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Mere Teacher · Previous Year Questions
JEE Main Mathematics · Three-Dimensional Geometry · Set 6 (Q151-180)
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151.
The shortest distance between the skew-lines −1x−3=2y−4=1z+2,1x−1=3y+7=2z+2 is
(1)6
(2)7
(3)33
(4)35
152.
The angle between the lines r=(4i−j)+s(2i+j−3k) and r=(i−j+2k)+t(i−3j+2k) is
(1)23π
(2)3π
(3)32π
(4)6π
153.
Equation of the line passing through the intersection of the plane x+2y+3z=4 and the line 2x−1=1y+1=−1z−1 and parallel to the vector (2i−3j)×(i+2j−k) is
(1)3x−5=2y−1=−7z+1
(2)−3x−5=−2y−1=7z−1
(3)−3x−5=−2y−1=−7z+1
(4)−3x−5=2y−1=7z+1
154.
Let π1 be a plane passing through the point i+j+k and perpendicular to the vector −j+2k. Let the line L passing through the points 3i−2j+k and −i+3j+k be a normal to the plane π2. If the angle between the planes π1 and π2 is θ then cosθ=
(1)415
(2)205−14
(3)4π
(4)2π
155.
The shortest distance between the lines 0x−1=−1y+1=1z and x+y+z+1=0,2x−y+z+3=0 is:
(1)21
(2)1
(3)31
(4)21
156.
If (a,b,c) are the direction ratios of a line joining the points (4,3,−5) and (−2,1,−8) then the point P(a,3b,2c) lies on the plane
(1)x+y+z=0
(2)x+y−2z=0
(3)x+2y+3z=0
(4)x−2y+3z=0
157.
A vector n is inclined to X-axis at 45∘,Y-axis at 60∘ and at an acute angle to Z-axis. If n is normal to a plane passing through the point (−2,1,1), then equation of the plane is
(1)2x+y+z=0
(2)x+2y+z=1
(3)−2x+y+2z=5
(4)x+y+2z=1
158.
If 1,m,n are the direction cosines of a line which makes angles α,β and γ with the coordinate axes X,Y,Z respectively, then lm+mn+nl takes the maximum value when
(1)α,β,γ are in Arithmetic progression
(2)α=β=γ
(3)any two of α,β,γ are the same
(4)one of α,β,γ is zero and the remaining two are non-zero and unequal
159.
If a line L is common to the planes x−y+z+2=0 and 2x+y−2z+5=0 then the direction cosines of the line L are
(1)(261,264,263)
(2)(31,32,32)
(3)(31,31,31)
(4)(6−1,65,610)
160.
The distance of the point (7,10,11) from the line 1x−4=0y−4=3z−2 along the line 2x−9=3y−13=6z−17 is
(1)16
(2)12
(3)18
(4)14
161.
The equation of the plane passing through the point i+2j−k and perpendicular to the line of intersection of the planes r⋅(3i−j+k)=1 and r(i+4j−2k)=2, is
(1)r⋅(−2i−5j+k)=0
(2)r⋅(i+7j+4k)=0
(3)r⋅(2i−7j−13k)=1
(4)r⋅(−2i+7j+13k)=0
162.
If the shortest distance between the lines 1x−4=2y+1=−3z and 2x−λ=4y+1=−5z−2 is 56, then the sum of all possible values of λ is :
(1)5
(2)8
(3)7
(4)10
163.
If the foot of the perpendicular drawn from the point (0,0,0) to the plane is (4,−2,−5) then the equation of the plane is
(1)4x+2y+5z=−13
(2)4x−2y−5z=45
(3)4x+2y−5z=37
(4)4x−2y+5z=−5
164.
The joint equation of pair of lines which bisects the angle between the lines x2+3xy+2y2=0 is
(1)3x2−2xy−3y2=0
(2)2x2−3xy−2y2=0
(3)3x2+2xy−3y2=0
(4)2x2+3xy−2y2=0
165.
