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JEE Main · Mathematics · Three Dimensional Geometry

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1
[JEE Main 2026, 8 Apr Shift 2, Q15] Let the foot of perpendicular from the point (λ,2,3)(\lambda,2,3) on the line x−41=y−92=z−51\frac{x-4}{1}=\frac{y-9}{2}=\frac{z-5}{1} be the point (1,μ,2)(1,\mu,2). Then the distance between the lines x−12=y−23=z+46\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6} and x−λ2=y−μ3=z+56\frac{x-\lambda}{2}=\frac{y-\mu}{3}=\frac{z+5}{6} is equal to:
Not attempted
A
127\frac{12}{7}
B
1457\frac{\sqrt{145}}{7}
C
1467\frac{\sqrt{146}}{7}
D
1437\frac{\sqrt{143}}{7}
2
[JEE Main 2026, 8 Apr Shift 2, Q23] Let a line L1L_1 pass through the origin and be perpendicular to the lines L2:r⃗=(3+t)i^+(2t−1)j^+(2t+4)k^L_2:\vec r=(3+t)\hat i+(2t-1)\hat j+(2t+4)\hat k and L3:r⃗=(3+2s)i^+(3+2s)j^+(2+s)k^L_3:\vec r=(3+2s)\hat i+(3+2s)\hat j+(2+s)\hat k, t,s∈Rt,s\in\mathbb{R}. If (a,b,c)(a,b,c), a∈Za\in\mathbb{Z}, is the point on L3L_3 at a distance of 17\sqrt{17} from the point of intersection of L1L_1 and L2L_2, then (a+b+c)2(a+b+c)^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 4
3
[JEE Main 2026, 6 Apr Shift 2, Q14] The shortest distance between the lines x−41=y−32=z−2−3\frac{x-4}{1}=\frac{y-3}{2}=\frac{z-2}{-3} and x+22=y−64=z−5−5\frac{x+2}{2}=\frac{y-6}{4}=\frac{z-5}{-5} is:
Not attempted
A
566\frac{5\sqrt6}{6}
B
252\sqrt5
C
353\sqrt5
D
454\sqrt5
4
[JEE Main 2026, 6 Apr Shift 2, Q24] Let the image of the point P(0,−5,0)P(0,-5,0) in the line x−12=y1=z+1−2\frac{x-1}{2}=\frac y1=\frac{z+1}{-2} be the point R and the image of the point Q(0,−12,0)Q\left(0,-\frac12,0\right) in the line x−1−1=y+94=z+11\frac{x-1}{-1}=\frac{y+9}{4}=\frac{z+1}{1} be the point S. Then the square of the area of the parallelogram PQRS is ______.
Not attempted
Your answer: —
Correct answer: 162
5
[JEE Main 2026, 6 Apr Shift 1, Q15] Let the image of the point P(1,6,a)P(1,6,a) in the line L:x1=y−12=z−a+1bL:\frac x1=\frac{y-1}{2}=\frac{z-a+1}{b}, b>0b>0, be (a3,0,a+c)\left(\frac a3,0,a+c\right). If S(α,β,γ)S(\alpha,\beta,\gamma), α>0\alpha>0, is the point on L such that the distance of S from the foot of perpendicular from the point P on L is 2142\sqrt{14}, then α+β+γ\alpha+\beta+\gamma is equal to:
Not attempted
A
1919
B
2020
C
2121
D
2222
6
[JEE Main 2026, 6 Apr Shift 1, Q16] Let a line L be perpendicular to both the lines L1:x+13=y+35=z+57L_1:\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7} and L2:x−21=y−44=z−67L_2:\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-6}{7}. If θ\theta is the acute angle between the lines L and L3:x−872=y−471=z2L_3:\frac{x-\frac87}{2}=\frac{y-\frac47}{1}=\frac z2, then tan⁡θ\tan\theta is equal to:
Not attempted
A
322\frac32\sqrt2
B
522\frac52\sqrt2
C
532\frac53\sqrt2
D
432\frac43\sqrt2
7
[JEE Main 2026, 5 Apr Shift 2, Q14] Let a triangle PQR be such that P and Q lie on the line x+38=y−42=z+12\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2} and are at a distance of 6 units from R(1,2,3)R(1,2,3). If (α,β,γ)(\alpha,\beta,\gamma) is the centroid of △PQR\triangle PQR, then α+β+γ\alpha+\beta+\gamma is equal to:
Not attempted
A
44
B
55
C
66
D
88
8
[JEE Main 2026, 5 Apr Shift 2, Q15] If the distance of the point (a,2,5)(a,2,5) from the image of the point (1,2,7)(1,2,7) in the line x1=y−11=z−22\frac x1=\frac{y-1}{1}=\frac{z-2}{2} is 4, then the sum of all possible values of a is equal to:
Not attempted
A
1111
B
99
C
66
D
44
9
[JEE Main 2026, 5 Apr Shift 1, Q13] The square of the distance of the point P(5,6,7)P(5,6,7) from the line x−22=y−53=z−24\frac{x-2}{2}=\frac{y-5}{3}=\frac{z-2}{4} is equal to:
Not attempted
A
33
B
55
C
66
D
88
10
[JEE Main 2026, 5 Apr Shift 1, Q15] The square of the distance of the point of intersection of the lines r⃗=(i^+j^−k^)+λ(ai^−j^)\vec r=(\hat i+\hat j-\hat k)+\lambda(a\hat i-\hat j), a≠0a\ne0 and r⃗=(4i^−k^)+μ(2i^+ak^)\vec r=(4\hat i-\hat k)+\mu(2\hat i+a\hat k) from the origin is:
Not attempted
A
55
B
1010
C
1717
D
2626
11
[JEE Main 2026, 4 Apr Shift 2, Q13] The shortest distance between the lines r⃗=(13i^+2j^+83k^)+λ(2i^−5j^+6k^)\vec r=\left(\frac13\hat i+2\hat j+\frac83\hat k\right)+\lambda(2\hat i-5\hat j+6\hat k) and r⃗=(−23i^−13k^)+μ(j^−k^)\vec r=\left(-\frac23\hat i-\frac13\hat k\right)+\mu(\hat j-\hat k), λ,μ∈R\lambda,\mu\in\mathbb{R}, is:
Not attempted
A
5\sqrt5
B
33
C
232\sqrt3
D
15\sqrt{15}
12
[JEE Main 2026, 4 Apr Shift 2, Q14] If (2α+1,α2−3α,α−12)\left(2\alpha+1,\alpha^2-3\alpha,\frac{\alpha-1}{2}\right) is the image of (α,2α,1)(\alpha,2\alpha,1) in the line x−23=y−12=z1\frac{x-2}{3}=\frac{y-1}{2}=\frac z1, then the possible value(s) of α\alpha is (are)
Not attempted
A
Only 3
B
Only 3 and −1-1
C
Only 3, 14\frac14 and −1-1
D
Only 3 and 14\frac14
13
[JEE Main 2026, 4 Apr Shift 1, Q14] A line with direction ratios 1,−1,21,-1,2 intersects the lines x2=y3=z+13\frac x2=\frac y3=\frac{z+1}{3} and x+1−1=y−21=z4\frac{x+1}{-1}=\frac{y-2}{1}=\frac z4 at the points P and Q, respectively. If the length of the line segment PQ is α\alpha, then 225α2225\alpha^2 is equal to:
Not attempted
A
10241024
B
10141014
C
11041104
D
12041204
14
[JEE Main 2026, 4 Apr Shift 1, Q15] The square of the distance of the point (−2,−8,6)(-2,-8,6) from the line x−11=y−12=z−1\frac{x-1}{1}=\frac{y-1}{2}=\frac{z}{-1} along the line x+51=y+5−1=z2\frac{x+5}{1}=\frac{y+5}{-1}=\frac z2 is equal to:
Not attempted
A
33
B
66
C
88
D
1212
15
[JEE Main 2026, 2 Apr Shift 2, Q15] Let the point A be the foot of perpendicular drawn from the point P(a,b,0)P(a,b,0) on the line x−12=y−21=z−α3\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-\alpha}{3}. If the midpoint of the line segment PA is (0,34,−14)\left(0,\frac34,-\frac14\right), then the value of a2+b2+α2a^2+b^2+\alpha^2 is equal to:
Not attempted
A
11
B
22
C
66
D
99
16
[JEE Main 2026, 2 Apr Shift 1, Q13] If the point of intersection of the lines x+13=y+a5=z+b+17\frac{x+1}{3}=\frac{y+a}{5}=\frac{z+b+1}{7} and x−21=y−b4=z−2a7\frac{x-2}{1}=\frac{y-b}{4}=\frac{z-2a}{7} lies on xy-plane, then the value of a+ba+b is:
Not attempted
A
22
B
55
C
77
D
99
17
[JEE Main 2026, 2 Apr Shift 1, Q15] Let a line L passing through the point (1,1,1)(1,1,1) be perpendicular to both the vectors 2i^+2j^+k^2\hat i+2\hat j+\hat k and i^+2j^+2k^\hat i+2\hat j+2\hat k. If P(a,b,c)P(a,b,c) is the foot of perpendicular from the origin on the line L, then the value of 34(a+b+c)34(a+b+c) is:
Not attempted
A
5050
B
8080
C
100100
D
120120
18
[JEE Main 2026, 28 Jan Shift 2, Q7] Let Q(a,b,c)Q(a,b,c) be the image of the point P(3,2,1)P(3,2,1) in the line x−11=y2=z−11\frac{x-1}{1}=\frac y2=\frac{z-1}{1}. Then the distance of Q from the line x−93=y−92=z−5−2\frac{x-9}{3}=\frac{y-9}{2}=\frac{z-5}{-2} is
Not attempted
A
88
B
77
C
66
D
55
19
[JEE Main 2026, 28 Jan Shift 2, Q25] If the distance of the point P(43,α,β)P(43,\alpha,\beta), β<0\beta<0, from the line r⃗=4i^−k^+μ(2i^+3k^)\vec r=4\hat i-\hat k+\mu(2\hat i+3\hat k), μ∈R\mu\in\mathbb{R} along a line with direction ratios 3,−1,03,-1,0 is 131013\sqrt{10}, then α2+β2\alpha^2+\beta^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 170
20
[JEE Main 2026, 28 Jan Shift 1, Q20] If the distances of the point (1,2,a)(1,2,a) from the line x−11=y2=z−11\frac{x-1}{1}=\frac y2=\frac{z-1}{1} along the lines L1:x−13=y−24=z−abL_1:\frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b} and L2:x−11=y−24=z−acL_2:\frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c} are equal, then a+b+ca+b+c is equal to
Not attempted
A
77
B
66
C
44
D
55
21
[JEE Main 2026, 24 Jan Shift 2, Q19] The sum of all values of α\alpha, for which the shortest distance between the lines x+1α=y−2−1=z−4−α\frac{x+1}{\alpha}=\frac{y-2}{-1}=\frac{z-4}{-\alpha} and xα=y−12=z−12α\frac{x}{\alpha}=\frac{y-1}{2}=\frac{z-1}{2\alpha} is 2\sqrt2, is
Not attempted
A
−8-8
B
66
C
−6-6
D
88
22
[JEE Main 2026, 24 Jan Shift 1, Q13] Let the lines L1:r⃗=i^+2j^+3k^+λ(2i^+3j^+4k^)L_1:\vec r=\hat i+2\hat j+3\hat k+\lambda(2\hat i+3\hat j+4\hat k), λ∈R\lambda\in\mathbb{R} and L2:r⃗=(4i^+j^)+μ(5i^+2j^+k^)L_2:\vec r=(4\hat i+\hat j)+\mu(5\hat i+2\hat j+\hat k), μ∈R\mu\in\mathbb{R}, intersect at the point R. Let P and Q be the points lying on lines L1L_1 and L2L_2, respectively, such that ∣PR→∣=29|\overrightarrow{PR}|=\sqrt{29} and ∣PQ→∣=473|\overrightarrow{PQ}|=\sqrt{\frac{47}{3}}. If the point P lies in the first octant, then 27(QR)227(QR)^2 is equal to
Not attempted
A
320320
B
340340
C
360360
D
348348
23
[JEE Main 2026, 24 Jan Shift 1, Q23] Let a line L passing through the point P(1,1,1)P(1,1,1) be perpendicular to the lines x−44=y−11=z−11\frac{x-4}{4}=\frac{y-1}{1}=\frac{z-1}{1} and x−171=y−711=z0\frac{x-17}{1}=\frac{y-71}{1}=\frac z0. Let the line L intersect the yz plane at the point Q. Another line parallel to L and passing through the point S(1,0,−1)S(1,0,-1) intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ____ .
Not attempted
Your answer: —
Correct answer: 6
24
[JEE Main 2026, 23 Jan Shift 2, Q23] If the image of the point P(a,2,a)P(a,2,a) in the line x2=y+a1=z1\frac x2=\frac{y+a}{1}=\frac z1 is Q and the image of Q in the line x−2b2=y−a1=z+2b−5\frac{x-2b}{2}=\frac{y-a}{1}=\frac{z+2b}{-5} is P, then a+ba+b is equal to ____ .
