JEE Main Mathematics - Three Dimensional Geometry
Exam Duration: 60 Mins Total Questions : 30
The equation of the plane through the point (1,1,1) and parallel to the plane
2x-3y+4z+5=0, is
- (a)
2x+3y-4z-1=0
- (b)
3x-2y+4z-5=0
- (c)
2x-3y+4z-3=0
- (d)
4x-3y+2z-3=0
The angle between the line
\(\frac { x-1 }{ 2 } =\frac { y-1 }{ 3 } =\frac { z-3 }{ 4 }\)
and the plane x+2y-2z=3, is
- (a)
00
- (b)
300
- (c)
600
- (d)
900
Equation of the plane passing through the point (1,-1,2) and containing the line
\(\frac { x-2 }{ 3 } =\frac { y-5 }{ 1 } =\frac { z+2 }{ 2 } \), is
- (a)
16x-14y-17z+4=0
- (b)
x+3y+z=0
- (c)
14x-16y-7z-16=0
- (d)
NONE OF THESE
The shortest distance between the lines
\(\frac { x-5 }{ 3 } =\frac { y-7 }{ -16 } =\frac { z-3 }{ 7 }\)
\(\frac { x-9 }{ 3 } =\frac { y-13 }{ 8 } =\frac { z-15 }{ -5 }\)
is
- (a)
10
- (b)
11
- (c)
12
- (d)
14
A straight line L on the XY-plane bisects the angle between OX and OY. What are the direction cosines of L?
- (a)
\(<({1\over \sqrt2}),({1\over\sqrt2}),0>\)
- (b)
\(<({1\over2}),({\sqrt3\over2}),0>\)
- (c)
<0,0,1>
- (d)
\(<({2\over3}),({2\over3}),({1\over3})>\)
Find the distance of the plane 2x-3y+4z-6=0 from origin.
- (a)
\(6\over\sqrt{28}\)
- (b)
\(5\over\sqrt{29}\)
- (c)
\(6\over\sqrt{29}\)
- (d)
\(7\over{\sqrt{29}}\)
The intercepts made on the axes by the plane which bisects the line joining the points (1,2,3) and (-3,4,5) at right angle are
- (a)
\((-{9\over2},9,9)\)
- (b)
\(({9\over2},-9,9)\)
- (c)
\((19,-{9\over2},9)\)
- (d)
\((9,{9\over2},19)\)
If the lines \({x-2\over1}={y-3\over1}={z-4\over{-k}}\) and \({x-1\over k}={y-4\over2}={z-5\over1}\) are coplanar, then k can have
- (a)
any value
- (b)
exactly one value
- (c)
exactly two vlaue
- (d)
exactly three values
The line passing through the points (5,1,a) and (3,b,1) crosses the yz-plane at the point \((0,{17\over2},{-{13\over2}})\). Then
- (a)
a=8, b=2
- (b)
a=2, b=8
- (c)
a=4, b=6
- (d)
a=6, b=4
Let L be the line of intersection of the planes 2x+3y+z=1 and x+3y+2z=2. If L makes an angle \(\alpha\) with the positive X-axis, then \(cos \alpha\) equals
- (a)
\(1\over\sqrt3\)
- (b)
1/2
- (c)
1
- (d)
\(1/\sqrt2\)
The line \(\frac { x-2 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ -1 } \) intersects the curve xy = c2, z = 0 if c is equal to
- (a)
\(\pm1\)
- (b)
\(\pm \frac {1}{3}\)
- (c)
\(\pm \sqrt{5}\)
- (d)
none of these
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 line joining the points (1, 1, 2) and (3, - 2,1)meets the plane 3x + 2y + z = 6 at the point
- (a)
( 1, 1, 2 )
- (b)
( 3, -2, 1)
- (c)
( 2, -3, 1 )
- (d)
( 3, 2, 1 )
The point equidistant from the points (a, 0, 0), (0, b, 0), (0,0, c) and (0,0,0) is
- (a)
\(\left( \frac { a }{ 3 } ,\frac { b }{ 3 } ,\frac { c }{ 3 } \right) \)
- (b)
( a, b, c )
- (c)
\(\left( \frac { a }{ 2 } ,\frac { b }{ 2 } ,\frac { c }{ 2 } \right) \)
- (d)
none of these
The plane 4x + 7y + 4z + 81 = 0 is rotated through a right angle about its line of intersection with the plane 5x + 3y + 10z = 25. The equation of the plane in its new position is x - 4y + 6z = k, where k is
- (a)
106
- (b)
-89
- (c)
73
- (d)
37
If P, Q, R, S are the points (4,5,3) (6,3,4), (2,4, -1), (0,5,1) the length of projetcion RS on PQ is
- (a)
4/3
- (b)
2/3
- (c)
4
- (d)
6
The equation to the plane through the points (2,-1,0), (3,-4,5) parallel to a line with direction cosines proportional to 2, 3, 4 is 9x - 2y - 3z = k, where k is
- (a)
20
- (b)
-20
- (c)
10
- (d)
-10
The value(s) of \(\lambda\), for which the triangle with vertices (6,10,10), (1, 0, - 5) and (6,-10, \(\lambda\)) will he a nght angled triangle is/are
- (a)
1
- (b)
\(\frac{70}{3}\)
- (c)
35
- (d)
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.