If the line 3x−2=2y+1=−1z−1 intersects the plane 2x+3y−z+13=0 at a point P and the plane 3x+y+4z=16 at a point Q then PQ is equal to:
(1)14
(2)14
(3)27
(4)214
166.
The cartesian equation of the plane whose vector equation is γ=(1+λ−μ)i+(2−λ)j+(3−2λ+2μ)k where λ,μ are scalars, is
(1)2x+y=5
(2)2x−y=5
(3)2x−z=5
(4)2x+z=5
167.
If a point R(4,y,z) lies on the line segment joining the points P(2,−3,4) and Q(8,0,10), then distance of R from the origin is :
(1)214
(2)221
(3)6
(4)53
168.
If the equation of the line passing through the point (0,−21,0) and perpendicular to the lines r=λ(i+aj+bk) and r=(i−j−6k)+μ(−bi+aj+5k) is −2x−1=dy+4=−4z−c, then a+b+c+d is equal to:
(1)13
(2)14
(3)12
(4)10
169.
Let PQR be a triangle with R(−1,4,2). Suppose M(2,1,2) is the mid point of PQ. The distance of the centroid of △PQR from the point of intersection of the line 0x−2=2y=−1z+3 and 1x−1=−3y+3=1z+1 is
(1)69
(2)9
(3)69
(4)99
170.
Direction ratios of the line which is perpendicular to the lines with direction ratios −1,2,2 and 0,2,1 are
(1)1,1,2
(2)2,−1,2
(3)−2,1,2
(4)2, 1,-2
171.
The shortest distance between the lines 3x−3=−1y−8=1z−8 and −3x+3=2y+7=4z−6 is
(1)330
(2)230
(3)30
(4)430
172.
If the planes 2x+3y+4z+7=0 and 4x+ky+8z+1=0 are parallel, then the equation of the plane passing through is point (k,k,k) and having the direction ratios of its normal as ( k−1,k,k+1 ) is
(1)x+2y+3z=36
(2)3x+4y+5z=72
(3)4x+5y+6z=90
(4)5x+6y+7z=108
173.
Let A(2,5,7) be the image of the point B(1,−2,3) with respect to a plane π. Let C be the point where AB meets the plane π. Let D=(2,1,6). Then the direction cosines of CD are
(1)111,113,11−1
(2)61,6−1,62
(3)463,46−1,466
(4)141,142,143
174.
If a plane passing through the points (2,3,0),(0,−5,2) and (−2,0,3) meets the X,Y,Z-axes in A,B,C respectively then A=
(1)(−,0,0)
(2)(37,0,0)
(3)(1321,0,0)
(4)(21,0,0)
175.
Let A and B be two distinct points on the line L:3x−6=2y−7=−2z−7. Both A and B are at a distance 217 from the foot of perpendicular drawn from the point (1,2,3) on the line L. If O is the origin, then OA⋅OB is equal to
(1)49
(2)21
(3)47
(4)62
176.
The value of k such that the line 1x−4=1y−2=2z−k lies on the plane 2x−4y+z=7 is
(1)-7
(2)4
(3)-4
(4)7
177.
If a line makes angles α,β,γ and δ with the four diagonals of a cube, then the value of sin2α+sin2β+sin2γ+sin2δ is
(1)34
(2)38
(3)37
(4)35
178.
The distance of the origin from the plane r⋅(3i+4j−12k)=7 measured parallel to the line r=(i+2j+3k)+t(6i+2j+3k) is
(1)845
(2)1049
(3)107
(4)53
179.
Let the co-ordinates of one vertex of ∆ABC be A(0,2,α) and the other two vertices lie on the line 5x+α=2y−1=3z+4. For αϵZ, if the area of ∆ABC is 21 sq. units and the line segment BC has length 221 units, then α2 is equal to ________
180.
Let L1:3x−1=−1y−1=0z+1 and L2:2x−2=0y=αz+4,α∈R, be two lines, which intersect at the point B. If P is the foot of perpendicular from the point A(1,1,−1) on L2, then the value of 26α(PB)2 is .