Not attempted
Your answer: —
Correct answer: 3
25
[JEE Main 2026, 23 Jan Shift 1, Q8] The vertices B and C of a triangle ABC lie on the line x1=1−y−2=z−23\frac x1=\frac{1-y}{-2}=\frac{z-2}{3}. The coordinates of A and B are (1,6,3)(1,6,3) and (4,9,α)(4,9,\alpha) respectively and C is at a distance of 10 units from B. The area (in sq. units) of △ABC\triangle ABC is :
Not attempted
A
151315\sqrt{13}
B
201320\sqrt{13}
C
5135\sqrt{13}
D
101310\sqrt{13}
26
[JEE Main 2026, 23 Jan Shift 1, Q15] Let the direction cosines of two lines satisfy the equations: 4l+m−n=04l+m-n=0 and 2mn+10nl+3lm=02mn+10nl+3lm=0. Then the cosine of the acute angle between these lines is :
Not attempted
A
10338\frac{10}{3\sqrt{38}}
B
10738\frac{10}{7\sqrt{38}}
C
1038\frac{10}{\sqrt{38}}
D
20338\frac{20}{3\sqrt{38}}
27
[JEE Main 2026, 22 Jan Shift 2, Q16] Let L be the line x+12=y+13=z+36\frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6} and let S be the set of all points (a,b,c)(a,b,c) on L, whose distance from the line x+12=y+13=z−90\frac{x+1}{2}=\frac{y+1}{3}=\frac{z-9}{0} along the line L is 7. Then ∑(a,b,c)∈S(a+b+c)\sum_{(a,b,c)\in S}(a+b+c) is equal to:
Not attempted
A
4040
B
2828
C
3434
D
66
28
[JEE Main 2026, 22 Jan Shift 1, Q1] If the image of the point P(1,2,a)P(1,2,a) in the line x−63=y−72=7−z2\frac{x-6}{3}=\frac{y-7}{2}=\frac{7-z}{2} is Q(5,b,c)Q(5,b,c), then a2+b2+c2a^2+b^2+c^2 is equal to
Not attempted
A
264264
B
293293
C
298298
D
283283
29
[JEE Main 2026, 22 Jan Shift 1, Q11] Let P(α,β,γ)P(\alpha,\beta,\gamma) be the point on the line x−12=y+1−3=z\frac{x-1}{2}=\frac{y+1}{-3}=z at a distance 4144\sqrt{14} from the point (1,−1,0)(1,-1,0) and nearer to the origin. Then the shortest distance, between the lines x−α1=y−β2=z−γ3\frac{x-\alpha}{1}=\frac{y-\beta}{2}=\frac{z-\gamma}{3} and x+52=y−101=z−31\frac{x+5}{2}=\frac{y-10}{1}=\frac{z-3}{1}, is equal to
Not attempted
A
4754\sqrt{\frac75}
B
4574\sqrt{\frac57}
C
7547\sqrt{\frac54}
D
2742\sqrt{\frac74}
30
[JEE Main 2026, 21 Jan Shift 2, Q3] Let the line L1L_1 be parallel to the vector −3i^+2j^+4k^-3\hat i+2\hat j+4\hat k and pass through the point (2,6,7)(2,6,7), and the line L2L_2 be parallel to the vector 2i^+j^+3k^2\hat i+\hat j+3\hat k and pass through the point (4,3,5)(4,3,5). If the line L3L_3 is parallel to the vector −3i^+5j^+16k^-3\hat i+5\hat j+16\hat k and intersects the lines L1L_1 and L2L_2 at the points C and D, respectively, then ∣CD→∣2|\overrightarrow{CD}|^2 is equal to:
Not attempted
A
312312
B
290290
C
171171
D
8989
31
[JEE Main 2026, 21 Jan Shift 2, Q10] Let the line LL pass through the point (−3,5,2)(-3,5,2) and make equal angles with the positive coordinate axes. If the distance of LL from the point (−2,r,1)(-2,r,1) is 143\sqrt{\frac{14}{3}}, then the sum of all possible values of rr is:
Not attempted
A
66
B
1010
C
1212
D
1616
32
[JEE Main 2026, 21 Jan Shift 1, Q2] Let (α,β,γ)(\alpha,\beta,\gamma) be the coordinates of the foot of the perpendicular drawn from the point (5,4,2)(5,4,2) on the line r⃗=(−i^+3j^+k^)+λ(2i^+3j^−k^)\vec r=(-\hat i+3\hat j+\hat k)+\lambda(2\hat i+3\hat j-\hat k). Then the length of the projection of the vector αi^+βj^+γk^\alpha\hat i+\beta\hat j+\gamma\hat k on the vector 6i^+2j^+3k^6\hat i+2\hat j+3\hat k is :
Not attempted
A
33
B
44
C
187\frac{18}{7}
D
157\frac{15}{7}
33
[JEE Main 2025, 8 Apr Shift 2, Q26] Let the values of λ\lambda for which the shortest distance between the lines x−12=y−23=z−34\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} and x−λ3=y−44=z−55\frac{x-\lambda }{3}=\frac{y-4}{4}=\frac{z-5}{5} is 16\frac{1}{\sqrt{6}} be λ1{\lambda}_1 and λ2{\lambda}_2 . Then the radius of the circle passing through the points (0,0),(λ1,λ2)(0,0),({\lambda}_1,{\lambda}_2) and (λ2,λ1)({\lambda}_2,{\lambda}_1) is
Not attempted
A
33
B
523\frac{5\sqrt{2}}{3}
C
23\frac{\sqrt{2}}{3}
D
44
34
[JEE Main 2025, 8 Apr Shift 2, Q49] Let the area of the triangle formed by the lines x+2=y−1=z,x−35=y−1=z−11x+2=y-1=z,\frac{x-3}{5}=\frac{y}{-1}=\frac{z-1}{1} and x−3=y−33=z−21\frac{x}{-3}=\frac{y-3}{3}=\frac{z-2}{1} be AA . Then A2A^2 is equal to ________.
Not attempted
Your answer: —
Correct answer: 56
35
[JEE Main 2025, 7 Apr Shift 2, Q35] If the equation of the line passing through the point (0,−12,0)(0,-\frac{1}{2},0) and perpendicular to the lines r⃗=λ(i^+aj^+bk^)\vec{r}=\lambda (\hat{i}+a\hat{j}+b\hat{k}) and r⃗=(i^−j^−6k^)+μ(−bi^+aj^+5k^)\vec{r}=(\hat{i}-\hat{j}-6\hat{k})+\mu (-b\hat{i}+a\hat{j}+5\hat{k}) is x−1−2=y+4d=z−c−4\frac{x-1}{-2}=\frac{y+4}{d}=\frac{z-c}{-4} , then a+b+c+da+b+c+d is equal to :
Not attempted
A
13
B
14
C
12
D
10
36
[JEE Main 2025, 7 Apr Shift 2, Q39] Consider the lines L1\mathrm{L}_{1} : x - 1 = y - 2 = z and L2\mathrm{L}_{2} : x - 2 = y = z - 1. Let the feet of the perpendiculars from the point P(5, 1, -3) on the lines L1\mathrm{L}_{1} and L2\mathrm{L}_{2} be Q and R respectively. If the area of the triangle PQR is A, then 4A2\mathrm{4A}^{2} is equal to :
Not attempted
A
151
B
147
C
139
D
143
37
[JEE Main 2025, 7 Apr Shift 1, Q40] Let the line L pass through (1,1,1)(1,1,1) and intersect the lines x−12=y+13=z−14\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4} and x−31=y−42=z1\frac{x-3}{1}=\frac{y-4}{2}=\frac{z}{1} . Then, which of the following points lies on the line LL ?
Not attempted
A
(7,15,13)(7,15,13)
B
(4,22,7)(4,22,7)
C
(10,−29,−50)(10,-29,-50)
D
(5,4,3)(5,4,3)
38
[JEE Main 2025, 7 Apr Shift 1, Q42] If the shortest distance between the lines x−12=y−23=z−34\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} and x1=yα=z−51\frac{x}{1}=\frac{y}{\alpha }=\frac{z-5}{1} is 56\frac{5}{\sqrt{6}} , then the sum of all possible values of α\alpha is
Not attempted
A
32\frac{3}{2}
B
33
C
−3-3
D
−32-\frac{3}{2}
39
[JEE Main 2025, 4 Apr Shift 2, Q39] Let A be the point of intersection of the lines L1:x−71=y−50=z−3−1L_1:\frac{x-7}{1}=\frac{y-5}{0}=\frac{z-3}{-1} and L2:x−13=y+34=z+75L_2:\frac{x-1}{3}=\frac{y+3}{4}=\frac{z+7}{5} . Let B and C be the points on the lines L1L_1 and L2L_2 respectively such that AB=AC=15AB=AC=\sqrt{15} . Then the square of the area of the triangle ABCABC is :
Not attempted
A
63
B
57
C
60
D
54
40
[JEE Main 2025, 4 Apr Shift 2, Q44] Let the values of p , for which the shortest distance between the lines x+13=y4=z5\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5} and r⃗=(pi^+2j^+k^)+λ(2i^+3j^+4k^)\vec{r}=(p\hat{i}+2\hat{j}+\hat{k})+\lambda (2\hat{i}+3\hat{j}+4\hat{k}) is 16\frac{1}{\sqrt{6}} , be a,b,(a<b)a,b,(a<b) . Then the length of the latus rectum of the ellipse x2a2+y2 b2=1\frac{x^2}{a^2}+\frac{y^2}{{\ b}^2}=1 is :
Not attempted
A
32\frac{3}{2}
B
9
C
18
D
23\frac{2}{3}
41
[JEE Main 2025, 4 Apr Shift 1, Q27] Let the shortest distance between the lines x−33=y−α−1=z−31\frac{x-3}{3}=\frac{y-\alpha }{-1}=\frac{z-3}{1} and x+3−3=y+72=z−β4\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-\beta }{4} be 3303\sqrt{30} . Then the positive value of 5α+β5\alpha +\beta is
Not attempted
A
42
B
40
C
48
D
46
42
[JEE Main 2025, 4 Apr Shift 1, Q40] Let AA and BB be two distinct points on the line L:x−63=y−72=z−7−2L:\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2} . Both AA and BB are at a distance 2172\sqrt{17} from the foot of perpendicular drawn from the point (1,2,3)(1,2,3) on the line LL . If OO is the origin, then OA⃗⋅OB⃗\vec{OA}\cdot \vec{OB} is equal to
Not attempted
A
49
B
21
C
47
D
62
43
[JEE Main 2025, 3 Apr Shift 2, Q28] Each of the angles β\beta and γ\gamma that a given line makes with the positive yy - and zz -axes, respectively, is half of the angle that this line makes with the positive xx -axes. Then the sum of all possible values of the angle β\beta is
Not attempted
A
π2\frac{\pi }{2}
B
π\pi
C
3π4\frac{3\pi }{4}
D
3π2\frac{3\pi }{2}
44
[JEE Main 2025, 3 Apr Shift 2, Q39] The distance of the point (7,10,11)(7,10,11) from the line x−41=y−40=z−23\frac{x-4}{1}=\frac{y-4}{0}=\frac{z-2}{3} along the line x−92=y−133=z−176\frac{x-9}{2}=\frac{y-13}{3}=\frac{z-17}{6} is
Not attempted
A
16
B
12
C
18
D
14
45
[JEE Main 2025, 3 Apr Shift 1, Q35] Let a line passing through the point (4,1,0)(4,1,0) intersect the line L1:x−12=y−23=z−34L_1:\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} at the point A(α,β,γ)A(\alpha ,\beta ,\gamma ) and the line L2:x−6=y=−z+4L_2:x-6=y=-z+4 at the point B(a,b,c)B(a,b,c) . Then ∣101αβγabc∣\begin{vmatrix}1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c\end{vmatrix} is equal to
Not attempted
A
16
B
6
C
8
D
12
46
[JEE Main 2025, 3 Apr Shift 1, Q38] Line L1L_1 passes through the point (1,2,3)(1,2,3) and is parallel to zz -axis. Line L2L_2 passes through the point (λ,5,6)(\lambda ,5,6) and is parallel to yy -axis. Let for λ=λ1,λ2,λ2<λ1\lambda ={\lambda}_1,{\lambda}_2,{\lambda}_2<{\lambda}_1 , the shortest distance between the two lines be 3 . Then the square of the distance of the point (λ1,λ2,7)({\lambda}_1,{\lambda}_2,7) from the line L1L_1 is
Not attempted
A
25
B
32
C
40
D
37
47
[JEE Main 2025, 2 Apr Shift 2, Q33] If the image of the point P(1,0,3)P(1,0,3) in the line joining the points A(4,7,1)A(4,7,1) and B(3,5,3)B(3,5,3) is Q(α,β,γ)Q(\alpha ,\beta ,\gamma ) , then α+β+γ\alpha +\beta +\gamma is equal to :
Not attempted
A
463\frac{46}{3}
B
18
C
13
D
473\frac{47}{3}
48
[JEE Main 2025, 2 Apr Shift 2, Q38] The line L1L_1 is parallel to the vector a⃗=−3i^+2j^+4k^\vec{a}=-3\hat{i}+2\hat{j}+4\hat{k} and passes through the point (7,6,2)(7,6,2) and the line L2L_2 is parallel to the vector b⃗=2i^+j^+3k^\vec{b}=2\hat{i}+\hat{j}+3\hat{k} and passes through the point (5,3,4)(5,3,4) . The shortest distance between the lines L1L_1 and L2L_2 is :
Not attempted
A
2338\frac{23}{\sqrt{38}}
B
2138\frac{21}{\sqrt{38}}
C
2357\frac{23}{\sqrt{57}}
D
2157\frac{21}{\sqrt{57}}
49
[JEE Main 2025, 2 Apr Shift 1, Q27] Let the vertices Q and R of the triangle PQR lie on the line x+35=y−12=z+43,QR=5\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3},\mathrm{QR}=5 and the coordinates of the point PP be (0,2,3)(0,2,3) . If the area of the triangle PQRPQR is mn\frac{m}{n} then :
Not attempted
A
2 m−521n=02\ m-5\sqrt{21}n=0
B
m−521n=0m-5\sqrt{21}n=0
C
5 m−212n=05\ m-21\sqrt{2}n=0
D
5 m−221n=05\ m-2\sqrt{21}n=0
50
[JEE Main 2025, 29 Jan Shift 2, Q28] Let a straight line LL pass through the point P(2,−1,3)P(2,-1,3) and be perpendicular to the lines x−12=y+11=z−3−2\frac{x-1}{2}=\frac{y+1}{1}=\frac{z-3}{-2} and x−31=y−23=z+24\frac{x-3}{1}=\frac{y-2}{3}=\frac{z+2}{4} . If the line LL intersects the yzyz -plane at the point QQ , then the distance between the points PP and QQ is:
Not attempted
A
10\sqrt{10}
B
22
C
232\sqrt{3}
D
33
51
[JEE Main 2025, 29 Jan Shift 2, Q43] Let P be the foot of the perpendicular from the point (1,2,2)(1,2,2) on the line L:x−11=y+1−1=z−22L:\frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2} . Let the line r⃗=(−i^+j^−2k^)+λ(i^−j^+k^),λ∈R\vec{r}=(-\hat{i}+\hat{j}-2\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k}),\lambda \in \mathbf{R} , intersect the line L at Q . Then 2(PQ)22(\mathrm{PQ})^2 is equal to :
Not attempted
A
25
B
27
C
19
D
29
52
[JEE Main 2025, 29 Jan Shift 1, Q26] Let L1:x−11=y−2−1=z−12L_1:\frac{x-1}{1}=\frac{y-2}{-1}=\frac{z-1}{2} and L2:x+1−1=y−22=z1L_2:\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z}{1} be two lines. Let L3L_3 be a line passing through the point (α,β,γ)(\alpha ,\beta ,\gamma ) and be perpendicular to both L1L_1 and L2L_2 . If L3L_3 intersects L1L_1 , then ∣5α−11β−8γ∣|5\alpha -11\beta -8\gamma | equals :
Not attempted
A
25
B
20
C
16
D
18
53
[JEE Main 2025, 29 Jan Shift 1, Q39] Let a⃗=i^+2j^+k^\vec{a}=\hat{i}+2\hat{j}+\hat{k} and b⃗=2i^+7j^+3k^\vec{b}=2\hat{i}+7\hat{j}+3\hat{k} . Let L1:r⃗=(−i^+2j^+k^)+λa⃗,λ∈RL_1:\vec{r}=(-\hat{i}+2\hat{j}+\hat{k})+\lambda \vec{a},\lambda \in \mathbf{R} and L2:r⃗=(j^+k^)+μb⃗,μ∈RL_2:\vec{r}=(\hat{j}+\hat{k})+\mu \vec{b},\mu \in R be two lines. If the line L3L_3 passes through the point of intersection of L1L_1 and LyL_y and is parallel to a⃗+b⃗\vec{a}+\vec{b} , then L3L_3 passes through the point :
Not attempted
A
(−1,−1,1)(-1,-1,1)
B
(2,8,5)(2,8,5)
C
(8,26,12)(8,26,12)
D
(5,17,4)(5,17,4)
54
[JEE Main 2025, 28 Jan Shift 2, Q30] The square of the distance of the point (157,327,7)(\frac{15}{7},\frac{32}{7},7) from the line x+13=y+35=z+57\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7} in the direction of the vector i^+4j^+7k^\hat{i}+4\hat{j}+7\hat{k} is:
Not attempted
A
66
B
54
C
41
D
44
55
[JEE Main 2025, 28 Jan Shift 1, Q26] Let A(x,y,z)A(x,y,z) be a point in xyxy -plane, which is equidistant from three points (0,3,2),(2,0,3)(0,3,2),(2,0,3) and (0,0,1)(0,0,1) . Let B=(1,4,−1)B=(1,4,-1) and C=(2,0,−2)C=(2,0,-2) . Then among the statements (S1) : △ABC\triangle \mathrm{ABC} is an isosceles right angled triangle, and (S2) : the area of △ABC\triangle \mathrm{ABC} is 922\frac{9\sqrt{2}}{2} ,
Not attempted
A
both are false
B
only (S2) is true
C
only (S1) is true
D
both are true
56
[JEE Main 2025, 28 Jan Shift 1, Q38] If the image of the point (4,4,3)(4,4,3) in the line x−12=y−21=z−13\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-1}{3} is (α,β,γ)(\alpha ,\beta ,\gamma ) , then α+β+γ\alpha +\beta +\gamma is equal to
Not attempted
A
12
B
9
C
7
D
8
57
[JEE Main 2025, 24 Jan Shift 2, Q46] Let P be the image of the point Q(7,−2,5)Q(7,-2,5) in the line L:x−12=y+13=z4L:\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4} and R(5,p,q)R(5,p,q) be a point on LL . Then the square of the area of △PQR\triangle PQR is _________.
