- (a)
If both Statement -I and Statement -II are true but Statement -II is the correct explanation of Statement -I.
- (b)
If both Statement -I and Statement -II are true and Statement -II is not the correct explanation of Statement -I.
- (c)
If Statement -I is true but Statement -II is false.
- (d)
If Statement -I is false and Statement -II is true.
Statement -I: Two systems of rectangular axis have the same origin. If a plane cuts them at distance a, b,c and a',b',c' respectively origin, then \(\frac { 1 }{ { a }^{ 2 } } +\frac { 1 }{ { b }^{ 2 } } +\frac { 1 }{ { c }^{ 2 } } =\frac { 1 }{ { a' }^{ 2 } } +\frac { 1 }{ b'^{ 2 } } +\frac { 1 }{ { c' }^{ 1 } } .\)
Statement -II: The points \((\hat { i } -\hat { j } +3\hat { k } )\)and \(3(\hat { i } +\hat { j } +\hat { k }
)\) are equidistant from the plane \(\left( 5\hat { i } +2\hat { j } -7\hat { k } \right) +9=0.\)
- (a)
If both Statement -I and Statement -II are true but Statement -II is the correct explanation of Statement -I.
- (b)
If both Statement -I and Statement -II are true and Statement -II is not the correct explanation of Statement -I.
- (c)
If Statement -I is true but Statement -II is false.
- (d)
If Statement -I is false and Statement -II is true.
Match the following.
Column I | Column II | |
(i) | The equation of the plane through the points (2,1,0) (3,-2,-2) and (3,1,7) is | (p) x+y+2z-19=0 |
(ii) | The equation of a plane which bisects perpendicularly the line joining the points A(2,3,4) and B(3,5,8) is | (q)3x-2y+6z-27=0 |
(iii) | The equation of the plane through the points(2,1,-1),(-1,3,4) and perpendicular to the plane x-2y+4z=10 is | (r)7x+3y-z-17=0 |
(iv) | If the line drawn from the point (-2,-1,-3) meets a plane at right angle at the point (1,-3,3), then the equation of the plane is | (s) 18x+17y+4z-49=0 |
- (a)
(i)⟶(q),(ii)⟶(r),(iii)⟶(s)(iv)⟶(p)
- (b)
(i)⟶(r),(ii)⟶(p),(iii)⟶(s)(iv)⟶(q)
- (c)
(i)⟶(p),(ii)⟶(q),(iii)⟶(r)(iv)⟶(s)
- (d)
(i)⟶(r),(ii)⟶(p),(iii)⟶(q)(iv)⟶(s)