Equation of the tangent to the curve x = t, y = t2 z = t3 ,at its point p(1, 1, 1)
- (a)
\(\frac { x-1 }{ 1 } =\frac { y-1 }{ 2 } =\frac { z-1 }{ 3 } \)
- (b)
\(\frac { x-1 }{ 3 } =\frac { y-1 }{ 2 } =\frac { z-1 }{ 1 } \)
- (c)
\(\frac { x-1 }{ 1 } =\frac { y-1 }{ 3 } =\frac { z-1 }{ 5 } \)
- (d)
all of the above
Let two planes P1 : 2x - y + z = 2 and P2 : x + 2y - z = 3 are given
Equation of the plane which passes through the point (-1, 3, 2) and is perpendicular to each of the planes P1 and P2 is
- (a)
x + 3y - 5z + 2 = 0
- (b)
x + 3y + 5z - 18 = 0
- (c)
x + 3y - 5z + 20 = 0
- (d)
x - 3y + 5z = 0
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)
\(\sqrt { \frac { 1 }{ 7 } } ,\pm \sqrt { \frac { 3 }{ 14 } } ,\frac { 1 }{ \sqrt { 14 } } \)
The vector and cartesian equations of the line which passes through the poin (5,2,-4) and is parallel to the vector \(3\hat { i } +2\hat { j } -8\hat { k } \) are
- (a)
\(\vec { r } =(3\hat { i } +2\hat { j } -8\hat { k } )+\lambda (5\hat { i } +2\hat { j } -4\hat { k) } ;\quad \frac { x-3 }{ 5 } =\frac { y-3 }{ 2 } =\frac { z+8 }{ -4 } \)
- (b)
\(\vec { r } =(5\hat { i } +2\hat { j } -4\hat { k) } +\lambda (3\hat { i } +2\hat { j } -8\hat { k } );\quad \frac { x-3 }{ 5 } =\frac { y-2 }{ 2 } =\frac { z+8 }{ -1 } \)
- (c)
\(\vec { r } =(5\hat { i } +2\hat { j } -4\hat { k) } +\lambda (3\hat { i } +2\hat { j } -8\hat { k } );\quad \frac { x-5 }{ 3 } =\frac { y-2 }{ 2 } =\frac { z+4 }{ -8 } \)
- (d)
\(\vec { r } =(3\hat { i } +2\hat { j } -8\hat { k } )+\lambda (5\hat { i } +2\hat { j } -4\hat { k) } ;\quad \frac { x-5 }{ 3 } =\frac { y-2 }{ 2 } =\frac { z+4 }{ -8 } \)
three vectors A(1,2,3), B(-1,-2,-1) and C(2,3,2) are three vertices of a parallelogram ABCD. Find the equation of CD.
- (a)
\(\frac { x }{ 1 } =\frac { y }{ 2 } =\frac { z }{ 2 } \)
- (b)
\(\frac { x+2 }{ 1 } =\frac { y+3 }{ 2 } =\frac { z-2 }{ 2 } \)
- (c)
\(\frac { x }{ 2 } =\frac { y }{ 3 } =\frac { z }{ 2 } \)
- (d)
\(\frac { x-2 }{ 1 } =\frac { y-3 }{ 2 } =\frac { z-2 }{ 2 } \)
The angle between a line whose direction ratios are in the ratio 2:2:1 and a line joining (3,1,4) to (7,2,12) is
- (a)
cos-1(2/3)
- (b)
cos-1(-2/3)
- (c)
tan-1(2/3)
- (d)
None of these
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 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)
\(\frac{4}{\sqrt{22}}\)
Statement -I: The equation of a plane which passes through (2,-3,1) and normal to the line joining the points (3,4,-1) and (2,-1,5) is given by
x+5y-6z+19=0
Statement -II: The length of perpendicular from the point (7,14,5) to the plane 2x+4y-z=2 is 2\(\sqrt{21}\) .
- (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.
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