Not attempted
Your answer: —
Correct answer: 957
58
[JEE Main 2025, 24 Jan Shift 1, Q35] Let in a △ABC\triangle ABC , the length of the side ACAC be 6 , the vertex BB be (1,2,3)(1,2,3) and the vertices A,CA,C lie on the line x−63=y−72=z−7−2\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2} . Then the area (in sq. units) of △ABC\triangle ABC is:
Not attempted
A
42
B
17
C
56
D
21
59
[JEE Main 2025, 24 Jan Shift 1, Q36] Let the line passing through the points (−1,2,1)(-1,2,1) and parallel to the line x−12=y+13=z4\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4} intersect the line x+23=y−32=z−41\frac{x+2}{3}=\frac{y-3}{2}=\frac{z-4}{1} at the point PP . Then the distance of PP from the point Q(4,−5,1)Q(4,-5,1) is
Not attempted
A
565\sqrt{6}
B
55
C
555\sqrt{5}
D
1010
60
[JEE Main 2025, 23 Jan Shift 2, Q31] If the square of the shortest distance between the lines x−21=y−12=z+3−3\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3} and x+12=y+34=z+5−5\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5} is mn\frac{m}{n} , where mm , nn are coprime numbers, then m+nm+n is equal to :
Not attempted
A
14
B
6
C
21
D
9
61
[JEE Main 2025, 23 Jan Shift 2, Q36] The distance of the line x−22=y−63=z−34\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4} from the point (1,4,0)(1,4,0) along the line x1=y−22=z+33\frac{x}{1}=\frac{y-2}{2}=\frac{z+3}{3} is :
Not attempted
A
17\sqrt{17}
B
13\sqrt{13}
C
15\sqrt{15}
D
14\sqrt{14}
62
[JEE Main 2025, 23 Jan Shift 1, Q27] Let P be the foot of the perpendicular from the point Q(10,−3,−1)Q(10,-3,-1) on the line x−37=y−2−1=z+1−2\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z+1}{-2} . Then the area of the right angled triangle PQRPQR , where RR is the point (3,−2,1)(3,-2,1) , is
Not attempted
A
30\sqrt{30}
B
9159\sqrt{15}
C
3303\sqrt{30}
D
8158\sqrt{15}
63
[JEE Main 2025, 22 Jan Shift 2, Q27] Let a line pass through two distinct points P(−2,−1,3)P(-2,-1,3) and QQ , and be parallel to the vector 3i^+2j^+2k^3\hat{i}+2\hat{j}+2\hat{k} . If the distance of the point Q from the point R(1,3,3)R(1,3,3) is 5 , then the square of the area of △PQR\triangle PQR is equal to :
Not attempted
A
148
B
144
C
136
D
140
64
[JEE Main 2025, 22 Jan Shift 2, Q31] The perpendicular distance, of the line x−12=y+2−1=z+32\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z+3}{2} from the point P(2,−10,1)P(2,-10,1) , is :
Not attempted
A
66
B
434\sqrt{3}
C
353\sqrt{5}
D
525\sqrt{2}
65
[JEE Main 2025, 22 Jan Shift 1, Q28] Let L1:x−12=y−23=z−34L_1:\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} and L2:x−23=y−44=z−55L_2:\frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5} be two lines. Then which of the following points lies on the line of the shortest distance between L1L_1 and L2L_2 ?
Not attempted
A
(143,−3,223)(\frac{14}{3},-3,\frac{22}{3})
B
(2,3,13)(2,3,\frac{1}{3})
C
(83,−1,13)(\frac{8}{3},-1,\frac{1}{3})
D
(−53,−7,1)(-\frac{5}{3},-7,1)
66
[JEE Main 2025, 22 Jan Shift 1, Q48] Let L1:x−13=y−1−1=z+10L_1:\frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0} and L2:x−22=y0=z+4α,α∈RL_2:\frac{x-2}{2}=\frac{y}{0}=\frac{z+4}{\alpha },\alpha \in \mathbf{R} , be two lines, which intersect at the point BB . If PP is the foot of perpendicular from the point A(1,1,−1)A(1,1,-1) on L2L_2 , then the value of 26α( PB)226\alpha (\ \mathrm{PB})^2 is _________ .
Not attempted
Your answer: —
Correct answer: 216
67
[JEE Main 2024, 9 Apr Shift 2, Q46] Consider the line LL passing through the points (1,2,3)(1,2,3) and (2,3,5)(2,3,5) . The distance of the point (113,113,193)(\frac{11}{3},\frac{11}{3},\frac{19}{3}) from the line LL along the line 3x−112=3y−111=3z−192\frac{3x-11}{2}=\frac{3y-11}{1}=\frac{3z-19}{2} is equal to
Not attempted
A
6
B
3
C
5
D
4
68
[JEE Main 2024, 9 Apr Shift 2, Q56] The square of the distance of the image of the point (6,1,5)(6,1,5) in the line x−13=y2=z−24\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4} , from the origin is __________.
Not attempted
Your answer: —
Correct answer: 62
69
[JEE Main 2024, 9 Apr Shift 1, Q44] The shortest distance between the lines x−34=y+7−11=z−15\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5} and x−53=y−9−6=z+21\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1} is:
Not attempted
A
185563\frac{185}{\sqrt{563}}
B
187563\frac{187}{\sqrt{563}}
C
178563\frac{178}{\sqrt{563}}
D
179563\frac{179}{\sqrt{563}}
70
[JEE Main 2024, 9 Apr Shift 1, Q49] Let the line LL intersect the lines x−2=−y=z−1,2(x+1)=2(y−1)=z+1x-2=-y=z-1,2(x+1)=2(y-1)=z+1 and be parallel to the line x−23=y−11=z−22\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2} . Then which of the following points lies on LL ?
Not attempted
A
(−13,1,−1)(-\frac{1}{3},1,-1)
B
(−13,−1,1)(-\frac{1}{3},-1,1)
C
(−13,−1,−1)(-\frac{1}{3},-1,-1)
D
(−13,1,1)(-\frac{1}{3},1,1)
71
[JEE Main 2024, 8 Apr Shift 2, Q32] If the shortest distance between the lines x−λ2=y−43=z−34\frac{x-\lambda }{2}=\frac{y-4}{3}=\frac{z-3}{4} and x−24=y−46=z−78\frac{x-2}{4}=\frac{y-4}{6}=\frac{z-7}{8} is 1329\frac{13}{\sqrt{29}} , then a value of λ\lambda is :
Not attempted
A
1325\frac{13}{25}
B
1
C
−- 1
D
−1325-\frac{13}{25}
72
[JEE Main 2024, 8 Apr Shift 2, Q55] Let P(α,β,γ)P(\alpha ,\beta ,\gamma ) be the image of the point Q(1,6,4)Q(1,6,4) in the line x1=y−12=z−23\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} . Then 2α+β+γ2\alpha +\beta +\gamma is equal to ________
Not attempted
Your answer: —
Correct answer: 11
73
[JEE Main 2024, 8 Apr Shift 1, Q42] Let P(x,y,z)P(x,y,z) be a point in the first octant, whose projection in the xyxy -plane is the point QQ . Let OP=γOP=\gamma ; the angle between OQOQ and the positive xx -axis be θ\theta ; and the angle between OPOP and the positive zz -axis be ϕ\phi , where OO is the origin. Then the distance of PP from the xx -axis is
Not attempted
A
γ1−sin⁡2ϕcos⁡2θ\gamma \sqrt{1-{\sin}^2\phi {\cos}^2\theta }
B
γ1+cos⁡2θsin⁡2ϕ\gamma \sqrt{1+{\cos}^2\theta {\sin}^2\phi }
C
γ1+cos⁡2ϕsin⁡2θ\gamma \sqrt{1+{\cos}^2\phi {\sin}^2\theta }
D
γ1−sin⁡2θcos⁡2ϕ\gamma \sqrt{1-{\sin}^2\theta {\cos}^2\phi }
74
[JEE Main 2024, 8 Apr Shift 1, Q49] If the shortest distance between the lines L1:r⃗=(2+λ)i^+(1−3λ)j^+(3+4λ)k^,λ∈RL2:r⃗=2(1+μ)i^+3(1+μ)j^+(5+μ)k^,μ∈R\begin{array}{cc}L_1:\vec{r}=(2+\lambda )\hat{i}+(1-3\lambda )\hat{j}+(3+4\lambda )\hat{k}, & \lambda \in \mathbb{R} \\ L_2:\vec{r}=2(1+\mu )\hat{i}+3(1+\mu )\hat{j}+(5+\mu )\hat{k}, & \mu \in \mathbb{R}\end{array} is mn\frac{m}{\sqrt{n}} , where gcd⁡(m,n)=1\gcd (m,n)=1 , then the value of m+nm+n equals
Not attempted
A
384
B
387
C
390
D
377
75
[JEE Main 2024, 6 Apr Shift 2, Q40] Let P(α,β,γ)P(\alpha ,\beta ,\gamma ) be the image of the point Q(3,−3,1)Q(3,-3,1) in the line x−01=y−31=z−1−1\frac{x-0}{1}=\frac{y-3}{1}=\frac{z-1}{-1} and RR be the point (2,5,−1)(2,5,-1) . If the area of the triangle PQR\mathrm{PQR} is λ\lambda and λ2=14 K{\lambda}^2=14\ K , then KK is equal to :
Not attempted
A
18
B
81
C
72
D
36
76
[JEE Main 2024, 6 Apr Shift 2, Q54] If the shortest distance between the lines x−λ3=y−2−1=z−11\frac{x-\lambda }{3}=\frac{y-2}{-1}=\frac{z-1}{1} and x+2−3=y+52=z−44\frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4} is 4430\frac{44}{\sqrt{30}} , then the largest possible value of ∣λ∣|\lambda | is equal to _________.
Not attempted
Your answer: —
Correct answer: 43
77
[JEE Main 2024, 6 Apr Shift 1, Q39] The shortest distance between the lines x−32=y+15−7=z−95\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5} and x+12=y−11=z−9−3\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3} is
Not attempted
A
838\sqrt{3}
B
636\sqrt{3}
C
535\sqrt{3}
D
434\sqrt{3}
78
[JEE Main 2024, 6 Apr Shift 1, Q55] Let PP be the point (10,−2,−1)(10,-2,-1) and QQ be the foot of the perpendicular drawn from the point R(1,7,6)R(1,7,6) on the line passing through the points (2,−5,11)(2,-5,11) and (−6,7,−5)(-6,7,-5) . Then the length of the line segment PQPQ is equal to _________.
Not attempted
Your answer: —
Correct answer: 13
79
[JEE Main 2024, 5 Apr Shift 2, Q34] Let (α,β,γ)(\alpha ,\beta ,\gamma ) be the image of the point (8,5,7)(8,5,7) in the line x−12=y+13=z−25\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{5} . Then α+β+γ\alpha +\beta +\gamma is equal to :
Not attempted
A
16
B
20
C
18
D
14
80
[JEE Main 2024, 5 Apr Shift 2, Q54] Let the point (−1,α,β)(-1,\alpha ,\beta ) lie on the line of the shortest distance between the lines x+2−3=y−24=z−52\frac{x+2}{-3}=\frac{y-2}{4}=\frac{z-5}{2} and x+2−1=y+62=z−10\frac{x+2}{-1}=\frac{y+6}{2}=\frac{z-1}{0} . Then (α−β)2(\alpha -\beta )^2 is equal to _________.
Not attempted
Your answer: —
Correct answer: 25
81
[JEE Main 2024, 5 Apr Shift 1, Q34] If the line 2−x3=3y−24λ+1=4−z\frac{2-x}{3}=\frac{3y-2}{4\lambda +1}=4-z makes a right angle with the line x+33μ=1−2y6=5−z7\frac{x+3}{3\mu }=\frac{1-2y}{6}=\frac{5-z}{7} , then 4λ+9μ4\lambda +9\mu is equal to :
Not attempted
A
4
B
13
C
5
D
6
82
[JEE Main 2024, 5 Apr Shift 1, Q43] Let dd be the distance of the point of intersection of the lines x+63=y2=z+11\frac{x+6}{3}=\frac{y}{2}=\frac{z+1}{1} and x−74=y−93=z−42\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2} from the point (7,8,9)(7,8,9) . Then d2+6d^2+6 is equal to :
Not attempted
A
75
B
78
C
72
D
69
83
[JEE Main 2024, 4 Apr Shift 2, Q35] Let PP be the point of intersection of the lines x−21=y−45=z−21\frac{x-2}{1}=\frac{y-4}{5}=\frac{z-2}{1} and x−32=y−23=z−32\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-3}{2} . Then, the shortest distance of PP from the line 4x=2y=z4x=2y=z is
Not attempted
A
3147\frac{3\sqrt{14}}{7}
B
5147\frac{5\sqrt{14}}{7}
C
147\frac{\sqrt{14}}{7}
D
6147\frac{6\sqrt{14}}{7}
84
[JEE Main 2024, 4 Apr Shift 2, Q59] Consider a line LL passing through the points P(1,2,1)P(1,2,1) and Q(2,1,−1)Q(2,1,-1) . If the mirror image of the point A(2,2,2)A(2,2,2) in the line LL is (α,β,γ)(\alpha ,\beta ,\gamma ) , then α+β+6γ\alpha +\beta +6\gamma is equal to __________.
Not attempted
Your answer: —
Correct answer: 6
85
[JEE Main 2024, 4 Apr Shift 1, Q40] Let the point, on the line passing through the points P(1,−2,3)P(1,-2,3) and Q(5,−4,7)Q(5,-4,7) , farther from the origin and at a distance of 9 units from the point PP , be (α,β,γ)(\alpha ,\beta ,\gamma ) . Then α2+β2+γ2{\alpha}^2+{\beta}^2+{\gamma}^2 is equal to :
Not attempted
A
150
B
155
C
160
D
165
86
[JEE Main 2024, 4 Apr Shift 1, Q52] If the shortest distance between the lines x+22=y+33=z−54\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4} and x−31=y−2−3=z+42\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2} is 3835k\frac{38}{3\sqrt{5}}k , and ∫0k[x2]dx=α−α{\int}_0^k[x^2]dx=\alpha -\sqrt{\alpha } , where [x][x] denotes the greatest integer function, then 6α36{\alpha}^3 is equal to _________.
