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Mere Teacher · Previous Year Questions
JEE Main Mathematics · Three-Dimensional Geometry · Set 3 (Q61-90)
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61.
XY-plane divides the line joining the points A(2,3,−5) and B(−1,−2,−3) in the ratio
(1)2:1 internally
(2)5:3 internally
(3)3:2 externally
(4)5:3 externally
62.
The length of the perpendicular drawn from the point (2,1,4)to the plane containing the linesr=(i+j)+λ(i+2j−k) and r=(i+j)+μ(−i+j−2k)is:
(1)3
(2)31
(3)3
(4)31
63.
The vector equation of the plane through the line 0 intersection of the planes x+y+z=1 and 2x+3y+4z=5 which is perpendicular to the plane x−y+z=0is :
(1)r×(i−k)+2=0
(2)r⋅(i−k)−2=0
(3)r×(i+k)+2=0
(4)r⋅(i−k)+2=0
64.
A vector v in the first octant is inclined to the x-axis at 60°, to the y-axis at 45° and to the z− axis at an acute angle. If a plane passing through the point (2,−1,1) and (a,b,c), is normal to v then
(1)2a+b+c=1
(2)a+b+2c=1
(3)a+2b+c=1
(4)2a−b+c=1
65.
Find the distance of the plane 3x−4y+5z+7=0 from the origin.
(1)527 units
(2)327 unit
(3)67 units
(4)52 units
66.
Let a=a1i+a2j+a3k where a1,a2,a3 and |a| are rational numbers. If a makes an angle 45∘ with b and b=2i+32j+4k then a lies in
(1)XY - plane
(2)YZ - plane
(3)XZ - plane
(4)along the bisector of the angle between k and −b
67.
The distance of the point (7,−3,−4) from the plane passing through the points ,(−1,1,−2)(2,−2,1) and (3,−4,2) is
(1)4
(2)5
(3)52
(4)42
68.
The plane containing the line 2x−3=−1y+2=3z−1 and also containing its projection on the plane 2x+3y−z=5, contains which one of the following points?
(1)(2,2,0)
(2)(−2,2,2)
(3)(0,−2,2)
(4)(2,0,−2)
69.
The distance of the point (1,−2,3) from the plane x−y+z=5 measured parallel to the line 2x=3y=−6z is:
(1)7
(2)57
(3)1
(4)71
70.
The equation of the line joining the points (−3,4,11) and (1,−2,7) is
(1)2x+3=3y−4=4z−11
(2)−2x+3=3y−4=2z−11
(3)−2x+3=3y+4=4z+11
(4)2x+3=−3y+4=2z+11
71.
Shortest distance between the lines 2x−1=−7y+8=5z−4 and 2x−1=1y−2=−3z−6
(1)23
(2)43
(3)33
(4)53
72.
The line 3x−2=4y−3=5z−4 is parallel to the plane
(1)2x+3y+4z=0
(2)3x+4y+5z=7
(3)2x+y−2z=0
(4)x+y+z=2
73.
If a,b and c are three non-collinear points and ka+2b+3c is a point in the plane of a,bc, then k=
(1)4
(2)5
(3)-5
(4)-4
74.
If the foot of the perpendicular from point (4,3,8)on the L1:lx−a=3y−2=4z−b,l=0is(3,5,7) is (3,5,7), thenthe shortest distance between the line L1 and lineL2:3x−2=4y−4=5z−5 is equal to:
(1)21
(2)61
(3)32
(4)31
75.
If the vectors 2i−3j+4k,2i+j−k and λi−j+2k are coplanar, then the value of λ is
(1)-5
(2)-6
(3)5
(4)6
76.
Let a unit vector OP make angle α,β,γ with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where βϵ(0,2π)OP is perpendicular to the plane through points (1,2,3),(2,3,4) and (1,5,7), then which one of the following is true?
(1)αϵ(2π,π) and γϵ(2π,π)
(2)αϵ(0,2π) and γϵ(0,2π)
(3)αϵ(2π,π) and γϵ(0,2π)
(4)αϵ(0,2π,) and γϵ(2π,π)
77.
Direction cosines of the line 2x+2=32y−5,z=−1 are
(1)54,53,0
(2)53,54,51
(3)−53,54,0
(4)54,−52,51
78.
Let 6x−3y+2z−6=0 be the given plane. If a,b, c are the intercepts made by the plane on X,Y,Z-axes respectively; 1,m,n are the direction cosines of a normal drawn to the plane and p is the perpendicular distance from the origin to the plane, then |al+bm+cn|=
(1)p
(2)2 p
(3)3 p
(4)4p
79.
A plane passing through the points (0,−1,0) and (0,0,1) and making an angle 4π with the plane y−z+5=0, also passes through the point:
(1)(−2,1,−4)
(2)(2,−1,4)
(3)(−2,−1,−4)
(4)(2,1,4)
80.
The perpendicular distance of the point P(6,7,8) from XY-plane is
(1)7
(2)6
(3)8
(4)5
81.
Let L be the line 2x+1=3y+1=6z+3 and let S be the set of all points (a,b,c) on L , whose distance from the line 2x+1=3y+1=0z−9 along the line L is 7 . Then ∑(a+b+c) is equal to:
(1)6
(2)34
(3)40
(4)28
82.
A(2,3,5),B(α,3,3) and C(7,5,β) are the vertices of a triangle. If the median through A is equally inclined with the co-ordinate axes, then cos−1(βα)=
(1)cos−1(9−1)
(2)2π
(3)3π
(4)cos−1(52)
83.
The distance of the point whose position vector is (2i+j−k) from the plane r⋅(i−2j+4k)=4 is
(1)218
(2)21−8
(3)821
(4)21−8
84.
Lines 1x−2=1y−3=−Kz−4 and Kx−1=2y−4=1z−5 are coplanar if
(1)K=2
(2)K=0
(3)K=3
(4)K=−1
85.
If the equation of the plane passing through the point (2,−1,3) and perpendicular to the planes 3x−2y+z=9 and x+y+z=9 is x+by+cz+d=0, then d=
(1)311
(2)0
(3)3
(4)31
86.
Consider the line L passing through the points (1,2,3) and (2,3,5). The distance of the point (311,311,319) from the line L along the line 23x−11=13y−11=23z−19 is equal to
(1)6
(2)5
(3)4
(4)3
87.
The distance of the point P(3, 4, 4) from the point of intersection of the line joining the points. Q(3,−4,−5) and R(2,−3, 1) and the plane 2x + y + z =7, is equal to
88.
Let the line 7x−3=−1y−2=−4z−3 intersect the plane containing the lines 1x−4=−2y+1=1z and 4ax-y + 5z-7a = 0 =2x-5y-z-3, a∈R at the point P(α,β,γ). Then the value of α+β+γequals______.
89.
Let P be an arbitrary point having sum of the squares of the distance from the planes x+y+z=0,lx−nz=0 and x−2y+z=0 equal to 9 . If the locus of the point Pis x2+y2+z2=9then the value of l−n is equal to
90.
Let the equation of the plane passing through the line x−2y−z−5=0=x+y+3z−5 and parallel to the line x+y+2z−7=0=2x+3y+z−2 be ax+by+cz=65. Then the distance of the point (a,b,c) from the plane 2x+2y−z+16=0 is _________
