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Mere Teacher · Previous Year Questions
JEE Main Mathematics · Three-Dimensional Geometry · Set 1 (Q1-30)
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1.
Let the point, on the line passing through the points P(1,−2,3) and Q(5,−4,7), farther from the origin and at distance of 9 units from the point P , be (α,β,γ). Then α2+β2+γ2 is equal to :
(1)165
(2)160
(3)155
(4)150
2.
The mirror image of P(2,4,−1) in the plane x−y+2z−2=0 is (a,b,c), then the value of a+b+c is
(1)4
(2)5
(3)7
(4)9
3.
If a plane meets the coordinate axes at A, B and C in such a way that the centroid of triangle ABC is at the point (1,2,3) then the equation of the plane is
(1)1x+2y+3z=1
(2)3x+6y+9z=1
(3)1x+2y+3z=31
(4)1x−2y+3z=−1
4.
The shortest distance between the lines 1x−5=2y−2=−3z−4 and 1x+3=4y+4=−5z−1
(1)73
(2)53
(3)63
(4)43
5.
If the line passing through the points (5,1,a) and (3,b,1) crosses the YZ plane at the point (0,217,2−13), then a+b=
(1)12
(2)10
(3)8
(4)4
6.
Let a=i+2j+k and b=2i+7j+3k. Let L1:r=(−i+2j+k)+λa,λ∈R and L2:r (j+k)+μb,μ∈R be two lines. If the line L3 passes through the point of intersection of L1 and L2, and is parallel to a+b, then L3 passes through the point:
(1)(5,17,4)
(2)(2,8,5)
(3)(8,26,12)
(4)(−1,−1,1)
7.
A plane cuts the coordinate axes X,Y,Z at A,B,C respectively such that the centroid of the △ABC is (6,6,3). Then the equation of that plane is
(1)x+y+z−6=0
(2)x+2y+z−18=0
(3)2x+y+z−18=0
(4)x+y+2z−18=0
8.
If the distance of point 2i+3j+λk from the plane r⋅(3i+2j+6k)=13 is 5 units then λ=
(1)6,−317
(2)6,317
(3)−6,−317
(4)−6,317
9.
Let the plane P : 4x - y + z = 10 be rotated by an angle 2π about its line of intersection with the plane x + y - z = 4. If α is the distance of the point (2, 3, -4) from the new position of the plane P. then 35α is equal to
(1)85
(2)105
(3)126
(4)90
10.
The shortest distance between the skew lines r=(6i+2j+2k)+t(i−2j+2k) and r=(−4i−k)+s(3i−2j−2k) is
(1)9
(2)740
(3)108
(4)120
11.
For real numbers α and β ≠ 0, if the point of intersection of the straight lines 1x−α=2y−1=3z−1 and βx−4=3y−6=3z−7,lies on the plane x + 2y − z = 8, then α − β is equal to :
(1)5
(2)9
(3)3
(4)7
12.
The distance of the point (1,2,1) from the line 2x−1=1y−2=2z−3 is
(1)523
(2)35
(3)325
(4)320
13.
If A and B are foot of perpendicular drawn from point Q(a,b,c) to the planes yz and zx, then equation of plane through the points A,B and O is
(1)ax+by−cz=0
(2)ax−by+cz=0
(3)ax−by−cz=0
(4)ax+by+cz=0
14.
A plane which bisects the angle between the two given planes 2x−y+2z−4=0 and x+2y+2z−2=0 passes through the point :
(1)(1,−4,1)
(2)(1,4,−1)
(3)(2,4,1)
(4)(2,−4,1)
15.
Find the co-ordinates of the foot of the perpendicular drawn from the origin to the plane 5y+8=0.
(1)(0,−518,2)
(2)(0,58,2)
(3)(258,0,0)
(4)(0,−58,0)
16.
Let L1:1x−1=−1y−2=2z−1 and L2:−1x+1=2y−2=1z be two lines. Let L3 be a line passing through the point (α,β,γ) and be perpendicular to both L1 and L2. If L3 intersects L1, then |5α−11β−8γ| equals:
(1)20
(2)18
(3)25
(4)16
17.
If (2,−1,2) and (K,3,5) are the triads of direction ratios of two lines and the angle between them is 45∘, then the value of K is
(1)2
(2)3
(3)4
(4)6
18.
A line L is parallel to both the planes 2x+3y+z=1 and x+3y+2z=2. If line L makes an angle α with the positive direction of X -axis, then cosα=
(1)31
(2)21
(3)21
(4)23
19.
If the shortest distance between the lines 2x−1=3y−2=λz−3 and 1x−2=4y−4=5z−5 is 31, the sum of possible value of λ is
(1)16
(2)11
(3)12
(4)15
20.
The equation of the plane through the points (2,1,0),(3,2,−2) and (3,1,7) is
(1)6x−3y+2z−7=0
(2)3x−2y+6z−27=0
(3)7x−9y−z−5=0
(4)2x−3y+4z−27=0
21.
The coordinates of the point (3,−7,5) in the new system, when the origin is shifted to the point (−1,−1,−1) by the translation coordinate of axes, is
(1)(4,−6,6)
(2)(4,6,6)
(3)(6,6,6)
(4)(4,4,4)
22.
A point on the plane that passes through the points (1,−1, 6),(0,0,7) and perpendicular to the plane x−2y+z=6 is
(1)(1,−1,2)
(2)(1,1,2)
(3)(−1,1,2)
(4)(1,1,−2)
23.
The mid points of the sides of triangle are (1,5,−1),(0,4,−2) and (2,3,4), then centroid of the triangle is
(1)(1,4,3)
(2)(1,4,31)
(3)(−1,4,3)
(4)(31,2,4)
24.
If the line passes through the points P(6,−1,2), Q(8,−7,2λ) and R(5,2,4) then value of λ is
(1)-3
(2)0
(3)-1
(4)2
25.
A line with positive direction cosines passes through the point P(2,−1,2) and makes equal angles with the co-ordinate axes. The line meets the plane 2x+y+z=9 at point Q. The length of the line scgment PQ equals
(1)3
(2)2
(3)3
(4)2
26.
Foot of perpendicular drawn from the origin to the plane 2x−3y+4z=29 is
(1)(2,−3,4)
(2)(5,−2,3)
(3)(5,−1,4)
(4)(7,−1,3)
27.
Equations of planes parallel to the plane x−2y+2z+4=0 which are at a distance of one unit from the point (1,2,3) are
(1)x+2y+2z=−6,x+2y+2z=5
(2)x−2y−6=0,x−2y+z=6
(3)x+2y+2z=6,x+2y+2z=0
(4)x−2y+2z=0,x−2y+2z−6=0
28.
If the equation of the plane passing through the point (1,1,2) and perpendicular to the line x−3y+2z−1=04x−y+z os Ax+By+Cz=1, then 140(C−B+A) is equal to ________
29.
The shortest distance between the lines 3x−2=2y+1=2z−6 and 3x−6=21−y=0z+8 is equal to _________
30.
Let a→,b→ and c→ be three vectors such that |a→|,=3,|b→|=5,b→⋅c→=10 and angle between b→ and c→ is 3π. If a→ is perpendicular to the vectorb→×c→ then ∣a→×(b→×c→) is equal to is equal to