Not attempted
Your answer: —
Correct answer: 48
87
[JEE Main 2024, 1 Feb Shift 2, Q45] Let PP and QQ be the points on the line x+38=y−42=z+12\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2} which are at a distance of 6 units from the point R(1,2,3)R(1,2,3) . If the centroid of the triangle PQR is (α,β,γ)(\alpha ,\beta ,\gamma ) , then α2+β2+γ2{\alpha}^2+{\beta}^2+{\gamma}^2 is :
Not attempted
A
18
B
24
C
26
D
36
88
[JEE Main 2024, 1 Feb Shift 2, Q47] If the mirror image of the point P(3,4,9)P(3,4,9) in the line x−13=y+12=z−21\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-2}{1} is (α,β,γ)(\alpha ,\beta ,\gamma ) , then 14 (α+β+γ)(\alpha +\beta +\gamma ) is :
Not attempted
A
102
B
138
C
132
D
108
89
[JEE Main 2024, 1 Feb Shift 1, Q50] If the shortest distance between the lines x−λ−2=y−21=z−11\frac{x-\lambda }{-2}=\frac{y-2}{1}=\frac{z-1}{1} and x−31=y−1−2=z−21\frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1} is 1 , then the sum of all possible values of λ\lambda is :
Not attempted
A
0
B
232\sqrt{3}
C
333\sqrt{3}
D
−23-2\sqrt{3}
90
[JEE Main 2024, 1 Feb Shift 1, Q59] Let the line of the shortest distance between the lines L1:r⃗=(i^+2j^+3k^)+λ(i^−j^+k^) and L2:r⃗=(4i^+5j^+6k^)+μ(i^+j^−k^)\begin{array}{cc} & L_1:\vec{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})\text{ and } \\ \\ & L_2:\vec{r}=(4\hat{i}+5\hat{j}+6\hat{k})+\mu (\hat{i}+\hat{j}-\hat{k})\end{array} intersect L1L_1 and L2L_2 at PP and QQ respectively. If (α,β,γ)(\alpha ,\beta ,\gamma ) is the mid point of the line segment PQ\mathrm{PQ} , then 2(α+β+γ)2(\alpha +\beta +\gamma ) is equal to ____________.
Not attempted
Your answer: —
Correct answer: 21
91
[JEE Main 2024, 31 Jan Shift 2, Q34] Let (α,β,γ)(\alpha ,\beta ,\gamma ) be the mirror image of the point (2,3,5)(2,3,5) in the line x−12=y−23=z−34\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} . Then, 2α+3β+4γ2\alpha +3\beta +4\gamma is equal to
Not attempted
A
32
B
31
C
33
D
34
92
[JEE Main 2024, 31 Jan Shift 2, Q38] The shortest distance, between lines L1L_1 and L2L_2 , where L1:x−12=y+1−3=z+42L_1:\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2} and L2L_2 is the line, passing through the points A(−4,4,3),B(−1,6,3)A(-4,4,3),B(-1,6,3) and perpendicular to the line x−3−2=y3=z−11\frac{x-3}{-2}=\frac{y}{3}=\frac{z-1}{1} , is
Not attempted
A
141221\frac{141}{\sqrt{221}}
B
24117\frac{24}{\sqrt{117}}
C
42117\frac{42}{\sqrt{117}}
D
121221\frac{121}{\sqrt{221}}
93
[JEE Main 2024, 31 Jan Shift 2, Q60] A line passes through A(4,−6,−2)A(4,-6,-2) and B(16,−2,4)B(16,-2,4) . The point P(a,b,c)P(a,b,c) , where a,b,ca,b,c are non-negative integers, on the line ABAB lies at a distance of 21 units, from the point AA . The distance between the points P(a,b,c)P(a,b,c) and Q(4,−12,3)Q(4,-12,3) is equal to __________.
Not attempted
Your answer: —
Correct answer: 22
94
[JEE Main 2024, 31 Jan Shift 1, Q42] The distance of the point Q(0,2,−2)Q(0,2,-2) form the line passing through the point P(5,−4,3)P(5,-4,3) and perpendicular to the lines r⃗=(−3i^+2k^)+λ(2i^+3j^+5k^),λ∈R\vec{r}=(-3\hat{i}+2\hat{k})+\lambda (2\hat{i}+3\hat{j}+5\hat{k}),\lambda \in \mathbb{R} and r⃗=(i^−2j^+k^)+μ(−i^+3j^+2k^),μ∈R\vec{r}=(\hat{i}-2\hat{j}+\hat{k})+\mu (-\hat{i}+3\hat{j}+2\hat{k}),\mu \in \mathbb{R} is :
Not attempted
A
74\sqrt{74}
B
86\sqrt{86}
C
54\sqrt{54}
D
20\sqrt{20}
95
[JEE Main 2024, 31 Jan Shift 1, Q56] Let QQ and RR be the feet of perpendiculars from the point P(a,a,a)P(a,a,a) on the lines x=y,z=1x=y,z=1 and x=−y,z=−1x=-y,z=-1 respectively. If ∠QPR\angle \mathrm{QPR} is a right angle, then 12a212a^2 is equal to _________.
Not attempted
Your answer: —
Correct answer: 12
96
[JEE Main 2024, 30 Jan Shift 2, Q41] Let L1:r⃗=(i^−j^+2k^)+λ(i^−j^+2k^),λ∈RL_1:\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\lambda (\hat{i}-\hat{j}+2\hat{k}),\lambda \in \mathbb{R} , L2:r⃗=(j^−k^)+μ(3i^+j^+pk^),μ∈R, and L3:r⃗=δ(ℓi^+mj^+nk^),δ∈RL_2:\vec{r}=(\hat{j}-\hat{k})+\mu (3\hat{i}+\hat{j}+p\hat{k}),\mu \in \mathbb{R}\text{, and }L_3:\vec{r}=\delta (\ell \hat{i}+m\hat{j}+n\hat{k}),\delta \in \mathbb{R} be three lines such that L1L_1 is perpendicular to L2L_2 and L3L_3 is perpendicular to both L1L_1 and L2L_2 . Then, the point which lies on L3L_3 is
Not attempted
A
(1,7,−4)(1,7,-4)
B
(1,−7,4)(1,-7,4)
C
(−1,7,4)(-1,7,4)
D
(−,1−7,4)(-,1-7,4)
97
[JEE Main 2024, 30 Jan Shift 2, Q55] Let a line passing through the point (−1,2,3)(-1,2,3) intersect the lines L1:x−13=y−22=z+1−2L_1:\frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2} at M(α,β,γ)M(\alpha ,\beta ,\gamma ) and L2:x+2−3=y−2−2=z−14L_2:\frac{x+2}{-3}=\frac{y-2}{-2}=\frac{z-1}{4} at N(a,b,c)N(a,b,c) . Then, the value of (α+β+γ)2(a+b+c)2\frac{(\alpha +\beta +\gamma )^2}{(a+b+c)^2} equals __________.
Not attempted
Your answer: —
Correct answer: 196
98
[JEE Main 2024, 30 Jan Shift 1, Q37] Let (α,β,γ)(\alpha ,\beta ,\gamma ) be the foot of perpendicular from the point (1,2,3)(1,2,3) on the line x+35=y−12=z+43\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3} . Then 19(α+β+γ)19(\alpha +\beta +\gamma ) is equal to :
Not attempted
A
99
B
102
C
101
D
100
99
[JEE Main 2024, 30 Jan Shift 1, Q57] If d1d_1 is the shortest distance between the lines x+1=2y=−12z,x=y+2=6z−6x+1=2y=-12z,x=y+2=6z-6 and d2d_2 is the shortest distance between the lines x−12=y+8−7=z−45,x−12=y−21=z−6−3\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5},\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3} , then the value of 323 d1 d2\frac{32\sqrt{3}{\ d}_1}{{\ d}_2} is :
Not attempted
Your answer: —
Correct answer: 16
100
[JEE Main 2024, 29 Jan Shift 2, Q51] Let O be the origin, and M and NN be the points on the lines x−54=y−41=z−53\frac{x-5}{4}=\frac{y-4}{1}=\frac{z-5}{3} and x+812=y+25=z+119\frac{x+8}{12}=\frac{y+2}{5}=\frac{z+11}{9} respectively such that MN\mathrm{MN} is the shortest distance between the given lines. Then OM⃗⋅ON⃗\vec{OM}\cdot \vec{ON} is equal to _________.
Not attempted
Your answer: —
Correct answer: 9
101
[JEE Main 2024, 29 Jan Shift 1, Q47] Let PQRPQR be a triangle with R(−1,4,2)R(-1,4,2) . Suppose M(2,1,2)M(2,1,2) is the mid point of PQ\mathrm{PQ} . The distance of the centroid of △PQR\triangle \mathrm{PQR} from the point of intersection of the lines x−20=y2=z+3−1\frac{x-2}{0}=\frac{y}{2}=\frac{z+3}{-1} and x−11=y+3−3=z+11\frac{x-1}{1}=\frac{y+3}{-3}=\frac{z+1}{1} is
Not attempted
A
69
B
99\sqrt{99}
C
69\sqrt{69}
D
9
102
[JEE Main 2024, 29 Jan Shift 1, Q54] A line with direction ratios 2,1,22,1,2 meets the lines x=y+2=zx=y+2=z and x+2=2y=2zx+2=2y=2z respectively at the points PP and QQ . If the length of the perpendicular from the point (1,2,12)(1,2,12) to the line PQ\mathrm{PQ} is ll , then l2l^2 is __________.
Not attempted
Your answer: —
Correct answer: 65
103
[JEE Main 2024, 27 Jan Shift 2, Q35] Let the image of the point (1,0,7)(1,0,7) in the line x1=y−12=z−23\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} be the point (α,β,γ)(\alpha ,\beta ,\gamma ) . Then which one of the following points lies on the line passing through (α,β,γ)(\alpha ,\beta ,\gamma ) and making angles 2π3\frac{2\pi }{3} and 3π4\frac{3\pi }{4} with yy -axis and zz -axis respectively and an acute angle with xx -axis ?
Not attempted
A
(1,−2,1+2)(1,-2,1+\sqrt{2})
B
(3,−4,3+22)(3,-4,3+2\sqrt{2})
C
(3,4,3−22)(3,4,3-2\sqrt{2})
D
(1,2,1−2)(1,2,1-\sqrt{2})
104
[JEE Main 2024, 27 Jan Shift 2, Q53] The lines x−22=y−2=z−716\frac{x-2}{2}=\frac{y}{-2}=\frac{z-7}{16} and x+34=y+23=z+21\frac{x+3}{4}=\frac{y+2}{3}=\frac{z+2}{1} intersect at the point PP . If the distance of PP from the line x+12=y−13=z−11\frac{x+1}{2}=\frac{y-1}{3}=\frac{z-1}{1} is ll , then 14l214l^2 is equal to __________.
Not attempted
Your answer: —
Correct answer: 108
105
[JEE Main 2024, 27 Jan Shift 1, Q37] The distance, of the point (7,−2,11)(7,-2,11) from the line x−61=y−40=z−83\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3} along the line x−52=y−1−3=z−56\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6} , is :
Not attempted
A
12
B
18
C
21
D
14
106
[JEE Main 2024, 27 Jan Shift 1, Q43] If the shortest distance between the lines x−41=y+12=z−3\frac{x-4}{1}=\frac{y+1}{2}=\frac{z}{-3} and x−λ2=y+14=z−2−5\frac{x-\lambda }{2}=\frac{y+1}{4}=\frac{z-2}{-5} is 65\frac{6}{\sqrt{5}} , then the sum of all possible values of λ\lambda is :
Not attempted
A
10
B
5
C
7
D
8
107
[JEE Main 2023, 15 Apr Shift 1, Q36] Let SS be the set of all values of λ\lambda , for which the shortest distance between the lines x−λ0=y−34=z+61\frac{x-\lambda }{0}=\frac{y-3}{4}=\frac{z+6}{1} and x+λ3=y−4=z−60\frac{x+\lambda }{3}=\frac{y}{-4}=\frac{z-6}{0} is 13. Then 8∣∑λ∈Sλ∣8|{\sum}_{\lambda \in S}\lambda | is equal to :
Not attempted
A
306
B
304
C
308
D
302
108
[JEE Main 2023, 13 Apr Shift 2, Q26] The line, that is coplanar to the line x+3−3=y−11=z−55\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5} , is :
Not attempted
A
x+1−1=y−22=z−54\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z-5}{4}
B
x+1−1=y−22=z−55\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z-5}{5}
C
x−1−1=y−22=z−55\frac{x-1}{-1}=\frac{y-2}{2}=\frac{z-5}{5}
D
x+11=y−22=z−55\frac{x+1}{1}=\frac{y-2}{2}=\frac{z-5}{5}
109
[JEE Main 2023, 12 Apr Shift 1, Q26] Let the lines l1:x+53=y+41=z−α−2l_1:\frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha }{-2} and l2:3x+2y+z−2=0=x−3y+2z−13l_2:3x+2y+z-2=0=x-3y+2z-13 be coplanar. If the point P(a,b,c)P(a,b,c) on l1l_1 is nearest to the point Q(−4,−3,2)Q(-4,-3,2) , then ∣a∣+∣b∣+∣c∣|a|+|b|+|c| is equal to
Not attempted
A
12
B
14
C
10
D
8
110
[JEE Main 2023, 11 Apr Shift 1, Q37] Let a⃗\vec{a} be a non-zero vector parallel to the line of intersection of the two planes described by i^+j^,i^+k^\hat{i}+\hat{j},\hat{i}+\hat{k} and i^−j^,j^−k^\hat{i}-\hat{j},\hat{j}-\hat{k} . If θ\theta is the angle between the vector a⃗\vec{a} and the vector b⃗=2i^−2j^+k^\vec{b}=2\hat{i}-2\hat{j}+\hat{k} and a⃗⋅b⃗=6\vec{a}\cdot \vec{b}=6 , then the ordered pair (θ,∣a⃗×b⃗∣)(\theta ,|\vec{a}\times \vec{b}|) is equal to :
Not attempted
A
(π3,36)(\frac{\pi }{3},3\sqrt{6})
B
(π3,6)(\frac{\pi }{3},6)
C
(π4,36)(\frac{\pi }{4},3\sqrt{6})
D
(π4,6)(\frac{\pi }{4},6)
111
[JEE Main 2023, 11 Apr Shift 1, Q45] Let a line ll pass through the origin and be perpendicular to the lines l1:r⃗=(ı^−11ȷ^−7k^)+λ(i^+2ȷ^+3k^),λ∈Rl_1:\vec{r}=(\hat{ı}-11\hat{ȷ}-7\hat{k})+\lambda (\hat{i}+2\hat{ȷ}+3\hat{k}),\lambda \in \mathbb{R} and l2:r⃗=(−ı^+k^)+μ(2ı^+2ȷ^+k^),μ∈Rl_2:\vec{r}=(-\hat{ı}+\hat{k})+\mu (2\hat{ı}+2\hat{ȷ}+\hat{k}),\mu \in \mathbb{R} . If PP is the point of intersection of ll and l1l_1 , and Q(∝,β,γ)Q(\propto ,\beta ,\gamma ) is the foot of perpendicular from P on l2l_2 , then 9(α+β+γ)9(\alpha +\beta +\gamma ) is equal to _____________.
Not attempted
Your answer: —
Correct answer: 5
112
[JEE Main 2023, 10 Apr Shift 1, Q25] The shortest distance between the lines x+21=y−2=z−52\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2} and x−41=y−12=z+30\frac{x-4}{1}=\frac{y-1}{2}=\frac{z+3}{0} is :
Not attempted
A
8
B
7
C
6
D
9
113
[JEE Main 2023, 8 Apr Shift 1, Q28] The shortest distance between the lines x−44=y+25=z+33\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3} and x−13=y−34=z−42\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2} is :
Not attempted
A
363\sqrt{6}
B
626\sqrt{2}
C
636\sqrt{3}
D
262\sqrt{6}
114
[JEE Main 2023, 6 Apr Shift 2, Q42] If the lines x−12=2−y−3=z−3α\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha } and x−45=y−12=zβ\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta } intersect, then the magnitude of the minimum value of 8αβ8\alpha \beta is _____________.
Not attempted
Your answer: —
Correct answer: 18
115
[JEE Main 2023, 6 Apr Shift 1, Q26] One vertex of a rectangular parallelopiped is at the origin OO and the lengths of its edges along x,yx,y and zz axes are 3,43,4 and 55 units respectively. Let PP be the vertex (3,4,5)(3,4,5) . Then the shortest distance between the diagonal OP and an edge parallel to zz axis, not passing through OO or PP is :
Not attempted
A
125\frac{12}{\sqrt{5}}
B
12512\sqrt{5}
C
125\frac{12}{5}
D
1255\frac{12}{5\sqrt{5}}
116
[JEE Main 2023, 1 Feb Shift 1, Q25] The shortest distance between the lines x−51=y−22=z−4−3\frac{x-5}{1}=\frac{y-2}{2}=\frac{z-4}{-3} and x+31=y+54=z−1−5\frac{x+3}{1}=\frac{y+5}{4}=\frac{z-1}{-5} is :
Not attempted
A
737\sqrt{3}
B
535\sqrt{3}
C
434\sqrt{3}
D
636\sqrt{3}
117
[JEE Main 2023, 31 Jan Shift 1, Q34] Let the shortest distance between the lines L:x−5−2=y−λ0=z+λ1,λ≥0L:\frac{x-5}{-2}=\frac{y-\lambda }{0}=\frac{z+\lambda }{1},\lambda \ge 0 and L1:x+1=y−1=4−zL_1:x+1=y-1=4-z be 262\sqrt{6} . If (α,β,γ)(\alpha ,\beta ,\gamma ) lies on LL , then which of the following is NOT possible?
Not attempted
A
α+2γ=24\alpha +2\gamma =24
B
2α+γ=72\alpha +\gamma =7
C
α−2γ=19\alpha -2\gamma =19
D
2α−γ=92\alpha -\gamma =9
118
[JEE Main 2023, 30 Jan Shift 2, Q41] Let a line LL pass through the point P(2,3,1)P(2,3,1) and be parallel to the line x+3y−2z−2=0=x−y+2zx+3y-2z-2=0=x-y+2z . If the distance of LL from the point (5,3,8)(5,3,8) is α\alpha , then 3α23{\alpha}^2 is equal to :
Not attempted
Your answer: —
Correct answer: 158
119
[JEE Main 2023, 30 Jan Shift 1, Q31] Let a unit vector OP^\hat{OP} make angles α,β,γ\alpha ,\beta ,\gamma with the positive directions of the co-ordinate axes OX\mathrm{OX} , OY,OZ\mathrm{OY},\mathrm{OZ} respectively, where β∈(0,π2)\beta \in (0,\frac{\pi }{2}) . If OP^\hat{\mathrm{OP}} is perpendicular to the plane through points (1,2,3),(2,3,4)(1,2,3),(2,3,4) and (1,5,7)(1,5,7) , then which one of the following is true?
Not attempted
A
α∈(π2,π)\alpha \in (\frac{\pi }{2},\pi ) and γ∈(π2,π)\gamma \in (\frac{\pi }{2},\pi )
B
α∈(0,π2)\alpha \in (0,\frac{\pi }{2}) and γ∈(π2,π)\gamma \in (\frac{\pi }{2},\pi )
C
α∈(π2,π)\alpha \in (\frac{\pi }{2},\pi ) and γ∈(0,π2)\gamma \in (0,\frac{\pi }{2})
D
α∈(0,π2)\alpha \in (0,\frac{\pi }{2}) and γ∈(0,π2)\gamma \in (0,\frac{\pi }{2})
120
[JEE Main 2023, 29 Jan Shift 2, Q37] The shortest distance between the lines x−12=y+8−7=z−45\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5} and x−12=y−21=z−6−3\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3} is :
Not attempted
A
232\sqrt{3}
B
333\sqrt{3}
C
434\sqrt{3}
D
535\sqrt{3}
121
[JEE Main 2023, 29 Jan Shift 1, Q48] Let the co-ordinates of one vertex of ΔABC\Delta ABC be A(0,2,α)A(0,2,\alpha ) and the other two vertices lie on the line x+α5=y−12=z+43\frac{x+\alpha }{5}=\frac{y-1}{2}=\frac{z+4}{3} . For α∈Z\alpha \in \mathbb{Z} , if the area of ΔABC\Delta ABC is 21 sq. units and the line segment BCBC has length 2212\sqrt{21} units, then α2{\alpha}^2 is equal to ___________.
Not attempted
Your answer: —
Correct answer: 9
122
[JEE Main 2023, 25 Jan Shift 2, Q23] The foot of perpendicular of the point (2, 0, 5) on the line x+12=y−15=z+1−1\frac{x+1}{2}=\frac{y-1}{5}=\frac{z+1}{-1} is ( α,β,γ\alpha ,\beta ,\gamma ). Then, which of the following is NOT correct?
Not attempted
A
αβ=−8\frac{\alpha }{\beta }=-8
B
αβγ=415\frac{\alpha \beta }{\gamma }=\frac{4}{15}
C
βγ=−5\frac{\beta }{\gamma }=-5
D
γα=58\frac{\gamma }{\alpha }=\frac{5}{8}
123
[JEE Main 2023, 25 Jan Shift 2, Q28] The shortest distance between the lines x+1=2y=−12zx+1=2y=-12z and x=y+2=6z−6x=y+2=6z-6 is :
Not attempted
A
3
B
52\frac{5}{2}
C
32\frac{3}{2}
D
2
124
[JEE Main 2023, 25 Jan Shift 2, Q45] If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line x−12=y+1−1=z−20\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-2}{0} is α\alpha , then 28 α2{\alpha}^2 is equal to ____________.
Not attempted
Your answer: —
Correct answer: 18
125
[JEE Main 2023, 25 Jan Shift 1, Q31] The distance of the point P(4, 6, −- 2) from the line passing through the point ( −- 3, 2, 3) and parallel to a line with direction ratios 3, 3, −- 1 is equal to :
Not attempted
A
3
B
14\sqrt{14}
C
6\sqrt{6}
D
232\sqrt{3}
126
[JEE Main 2023, 25 Jan Shift 1, Q33] Consider the lines L1L_1 and L2L_2 given by L1:x−12=y−31=z−22L_1:\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-2}{2} L2:x−21=y−22=z−33L_2:\frac{x-2}{1}=\frac{y-2}{2}=\frac{z-3}{3} . A line L3L_3 having direction ratios 1, −- 1, −- 2, intersects L1L_1 and L2L_2 at the points PP and QQ respectively. Then the length of line segment PQPQ is
Not attempted
A
434\sqrt{3}
B
262\sqrt{6}
C
4
D
323\sqrt{2}
127
[JEE Main 2023, 24 Jan Shift 2, Q38] If the shortest between the lines x+62=y−63=z−64\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4} and x−λ3=y−264=z+265\frac{x-\lambda }{3}=\frac{y-2\sqrt{6}}{4}=\frac{z+2\sqrt{6}}{5} is 6, then the square of sum of all possible values of λ\lambda is :
Not attempted
Your answer: —
Correct answer: 384
128
[JEE Main 2023, 24 Jan Shift 1, Q38] The shortest distance between the lines x−23=y+12=z−62\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-6}{2} and x−63=1−y2=z+80\frac{x-6}{3}=\frac{1-y}{2}=\frac{z+8}{0} is equal to ________
Not attempted
Your answer: —
Correct answer: 14
129
[JEE Main 2022, 28 July Shift 1, Q39] Let P(−2,−1,1)P(-2,-1,1) and Q(5617,4317,11117)Q(\frac{56}{17},\frac{43}{17},\frac{111}{17}) be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are α,−1,β\alpha ,-1,\beta , where both α\alpha and β\beta are integers of minimum absolute values, then α2+β2{\alpha}^2+{\beta}^2 is equal to ____________.
Not attempted
Your answer: —
Correct answer: 450
130
[JEE Main 2022, 27 July Shift 2, Q32] If the length of the perpendicular drawn from the point P(a,4,2)P(a,4,2) , a >0>0 on the line x+12=y−33=z−1−1\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-1}{-1} is 262\sqrt{6} units and Q(α1,α2,α3)Q({\alpha}_1,{\alpha}_2,{\alpha}_3) is the image of the point P in this line, then a+∑i=13αia+{\sum}_{i=1}^3{\alpha}_i is equal to :
Not attempted
A
7
B
8
C
12
D
14
131
[JEE Main 2022, 26 July Shift 1, Q35] The length of the perpendicular from the point (1,−2,5)(1,-2,5) on the line passing through (1,2,4)(1,2,4) and parallel to the line x+y−z=0=x−2y+3z−5x+y-z=0=x-2y+3z-5 is :
Not attempted
A
212\sqrt{\frac{21}{2}}
B
92\sqrt{\frac{9}{2}}
C
732\sqrt{\frac{73}{2}}
D
1
132
[JEE Main 2022, 26 July Shift 1, Q45] Let QQ and RR be two points on the line x+12=y+23=z−12\frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2} at a distance 26\sqrt{26} from the point P(4,2,7)P(4,2,7) . Then the square of the area of the triangle PQRPQR is ___________.
Not attempted
Your answer: —
Correct answer: 153
133
[JEE Main 2022, 25 July Shift 2, Q33] The shortest distance between the lines x+7−6=y−67=z\frac{x+7}{-6}=\frac{y-6}{7}=z and 7−x2=y−2=z−6\frac{7-x}{2}=y-2=z-6 is :
Not attempted
A
2292\sqrt{29}
B
1
C
3729\sqrt{\frac{37}{29}}
D
292\frac{\sqrt{29}}{2}
134
[JEE Main 2022, 30 June Shift 1, Q40] Consider a triangle ABC whose vertices are A(0, α\alpha , α\alpha ), B( α\alpha , 0, α\alpha ) and C( α\alpha , α\alpha , 0), α\alpha > 0. Let D be a point moving on the line x + z −- 3 = 0 = y and G be the centroid of Δ\Delta ABC. If the minimum length of GD is 572\sqrt{\frac{57}{2}} , then α\alpha is equal to ____________.
Not attempted
Your answer: —
Correct answer: 6
135
[JEE Main 2022, 28 June Shift 2, Q41] Let the image of the point P(1, 2, 3) in the line L:x−63=y−12=z−23L:\frac{x-6}{3}=\frac{y-1}{2}=\frac{z-2}{3} be Q. Let R ( α\alpha , β\beta , γ\gamma ) be a point that divides internally the line segment PQ in the ratio 1 : 3. Then the value of 22 ( α\alpha + β\beta + γ\gamma ) is equal to __________.
Not attempted
Your answer: —
Correct answer: 125
136
[JEE Main 2022, 27 June Shift 2, Q31] The shortest distance between the lines x−32=y−23=z−1−1\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1} and x+32=y−61=z−53\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3} , is :
Not attempted
A
185\frac{18}{\sqrt{5}}
B
2235\frac{22}{3\sqrt{5}}
C
4635\frac{46}{3\sqrt{5}}
D
636\sqrt{3}
137
[JEE Main 2022, 27 June Shift 1, Q31] If two straight lines whose direction cosines are given by the relations l+m−n=0l+m-n=0 , 3l2+m2+cnl=03l^2+m^2+cnl=0 are parallel, then the positive value of c is :
Not attempted
A
6
B
4
C
3
D
2
138
[JEE Main 2022, 26 June Shift 1, Q30] If the two lines l1:x−23=y+1−2,z=2l_1:\frac{x-2}{3}=\frac{y+1}{-2}, z=2 and l2:x−11=2y+3α=z+52l_2:\frac{x-1}{1}=\frac{2y+3}{\alpha }=\frac{z+5}{2} are perpendicular, then an angle between the lines l2\mathrm{l}_{2} and l3:1−x3=2y−1−4=z4l_3:\frac{1-x}{3}=\frac{2y-1}{-4}=\frac{z}{4} is :
Not attempted
A
cos⁡−1(294){\cos}^{-1}(\frac{29}{4})
B
sec⁡−1(294){\sec}^{-1}(\frac{29}{4})
C
cos⁡−1(229){\cos}^{-1}(\frac{2}{29})
D
cos⁡−1(229){\cos}^{-1}(\frac{2}{\sqrt{29}})
139
[JEE Main 2022, 25 June Shift 2, Q45] Let l1\mathrm{l}_{1} be the line in xy-plane with x and y intercepts 18\frac{1}{8} and 142\frac{1}{4\sqrt{2}} respectively, and l2\mathrm{l}_{2} be the line in zx-plane with x and z intercepts −18-\frac{1}{8} and −163-\frac{1}{6\sqrt{3}} respectively. If d is the shortest distance between the line l1\mathrm{l}_{1} and l2\mathrm{l}_{2} , then d−2\mathrm{d}^{-2} is equal to _______________.
Not attempted
Your answer: —
Correct answer: 51
140
[JEE Main 2022, 24 June Shift 2, Q33] If the shortest distance between the lines x−12=y−23=z−3λ\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda } and x−21=y−44=z−55\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5} is 13\frac{1}{\sqrt{3}} , then the sum of all possible value of λ\lambda is :
Not attempted
A
16
B
6
C
12
D
15
141
[JEE Main 2022, 24 June Shift 1, Q38] Let a line having direction ratios, 1, −- 4, 2 intersect the lines x−73=y−1−1=z+21\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1} and x2=y−73=z1\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1} at the points A and B. Then (AB)2\mathrm{(AB)}^{2} is equal to ___________.
Not attempted
Your answer: —
Correct answer: 84
142
[JEE Main 2022, 24 June Shift 1, Q43] If the shortest distance between the lines r⃗=(−i^+3k^)+λ(i^−aj^)\vec{r}=(-\hat{i}+3\hat{k})+\lambda (\hat{i}-a\hat{j}) and r⃗=(−j^+2k^)+μ(i^−j^+k^)\vec{r}=(-\hat{j}+2\hat{k})+\mu (\hat{i}-\hat{j}+\hat{k}) is 23\sqrt{\frac{2}{3}} , then the integral value of a is equal to ___________.
Not attempted
Your answer: —
Correct answer: 2
143
[JEE Main 2021, 1 September Shift 2, Q64] Let the acute angle bisector of the two planes x−2y−2z+1=0x-2y-2z+1=0 and 2x−3y−6z+1=02x-3y-6z+1=0 be the plane PP. Then which of the following points lies on PP?
Not attempted
A
(3,1,−12)\left(3,1,-\dfrac{1}{2}\right)
B
(−2,0,−12)\left(-2,0,-\dfrac{1}{2}\right)
C
(0,2,−4)\left(0,2,-4\right)
D
(4,0,−2)\left(4,0,-2\right)
144
[JEE Main 2021, 1 September Shift 2, Q72] The distance of line 3y−2z−1=0=3x−z+43y-2z-1=0=3x-z+4 from the point (2,−1,6)(2,-1,6) is :
Not attempted
A
26\sqrt{26}
B
252\sqrt{5}
C
262\sqrt{6}
D
424\sqrt{2}
145
[JEE Main 2021, 31 August Shift 2, Q66] The distance of the point (−1,2,−2)(-1,2,-2) from the line of intersection of the planes 2x+3y+2z=02x+3y+2z=0 and x−2y+z=0x-2y+z=0 is :
Not attempted
A
12\dfrac{1}{\sqrt2}
B
52\dfrac52
C
422\dfrac{\sqrt{42}}{2}
D
342\dfrac{\sqrt{34}}{2}
146
[JEE Main 2021, 31 August Shift 2, Q82] Suppose the line x−2α=y−2−5=z+22\dfrac{x-2}{\alpha}=\dfrac{y-2}{-5}=\dfrac{z+2}{2} lies on the plane x+3y−2z+β=0x+3y-2z+\beta=0. Then (α+β)(\alpha+\beta) is equal to _______.
Not attempted
Your answer: —
Correct answer: 7
147
[JEE Main 2021, 31 August Shift 1, Q64] Let the equation of the plane, that passes through the point (1,4,−3)(1,4,-3) and contains the line of intersection of the planes 3x−2y+4z−7=03x-2y+4z-7=0 and x+5y−2z+9=0x+5y-2z+9=0, be αx+βy+γz+3=0\alpha x+\beta y+\gamma z+3=0, then α+β+γ\alpha+\beta+\gamma is equal to :
Not attempted
A
−23-23
B
−15-15
C
2323
D
1515
148
[JEE Main 2021, 31 August Shift 1, Q83] The square of the distance of the point of intersection of the line x−12=y−23=z+16\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z+1}{6} and the plane 2x−y+z=62x-y+z=6 from the point (−1,−1,2)(-1,-1,2) is _______.
Not attempted
Your answer: —
Correct answer: 61
149
[JEE Main 2021, 27 August Shift 2, Q61] The angle between the straight lines, whose direction cosines are given by the equations 2l+2m−n=02l + 2m - n = 0 and mn+nl+lm=0mn + nl + lm = 0, is :
Not attempted
A
π2\frac{\pi}{2}
B
π−cos⁡−1(49)\pi - \cos^{-1}\left(\frac{4}{9}\right)
C
cos⁡−1(89)\cos^{-1}\left(\frac{8}{9}\right)
D
π3\frac{\pi}{3}
150
[JEE Main 2021, 27 August Shift 2, Q67] The equation of the plane passing through the line of intersection of the planes r⃗⋅(i^+j^+k^)=1\vec{r}\cdot(\hat{i} + \hat{j} + \hat{k}) = 1 and r⃗⋅(2i^+3j^−k^)+4=0\vec{r}\cdot(2\hat{i} + 3\hat{j} - \hat{k}) + 4 = 0 and parallel to the xx-axis is :
Not attempted
A
r⃗⋅(j^−3k^)+6=0\vec{r}\cdot(\hat{j} - 3\hat{k}) + 6 = 0
B
r⃗⋅(i^+3k^)+6=0\vec{r}\cdot(\hat{i} + 3\hat{k}) + 6 = 0
C
r⃗⋅(i^−3k^)+6=0\vec{r}\cdot(\hat{i} - 3\hat{k}) + 6 = 0
D
r⃗⋅(j^−3k^)−6=0\vec{r}\cdot(\hat{j} - 3\hat{k}) - 6 = 0
151
[JEE Main 2021, 27 August Shift 2, Q82] Let SS be the mirror image of the point Q(1,3,4)Q(1, 3, 4) with respect to the plane 2x−y+z+3=02x - y + z + 3 = 0 and let R(3,5,γ)R(3, 5, \gamma) be a point of this plane. Then the square of the length of the line segment SRSR is ________.
Not attempted
Your answer: —
Correct answer: 72
152
[JEE Main 2021, 27 August Shift 1, Q66] The distance of the point (1,−2,3)(1,-2,3) from the plane x−y+z=5x-y+z=5 measured parallel to a line, whose direction ratios are 2,3,−62,3,-6 is :
Not attempted
A
33
B
55
C
22
D
11
153
[JEE Main 2021, 27 August Shift 1, Q69] Equation of a plane at a distance 221\sqrt{\dfrac{2}{21}} from the origin, which contains the line of intersection of the planes x−y−z−1=0x-y-z-1=0 and 2x+y−3z+4=02x+y-3z+4=0, is :
Not attempted
A
3x−y−5z+2=03x-y-5z+2=0
B
3x−4z+3=03x-4z+3=0
C
−x+2y+2z−3=0-x+2y+2z-3=0
D
4x−y−5z+2=04x-y-5z+2=0
154
[JEE Main 2021, 26 August Shift 2, Q75] A hall has a square floor of dimension 1010 m ×\times 1010 m (see the figure) and vertical walls. If the angle GPHGPH between the diagonals AGAG and BHBH is cos⁡−115\cos^{-1}\dfrac15, then the height of the hall (in meters) is :
Not attempted
HGEFDCABP10 m10 m
A
55
B
2102\sqrt{10}
C
535\sqrt3
D
525\sqrt2
155
[JEE Main 2021, 26 August Shift 2, Q76] Let PP be the plane passing through the point (1,2,3)(1,2,3) and the line of intersection of the planes r⃗⋅(i^+j^+4k^)=16\vec r\cdot\left(\hat i+\hat j+4\hat k\right)=16 and r⃗⋅(−i^+j^+k^)=6\vec r\cdot\left(-\hat i+\hat j+\hat k\right)=6. Then which of the following points does NOT\textbf{NOT} lie on PP ?
Not attempted
A
(3,3,2)(3,3,2)
B
(6,−6,2)(6,-6,2)
C
(4,2,2)(4,2,2)
D
(−8,8,6)(-8,8,6)
156
[JEE Main 2021, 26 August Shift 2, Q85] Let QQ be the foot of the perpendicular from the point P(7,−2,13)P(7,-2,13) on the plane containing the lines x+16=y−17=z−38\dfrac{x+1}6=\dfrac{y-1}7=\dfrac{z-3}8 and x−13=y−25=z−37\dfrac{x-1}3=\dfrac{y-2}5=\dfrac{z-3}7. Then (PQ)2(PQ)^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 96
157
[JEE Main 2021, 26 August Shift 1, Q75] A plane PP contains the line x+2y+3z+1=0=x−y−z−6x+2y+3z+1=0=x-y-z-6, and is perpendicular to the plane −2x+y+z+8=0-2x+y+z+8=0. Then which of the following points lies on PP ?
Not attempted
A
(−1,1,2)(-1,1,2)
B
(0,1,1)(0,1,1)
C
(1,0,1)(1,0,1)
D
(2,−1,1)(2,-1,1)
158
[JEE Main 2021, 26 August Shift 1, Q83] Let the line LL be the projection of the line x−12=y−31=z−42\dfrac{x-1}2=\dfrac{y-3}1=\dfrac{z-4}2 in the plane x−2y−z=3x-2y-z=3. If dd is the distance of the point (0,0,6)(0,0,6) from LL, then d2d^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 26
159
[JEE Main 2021, 27 July Shift 2, Q76] For real numbers α\alpha and β≠0\beta\neq0, if the point of intersection of the straight lines x−α1=y−12=z−13\dfrac{x-\alpha}1=\dfrac{y-1}2=\dfrac{z-1}3 and x−4β=y−63=z−73\dfrac{x-4}\beta=\dfrac{y-6}3=\dfrac{z-7}3, lies on the plane x+2y−z=8x+2y-z=8, then α−β\alpha-\beta is equal to:
Not attempted
A
33
B
99
C
77
D
55
160
[JEE Main 2021, 27 July Shift 2, Q84] The distance of the point P(3,4,4)P(3,4,4) from the point of intersection of the line joining the points Q(3,−4,−5)Q(3,-4,-5) and R(2,−3,1)R(2,-3,1) and the plane 2x+y+z=72x+y+z=7, is equal to ____.
Not attempted
Your answer: —
Correct answer: 7
161
[JEE Main 2021, 27 July Shift 1, Q64] Let the plane passing through the point (−1,0,−2)(-1,0,-2) and perpendicular to each of the planes 2x+y−z=22x+y-z=2 and x−y−z=3x-y-z=3 be ax+by+cz+8=0ax+by+cz+8=0. Then the value of a+b+ca+b+c is equal to:
Not attempted
A
44
B
88
C
55
D
33
162
[JEE Main 2021, 27 July Shift 1, Q83] Let a plane PP pass through the point (3,7,−7)(3,7,-7) and contain the line x−2−3=y−32=z+21\dfrac{x-2}{-3}=\dfrac{y-3}{2}=\dfrac{z+2}{1}. If the distance of the plane PP from the origin is dd, then d2d^2 is equal to ________.
Not attempted
Your answer: —
Correct answer: 3
163
[JEE Main 2021, 25 July Shift 2, Q84] If the lines x−k1=y−22=z−33\dfrac{x-k}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{3} and x+13=y+22=z+31\dfrac{x+1}{3}=\dfrac{y+2}{2}=\dfrac{z+3}{1} are co-planar, then the value of kk is .......
Not attempted
Your answer: —
Correct answer: 1
164
[JEE Main 2021, 25 July Shift 1, Q66] Let the foot of perpendicular from a point P(1,2,−1)P(1,2,-1) to the straight line L:x1=y0=z−1L:\dfrac{x}{1}=\dfrac{y}{0}=\dfrac{z}{-1} be NN. Let a line be drawn from PP parallel to the plane x+y+2z=0x+y+2z=0 which meets LL at the point QQ. If α\alpha is the acute angle between the lines PNPN and PQPQ, then cos⁡α\cos\alpha is equal to ......
Not attempted
A
15\dfrac{1}{\sqrt5}
B
32\dfrac{\sqrt3}{2}
C
13\dfrac{1}{\sqrt3}
D
123\dfrac{1}{2\sqrt3}
165
[JEE Main 2021, 22 July Shift 2, Q62] If the shortest distance between the straight lines 3(x−1)=6(y−2)=2(z−1)3(x-1)=6(y-2)=2(z-1) and 4(x−2)=2(y−λ)=(z−3), λ∈R4(x-2)=2(y-\lambda)=(z-3),\ \lambda\in R is 138\dfrac1{\sqrt{38}}, then the integral value of λ\lambda is equal to :
Not attempted
A
33
B
22
C
−1-1
D
55
166
[JEE Main 2021, 22 July Shift 2, Q75] Let LL be the line of intersection of planes r⃗⋅(i^−j^+2k^)=2\vec r\cdot\left(\hat i-\hat j+2\hat k\right)=2 and r⃗⋅(2i^+j^−k^)=2\vec r\cdot\left(2\hat i+\hat j-\hat k\right)=2. If P(α,β,γ)P(\alpha,\beta,\gamma) is the foot of perpendicular on LL from the point (1,2,0)(1,2,0), then the value of 35(α+β+γ)35(\alpha+\beta+\gamma) is equal to :
Not attempted
A
101101
B
143143
C
134134
D
119119
167
[JEE Main 2021, 20 July Shift 2, Q61] Consider the line L given by the equation x−32=y−11=z−21\dfrac{x-3}{2}=\dfrac{y-1}{1}=\dfrac{z-2}{1}. Let Q be the mirror image of the point (2,3,−1)(2,3,-1) with respect to L. Let a plane P be such that it passes through Q, and the line L is perpendicular to P. Then which of the following points is on the plane P ?
Not attempted
A
(1,2,2)(1,2,2)
B
(−1,1,2)(-1,1,2)
C
(1,1,1)(1,1,1)
D
(1,1,2)(1,1,2)
168
[JEE Main 2021, 20 July Shift 2, Q73] The lines x=ay−1=z−2x=ay-1=z-2 and x=3y−2=bz−2x=3y-2=bz-2, (ab≠0)(ab\neq0) are coplanar, if :
Not attempted
A
a=2, b=2a=2,\ b=2
B
b=1, a∈R−{0}b=1,\ a\in R-\{0\}
C
a=2, b=3a=2,\ b=3
D
a=1, b∈R−{0}a=1,\ b\in R-\{0\}
169
[JEE Main 2021, 20 July Shift 1, Q88] Let PP be a plane passing through the points (1,0,1),(1,−2,1)(1,0,1),(1,-2,1) and (0,1,−2)(0,1,-2). Let a vector a⃗=αi^+βj^+γk^\vec a=\alpha\hat i+\beta\hat j+\gamma\hat k be such that a⃗\vec a is parallel to the plane PP, perpendicular to (i^+2j^+3k^)\left(\hat i+2\hat j+3\hat k\right) and a⃗⋅(i^+j^+2k^)=2\vec a\cdot\left(\hat i+\hat j+2\hat k\right)=2, then (α−β+γ)2(\alpha-\beta+\gamma)^2 equals ________.
Not attempted
Your answer: —
Correct answer: 81
170
[JEE Main 2021, 20 July Shift 1, Q90] If the shortest distance between the lines r⃗1=αi^+2j^+2k^+λ(i^−2j^+2k^),λ∈R\vec r_1=\alpha\hat i+2\hat j+2\hat k+\lambda\left(\hat i-2\hat j+2\hat k\right),\lambda\in R, α>0\alpha>0 and r⃗2=−4i^−k^+μ(3i^−2j^−2k^),μ∈R\vec r_2=-4\hat i-\hat k+\mu\left(3\hat i-2\hat j-2\hat k\right),\mu\in R is 99, then α\alpha is equal to ________.
Not attempted
Your answer: —
Correct answer: 6
171
[JEE Main 2021, 18 March Shift 2, Q81] Let PP be a plane containing the line x−13=y+64=z+52\dfrac{x-1}{3}=\dfrac{y+6}{4}=\dfrac{z+5}{2} and parallel to the line x−34=y−2−3=z+57\dfrac{x-3}{4}=\dfrac{y-2}{-3}=\dfrac{z+5}{7}. If the point (1,−1,α)(1,-1,\alpha) lies on the plane PP, then the value of ∣5α∣|5\alpha| is equal to __________.
Not attempted
Your answer: —
Correct answer: 38
172
[JEE Main 2021, 18 March Shift 2, Q86] Let the mirror image of the point (1,3,a)(1,3,a) with respect to the plane r⃗⋅(2i^−j^+k^)−b=0\vec r\cdot\left(2\hat i-\hat j+\hat k\right)-b=0 be (−3,5,2)(-3,5,2). Then, the value of ∣a+b∣|a+b| is equal to ______.
Not attempted
Your answer: —
Correct answer: 1
173
[JEE Main 2021, 18 March Shift 1, Q82] The equation of the planes parallel to the plane x−2y+2z−3=0x-2y+2z-3=0 which are at unit distance from the point (1,2,3)(1,2,3) is ax+by+cz+d=0ax+by+cz+d=0. If (b−d)=K(c−a)(b-d)=K(c-a), then the positive value of KK is ______.
Not attempted
Your answer: —
Correct answer: 4
174
[JEE Main 2021, 18 March Shift 1, Q89] Let the plane ax+by+cz+d=0ax+by+cz+d=0 bisect the line joining the points (4,−3,1)(4,-3,1) and (2,3,−5)(2,3,-5) at right angles. If a,b,c,da,b,c,d are integers, then the minimum value of (a2+b2+c2+d2)(a^2+b^2+c^2+d^2) is ______.
Not attempted
Your answer: —
Correct answer: 28
175
[JEE Main 2021, 17 March Shift 2, Q65] If the equation of the plane passing through the mirror image of a point (2,3,1)(2, 3, 1) with respect to line x+12=y−31=z+2−1\dfrac{x+1}{2} = \dfrac{y-3}{1} = \dfrac{z+2}{-1} and containing the line x−23=1−y2=z+11\dfrac{x-2}{3} = \dfrac{1-y}{2} = \dfrac{z+1}{1} is αx+βy+γz=24\alpha x + \beta y + \gamma z = 24, then α+β+γ\alpha + \beta + \gamma is equal to :
Not attempted
A
2121
B
1919
C
1818
D
2020
176
[JEE Main 2021, 17 March Shift 2, Q89] Let PP be an arbitrary point having the sum of the squares of the distances from the planes x+y+z=0x+y+z=0, lx−nz=0lx-nz=0 and x−2y+z=0x-2y+z=0, equal to 99. If the locus of the point PP is x2+y2+z2=9x^2+y^2+z^2=9, then the value of l−nl-n is equal to ____.
Not attempted
Your answer: —
Correct answer: 0
177
[JEE Main 2021, 17 March Shift 1, Q68] The equation of the plane which contains the yy-axis and passes through the point (1,2,3)(1, 2, 3) is:
Not attempted
A
3x+z=63x + z = 6
B
3x−z=03x - z = 0
C
x+3z=10x + 3z = 10
D
x+3z=0x + 3z = 0
178
[JEE Main 2021, 17 March Shift 1, Q82] If the equation of the plane passing through the line of intersection of the planes 2x−7y+4z−3=02x - 7y + 4z - 3 = 0, 3x−5y+4z+11=03x - 5y + 4z + 11 = 0 and the point (−2,1,3)(-2, 1, 3) is ax+by+cz−7=0ax + by + cz - 7 = 0, then the value of 2a+b+c−72a + b + c - 7 is ____
Not attempted
Your answer: —
Correct answer: 4
179
[JEE Main 2021, 16 March Shift 2, Q64] If the foot of the perpendicular from point (4,3,8)(4,3,8) on the line L1:x−aℓ=y−23=z−b4L_1:\dfrac{x-a}{\ell}=\dfrac{y-2}{3}=\dfrac{z-b}{4}, ℓ≠0\ell\neq0 is (3,5,7)(3,5,7), then the shortest distance between the line L1L_1 and line L2:x−23=y−44=z−55L_2:\dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5} is equal to :
Not attempted
A
23\sqrt{\dfrac{2}{3}}
B
13\dfrac{1}{\sqrt{3}}
C
12\dfrac{1}{2}
D
16\dfrac{1}{\sqrt{6}}
180
[JEE Main 2021, 16 March Shift 2, Q76] If (x,y,z)(x,y,z) be an arbitrary point lying on a plane PP which passes through the points (42,0,0)(42,0,0), (0,42,0)(0,42,0) and (0,0,42)(0,0,42), then the value of the expression 3+x−11(y−19)2(z−12)2+y−19(x−11)2(z−12)2+z−12(x−11)2(y−19)2−x+y+z14(x−11)(y−19)(z−12)3+\dfrac{x-11}{(y-19)^2(z-12)^2}+\dfrac{y-19}{(x-11)^2(z-12)^2}+\dfrac{z-12}{(x-11)^2(y-19)^2}-\dfrac{x+y+z}{14(x-11)(y-19)(z-12)} is equal to :
Not attempted
A
33
B
00
C
3939
D
−45-45
181
[JEE Main 2021, 16 March Shift 2, Q84] If the distance of the point (1,−2,3)(1,-2,3) from the plane x+2y−3z+10=0x+2y-3z+10=0 measured parallel to the line x−13=2−ym=z+31\dfrac{x-1}{3}=\dfrac{2-y}{m}=\dfrac{z+3}{1} is 72\sqrt{\dfrac{7}{2}}, then the value of ∣m∣|m| is equal to ____
Not attempted
Your answer: —
Correct answer: 2
182
[JEE Main 2021, 16 March Shift 1, Q63] If for a>0a > 0, the feet of perpendiculars from the points A(a,−2a,3)A(a, -2a, 3) and B(0,4,5)B(0, 4, 5) on the plane lx+my+nz=0lx + my + nz = 0 are points C(0,−a,−1)C(0, -a, -1) and DD respectively, then the length of line segment CDCD is equal to :
Not attempted
A
41\sqrt{41}
B
55\sqrt{55}
C
31\sqrt{31}
D
66\sqrt{66}
183
[JEE Main 2021, 16 March Shift 1, Q69] Let the position vectors of two points PP and QQ be 3i^−j^+2k^3\hat{i} - \hat{j} + 2\hat{k} and i^+2j^−4k^\hat{i} + 2\hat{j} - 4\hat{k}, respectively. Let RR and SS be two points such that the direction ratios of lines PRPR and QSQS are (4,−1,2)(4, -1, 2) and (−2,1,−2)(-2, 1, -2) respectively. Let lines PRPR and QSQS intersect at TT. If the vector TA→\overrightarrow{TA} is perpendicular to both PR→\overrightarrow{PR} and QS→\overrightarrow{QS} and the length of vector TA→\overrightarrow{TA} is 5\sqrt{5} units, then the modulus of a position vector of AA is :
Not attempted
A
5\sqrt{5}
B
171\sqrt{171}
C
227\sqrt{227}
D
482\sqrt{482}
184
[JEE Main 2021, 16 March Shift 1, Q79] Let PP be a plane lx+my+nz=0lx + my + nz = 0 containing the line, 1−x1=y+42=z+23\dfrac{1 - x}{1} = \dfrac{y + 4}{2} = \dfrac{z + 2}{3}. If plane PP divides the line segment ABAB joining points A(−3,−6,1)A(-3, -6, 1) and B(2,4,−3)B(2, 4, -3) in ratio k:1k : 1 then the value of kk is equal to :
Not attempted
A
1.51.5
B
22
C
44
D
33
185
[JEE Main 2021, 26 February Shift 2, Q61] Let LL be a line obtained from the intersection of two planes x+2y+z=6x+2y+z=6 and y+2z=4y+2z=4. If point P(α,β,γ)P(\alpha,\beta,\gamma) is the foot of perpendicular from (3,2,1)(3,2,1) on LL, then the value of 21(α+β+γ)21(\alpha+\beta+\gamma) equals:
Not attempted
A
142142
B
6868
C
136136
D
102102
186
[JEE Main 2021, 26 February Shift 2, Q78] If the mirror image of the point (1,3,5)(1,3,5) with respect to the plane 4x−5y+2z=84x-5y+2z=8 is (α,β,γ)(\alpha,\beta,\gamma), then 5(α+β+γ)5(\alpha+\beta+\gamma) equals:
Not attempted
A
4747
B
3939
C
4343
D
4141
187
[JEE Main 2021, 26 February Shift 1, Q67] If (1,5,35)(1,5,35), (7,5,5)(7,5,5), (1,λ,7)(1,\lambda,7) and (2λ,1,2)(2\lambda,1,2) are coplanar, then the sum of all possible values of λ\lambda is :
Not attempted
A
445\dfrac{44}{5}
B
395\dfrac{39}{5}
C
−445-\dfrac{44}{5}
D
−395-\dfrac{39}{5}
188
[JEE Main 2021, 26 February Shift 1, Q78] Consider the three planes P1:3x+15y+21z=9P_1: 3x+15y+21z=9 P2:x−3y−z=5P_2: x-3y-z=5, and P3:2x+10y+14z=5P_3: 2x+10y+14z=5 Then, which one of the following is true?
Not attempted
A
P1P_1 and P2P_2 are parallel
B
P1P_1, P2P_2 and P3P_3 are parallel
C
P1P_1 and P3P_3 are parallel
D
P2P_2 and P3P_3 are parallel
189
[JEE Main 2021, 26 February Shift 1, Q87] Let (λ,2,1)(\lambda,2,1) be a point on the plane which passes through the point (4,−2,2)(4,-2,2). If the plane is perpendicular to the line joining the points (−2,−21,29)(-2,-21,29) and (−1,−16,23)(-1,-16,23), then (λ11)2−4λ11−4\left(\dfrac{\lambda}{11}\right)^2-\dfrac{4\lambda}{11}-4 is equal to ______.
Not attempted
Your answer: —
Correct answer: 8
190
[JEE Main 2021, 25 February Shift 2, Q78] A plane passes through the points A(1,2,3)A(1,2,3), B(2,3,1)B(2,3,1) and C(2,4,2)C(2,4,2). If OO is the origin and PP is (2,−1,1)(2,-1,1), then the projection of OP→\overrightarrow{OP} on this plane is of length:
Not attempted
A
23\sqrt{\frac23}
B
25\sqrt{\frac25}
C
27\sqrt{\frac27}
D
211\sqrt{\frac{2}{11}}
191
[JEE Main 2021, 25 February Shift 2, Q90] A line ll passing through origin is perpendicular to the lines l1: r⃗=(3+t)i^+(−1+2t)j^+(4+2t)k^l_1:\ \vec r=(3+t)\hat i+(-1+2t)\hat j+(4+2t)\hat k and l2: r⃗=(3+2s)i^+(3+2s)j^+(2+s)k^l_2:\ \vec r=(3+2s)\hat i+(3+2s)\hat j+(2+s)\hat k. If the coordinates of the point in the first octant on l2l_2 at a distance of 17\sqrt{17} from the point of intersection of ll and l1l_1 are (a,b,c)(a,b,c), then 18(a+b+c)18(a+b+c) is equal to ______.
Not attempted
Your answer: —
Correct answer: 44
192
[JEE Main 2021, 25 February Shift 1, Q72] Let α\alpha be the angle between the lines whose direction cosines satisfy the equations l+m−n=0l+m-n=0 and l2+m2−n2=0l^2+m^2-n^2=0. Then the value of sin⁡4α+cos⁡4α\sin^4\alpha+\cos^4\alpha is:
Not attempted
A
58\frac58
B
34\frac34
C
12\frac12
D
38\frac38
193
[JEE Main 2021, 25 February Shift 1, Q74] The equation of the line through the point (0,1,2)(0,1,2) and perpendicular to the line x−12=y+13=z−1−2\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2} is:
Not attempted
A
x−3=y−14=z−23\frac{x}{-3}=\frac{y-1}{4}=\frac{z-2}{3}
B
x3=y−1−4=z−23\frac{x}{3}=\frac{y-1}{-4}=\frac{z-2}{3}
C
x3=y−14=z−23\frac{x}{3}=\frac{y-1}{4}=\frac{z-2}{3}
D
x3=y−14=z−2−3\frac{x}{3}=\frac{y-1}{4}=\frac{z-2}{-3}
194
[JEE Main 2021, 24 February Shift 2, Q78] The vector equation of the plane passing through the intersection of the planes r⃗⋅(i^+j^+k^)=1\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1 and r⃗⋅(i^−2j^)=−2\vec{r} \cdot (\hat{i} - 2\hat{j}) = -2, and the point (1,0,2)(1, 0, 2) is:
Not attempted
A
r⃗⋅(i^+7j^+3k^)=7\vec{r} \cdot (\hat{i} + 7\hat{j} + 3\hat{k}) = 7
B
r⃗⋅(i^−7j^+3k^)=73\vec{r} \cdot (\hat{i} - 7\hat{j} + 3\hat{k}) = \frac{7}{3}
C
r⃗⋅(i^+7j^+3k^)=73\vec{r} \cdot (\hat{i} + 7\hat{j} + 3\hat{k}) = \frac{7}{3}
D
r⃗⋅(3i^+7j^+3k^)=7\vec{r} \cdot (3\hat{i} + 7\hat{j} + 3\hat{k}) = 7
195
[JEE Main 2021, 24 February Shift 2, Q79] Let a,b∈Ra, b \in R. If the mirror image of the point P(a,6,9)P(a, 6, 9) with respect to the line x−37=y−25=z−1−9\frac{x - 3}{7} = \frac{y - 2}{5} = \frac{z - 1}{-9} is (20,b,−a−9)(20, b, -a - 9), then ∣a+b∣|a + b| is equal to:
Not attempted
A
8686
B
9090
C
8484
D
8888
196
[JEE Main 2021, 24 February Shift 2, Q90] Let λ\lambda be an integer. If the shortest distance between the lines x−λ=2y−1=−2zx - \lambda = 2y - 1 = -2z and x=y+2λ=z−λx = y + 2\lambda = z - \lambda is 722\frac{\sqrt{7}}{2\sqrt{2}}, then the value of ∣λ∣|\lambda| is ______.
Not attempted
Your answer: —
Correct answer: 1
197
[JEE Main 2021, 24 February Shift 1, Q45] The equation of the plane passing through the point (1,2,−3)(1,2,-3) and perpendicular to the planes 3x+y−2z=53x+y-2z=5 and 2x−5y−z=72x-5y-z=7, is :
Not attempted
A
3x−10y−2z+11=03x-10y-2z+11=0
B
6x−5y−2z−2=06x-5y-2z-2=0
C
11x+y+17z+38=011x+y+17z+38=0
D
6x−5y+2z+10=06x-5y+2z+10=0
198
[JEE Main 2021, 24 February Shift 1, Q47] The distance of the point (1,1,9)(1,1,9) from the point of intersection of the line x−31=y−42=z−52\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2} and the plane x+y+z=17x+y+z=17 is :
Not attempted
A
38\sqrt{38}
B
19219\sqrt{2}
C
2192\sqrt{19}
D
3838
199
If (a,b,c)(a,b,c) is the image of the point (1,2,−3)(1,2,-3) in the line x+12=y−3−2=z−1\dfrac{x+1}{2}=\dfrac{y-3}{-2}=\dfrac{z}{-1}, then a+b+ca+b+c is equal to:
Not attempted
A
-1
B
1
C
2
D
3
200
The distance of the point (1,−2,3)(1,-2,3) from the plane x−y+z=5x-y+z=5, measured parallel to the line x2=y3=z−6\dfrac{x}2=\dfrac{y}3=\dfrac{z}{-6}, is:
Not attempted
A
77
B
75\dfrac75
C
11
D
17\dfrac17
201
If the equation of a plane PP, passing through the intersection of the planes x+4y−z+7=0x+4y-z+7=0 and 3x+y+5z=83x+y+5z=8, is ax+by+6z=15ax+by+6z=15 for some a,b∈Ra,b\in\mathbb R, then the distance of the point (3,2,−1)(3,2,-1) from the plane PP is ______.
Not attempted
Your answer: —
Correct answer: 3
202
The plane which bisects the line joining the points (4,−2,3)(4,-2,3) and (2,4,−1)(2,4,-1) at right angles also passes through the point:
Not attempted
A
(4,0,1)(4,0,1)
B
(4,0,−1)(4,0,-1)
C
(0,−1,1)(0,-1,1)
D
(0,1,−1)(0,1,-1)
203
Let a plane PP contain two lines r⃗=i^+λ(i^+j^)\vec r = \hat i + \lambda(\hat i+\hat j), λ∈R\lambda \in \mathbb{R} and r⃗=−j^+μ(j^−k^)\vec r = -\hat j + \mu(\hat j - \hat k), μ∈R\mu \in \mathbb{R}. If Q(α,β,γ)Q(\alpha,\beta,\gamma) is the foot of the perpendicular drawn from the point M(1,0,1)M(1,0,1) to PP, then 3(α+β+γ)3(\alpha+\beta+\gamma) equals ________.
Not attempted
Your answer: —
Correct answer: 5
204
The lines r⃗=(i^−j^)+l(2i^+k^)\vec{r}=(\hat{i}-\hat{j})+l(2\hat{i}+\hat{k}) and r⃗=(2i^−j^)+m(i^+j^+k^)\vec{r}=(2\hat{i}-\hat{j})+m(\hat{i}+\hat{j}+\hat{k}):
Not attempted
A
do not intersect for any values of ll and mm
B
intersect for all values of ll and mm
C
intersect when l=2l=2 and m=12m=\dfrac12
D
intersect when l=1l=1 and m=2m=2
205
A plane passing through the point (3,1,1)(3,1,1) contains two lines whose direction ratios are 1,−2,21,-2,2 and 2,3,−12,3,-1 respectively. If this plane also passes through the point (α,−3,5)(\alpha, -3, 5), then α\alpha is equal to:
Not attempted
A
1010
B
−5-5
C
55
D
−10-10
206
The plane passing through the points (1,2,1)(1,2,1), (2,1,2)(2,1,2) and parallel to the line 2x=3y, z=12x=3y,\ z=1 also passes through the point:
Not attempted
A
(2,0,−1)(2,0,-1)
B
(−2,0,1)(-2,0,1)
C
(0,6,−2)(0,6,-2)
D
(0,−6,2)(0,-6,2)
207
If the distance between the plane 23x−10y−2z+48=023x - 10y - 2z + 48 = 0 and the plane containing the lines x+12=y−32=z+18\dfrac{x+1}{2} = \dfrac{y-3}{2} = \dfrac{z+1}{8} and x+32=y+21=z−1λ\dfrac{x+3}{2} = \dfrac{y+2}{1} = \dfrac{z-1}{\lambda} (which are coplanar) is k633\dfrac{k}{\sqrt{633}}, then kk is equal to ______.
Not attempted
Your answer: —
Correct answer: 3
208
If for some α\alpha and β\beta in R\mathbb{R}, the intersection of the following three planes x+4y−2z=1x+4y-2z=1, x+7y−5z=βx+7y-5z=\beta, x+5y+αz=5x+5y+\alpha z=5 is a line in R3\mathbb{R}^3, then α+β\alpha+\beta is equal to:
Not attempted
A
00
B
1010
C
−10-10
D
22
209
The mirror image of the point (1,2,3)(1,2,3) in a plane is (−73,−43,−13)\left(-\dfrac{7}{3},-\dfrac{4}{3},-\dfrac{1}{3}\right). Which of the following points lies on this plane?
Not attempted
A
(1,1,1)(1,1,1)
B
(1,−1,1)(1,-1,1)
C
(−1,−1,1)(-1,-1,1)
D
(−1,−1,−1)(-1,-1,-1)
210
The shortest distance between the lines x−33=y−8−1=z−31\dfrac{x-3}{3}=\dfrac{y-8}{-1}=\dfrac{z-3}{1} and x+3−3=y+72=z−64\dfrac{x+3}{-3}=\dfrac{y+7}{2}=\dfrac{z-6}{4} is
Not attempted
A
2302\sqrt{30}
B
7230\dfrac72\sqrt{30}
C
3303\sqrt{30}
D
33
211
If the foot of the perpendicular drawn from the point (1,0,3)(1,0,3) on a line passing through (α,7,1)(\alpha,7,1) is (53,73,173)\left(\dfrac53,\dfrac73,\dfrac{17}{3}\right), then α\alpha is equal to:
Not attempted
Your answer: —
Correct answer: 4
212
Let PP be a plane passing through the points (2,1,0)(2,1,0), (4,1,1)(4,1,1) and (5,0,1)(5,0,1) and RR be any point (2,1,6)(2,1,6). Then the image of RR in the plane PP is:
Not attempted
A
(6,5,2)
B
(6,5,-2)
C
(4,3,2)
D
(3,4,-2)
213
[JEE Main 2019, 12 April Shift 2, Q77] A plane which bisects the angle between the two given planes 2x−y+2z−4=02x-y+2z-4=0 and x+2y+2z−2=0x+2y+2z-2=0, passes through the point:
Not attempted
A
(2,4,1)(2,4,1)
B
(2,−4,1)(2,-4,1)
C
(1,4,−1)(1,4,-1)
D
(1,−4,1)(1,-4,1)
214
[JEE Main 2019, 12 April Shift 2, Q78] The length of the perpendicular drawn from the point (2,1,4)(2,1,4) to the plane containing the lines r⃗=(i^+j^)+λ(i^+2j^−k^)\vec r=(\hat i+\hat j)+\lambda(\hat i+2\hat j-\hat k) and r⃗=(i^+j^)+μ(−i^+j^−2k^)\vec r=(\hat i+\hat j)+\mu(-\hat i+\hat j-2\hat k) is:
Not attempted
A
3\sqrt3
B
13\dfrac1{\sqrt3}
C
13\dfrac13
D
33
215
[JEE Main 2019, 12 April Shift 1, Q68] If the line x−23=y+12=z−1−1\dfrac{x-2}{3}=\dfrac{y+1}{2}=\dfrac{z-1}{-1} intersects the plane 2x+3y−z+13=02x+3y-z+13=0 at a point PP and the plane 3x+y+4z=163x+y+4z=16 at a point QQ, then PQPQ is equal to:
Not attempted
A
2142\sqrt{14}
B
14\sqrt{14}
C
272\sqrt7
D
1414
216
[JEE Main 2019, 10 April Shift 2, Q70] A perpendicular is drawn from a point on the line x−12=y+1−1=z1\dfrac{x-1}{2}=\dfrac{y+1}{-1}=\dfrac z1 to the plane x+y+z=3x+y+z=3 such that the foot of the perpendicular QQ also lies on the plane x−y+z=3x-y+z=3. Then the coordinates of QQ are:
Not attempted
A
(2,0,1)(2,0,1)
B
(4,0,−1)(4,0,-1)
C
(1,0,2)(1,0,2)
D
(−1,0,4)(-1,0,4)
217
[JEE Main 2019, 10 April Shift 2, Q80] The distance of the point having position vector −i^+2j^+6k^-\hat i+2\hat j+6\hat k from the straight line passing through the point (2,3,−4)(2,3,-4) and parallel to the vector 6i^+3j^−4k^6\hat i+3\hat j-4\hat k is:
Not attempted
A
77
B
66
C
2132\sqrt{13}
D
434\sqrt3
218
[JEE Main 2019, 10 April Shift 2, Q89] If the plane 2x−y+2z+3=02x-y+2z+3=0 has the distances 13\dfrac13 and 23\dfrac23 units from the planes 4x−2y+4z+λ=04x-2y+4z+\lambda=0 and 2x−y+2z+μ=02x-y+2z+\mu=0, respectively, then the maximum value of λ+μ\lambda+\mu is equal to:
Not attempted
A
1313
B
99
C
55
D
1515
219
[JEE Main 2019, 10 April Shift 1, Q66] If the length of the perpendicular from the point (β,0,β)(\beta,0,\beta) (β≠0)(\beta\ne0) to the line x1=y−10=z+1−1\dfrac{x}{1}=\dfrac{y-1}{0}=\dfrac{z+1}{-1} is 32\sqrt{\dfrac32}, then β\beta is equal to:
Not attempted
A
22
B
11
C
−2-2
D
−1-1
220
[JEE Main 2019, 10 April Shift 1, Q78] If Q(0,−1,−3)Q(0,-1,-3) is the image of the point PP in the plane 3x−y+4z=23x-y+4z=2 and RR is the point (3,−1,−2)(3,-1,-2), then the area (in sq. units) of △PQR\triangle PQR is:
Not attempted
A
652\dfrac{\sqrt{65}}{2}
B
2132\sqrt{13}
C
912\dfrac{\sqrt{91}}{2}
D
914\dfrac{\sqrt{91}}{4}
221
[JEE Main 2019, 9 April Shift 2, Q61] The vertices BB and CC of a △ABC\triangle ABC lie on the line x+23=y−10=z4\dfrac{x+2}{3}=\dfrac{y-1}{0}=\dfrac{z}{4} such that BC=5BC=5 units. Then the area (in sq. units) of this triangle, given that the point A(1,−1,2)A(1,-1,2), is:
Not attempted
A
5175\sqrt{17}
B
66
C
2342\sqrt{34}
D
34\sqrt{34}
222
[JEE Main 2019, 9 April Shift 2, Q80] Let PP be the plane, which contains the line of intersection of the planes x+y+z−6=0x+y+z-6=0 and 2x+3y+z+5=02x+3y+z+5=0 and it is perpendicular to the xyxy-plane. Then the distance of the point (0,0,256)(0,0,256) from PP is equal to:
Not attempted
A
63563\sqrt5
B
2055205\sqrt5
C
115\dfrac{11}{\sqrt5}
D
175\dfrac{17}{\sqrt5}
223
[JEE Main 2019, 9 April Shift 1, Q71] A plane passing through the points (0,−1,0)(0,-1,0) and (0,0,1)(0,0,1) and making an angle π4\dfrac\pi4 with the plane y−z+5=0y-z+5=0, also passes through the point:
Not attempted
A
(2,1,4)\left(\sqrt2,1,4\right)
B
(−2,−1,−4)\left(-\sqrt2,-1,-4\right)
C
(−2,1,−4)\left(-\sqrt2,1,-4\right)
D
(2,−1,4)\left(\sqrt2,-1,4\right)
224
[JEE Main 2019, 9 April Shift 1, Q89] If the line x−12=y+13=z−24\dfrac{x-1}2=\dfrac{y+1}3=\dfrac{z-2}4 meets the plane x+2y+3z=15x+2y+3z=15 at a point PP, then the distance of PP from the origin is:
Not attempted
A
92\dfrac92
B
252\sqrt5
C
52\dfrac{\sqrt5}2
D
72\dfrac72
225
[JEE Main 2019, 8 April Shift 1, Q64] The equation of a plane containing the line of intersection of the planes 2x−y−4=02x-y-4=0 and y+2z−4=0y+2z-4=0 and passing through the point (1,1,0)(1,1,0) is:
Not attempted
A
2x−z=22x-z=2
B
x+3y+z=4x+3y+z=4
C
x−3y−2z=−2x-3y-2z=-2
D
x−y−z=0x-y-z=0
226
[JEE Main 2019, 8 April Shift 1, Q69] The length of the perpendicular from the point (2,−1,4)(2,-1,4) on the straight line x+310=y−2−7=z1\dfrac{x+3}{10}=\dfrac{y-2}{-7}=\dfrac z1 is:
Not attempted
A
greater than 44
B
less than 22
C
greater than 22 but less than 33
D
greater than 33 but less than 44
227
[JEE Main 2019, 12 January Shift 2, Q63] If an angle between the line x+12=y−21=z−3−2\dfrac{x+1}2=\dfrac{y-2}1=\dfrac{z-3}{-2} and the plane x−2y−kz=3x-2y-kz=3 is cos⁡−1(223)\cos^{-1}\left(\dfrac{2\sqrt2}3\right), then a value of kk is:
Not attempted
A
−53-\dfrac53
B
35\sqrt{\dfrac35}
C
53\sqrt{\dfrac53}
D
−35-\dfrac35
228
[JEE Main 2019, 12 January Shift 2, Q70] Let SS be the set of all real values of λ\lambda such that a plane passing through the points (−λ2,1,1)(-\lambda^2,1,1), (1,−λ2,1)(1,-\lambda^2,1) and (1,1,−λ2)(1,1,-\lambda^2) also passes through the point (−1,−1,1)(-1,-1,1). Then SS is equal to:
Not attempted
A
{3}\{\sqrt3\}
B
{3,−3}\{\sqrt3,-\sqrt3\}
C
{1,−1}\{1,-1\}
D
{3,−3}\{3,-3\}
229
[JEE Main 2019, 8 April Shift 2, Q12] The vector equation of the plane through the line of intersection of the planes x+y+z=1x+y+z=1 and 2x+3y+4z=52x+3y+4z=5, which is perpendicular to the plane x−y+z=0x-y+z=0, is:
Not attempted
A
r⃗⋅(i^+k^)+2=0\vec r\cdot(\hat i+\hat k)+2=0
B
r⃗⋅(i^−k^)−2=0\vec r\cdot(\hat i-\hat k)-2=0
C
r⃗⋅(i^−k^)+2=0\vec r\cdot(\hat i-\hat k)+2=0
D
r⃗⋅(i^+k^)−2=0\vec r\cdot(\hat i+\hat k)-2=0
230
[JEE Main 2019, 8 April Shift 2, Q19] If a point R(4,y,z)R(4,y,z) lies on the line segment joining the points P(2,−3,4)P(2,-3,4) and Q(8,0,10)Q(8,0,10), then the distance of RR from the origin is:
Not attempted
A
2142\sqrt{14}
B
66
C
53\sqrt{53}
D
2212\sqrt{21}
231
[JEE Main 2019, 12 January Shift 1, Q85] The perpendicular distance from the origin to the plane containing the two lines x+23=y−25=z+57\dfrac{x+2}3=\dfrac{y-2}5=\dfrac{z+5}7 and x−11=y−44=z+47\dfrac{x-1}1=\dfrac{y-4}4=\dfrac{z+4}7, is:
Not attempted
A
116\dfrac{11}{\sqrt6}
B
6116\sqrt{11}
C
1111
D
11611\sqrt6
232
[JEE Main 2019, 12 January Shift 1, Q87] A tetrahedron has vertices P(1,2,1)P(1,2,1), Q(2,1,3)Q(2,1,3), R(−1,1,2)R(-1,1,2) and O(0,0,0)O(0,0,0). The angle between the faces OPQOPQ and PQRPQR is:
Not attempted
A
cos⁡−1(935)\cos^{-1}\left(\dfrac9{35}\right)
B
cos⁡−1(1935)\cos^{-1}\left(\dfrac{19}{35}\right)
C
cos⁡−1(1731)\cos^{-1}\left(\dfrac{17}{31}\right)
D
cos⁡−1(731)\cos^{-1}\left(\dfrac7{31}\right)
233
[JEE Main 2019, 11 January Shift 2, Q61] If the point (2,α,β)(2,\alpha,\beta) lies on the plane which passes through the points (3,4,2)(3,4,2) and (7,0,6)(7,0,6) and is perpendicular to the plane 2x−5y=152x-5y=15, then 2α−3β2\alpha-3\beta is equal to:
Not attempted
A
55
B
1717
C
1212
D
77
234
[JEE Main 2019, 11 January Shift 2, Q70] Two lines x−31=y+13=z−6−1\dfrac{x-3}1=\dfrac{y+1}3=\dfrac{z-6}{-1} and x+57=y−2−6=z−34\dfrac{x+5}7=\dfrac{y-2}{-6}=\dfrac{z-3}4 intersect at the point RR. The reflection of RR in the xyxy-plane has coordinates:
Not attempted
A
(2,4,7)(2,4,7)
B
(−2,4,7)(-2,4,7)
C
(2,−4,−7)(2,-4,-7)
D
(2,−4,7)(2,-4,7)
235
[JEE Main 2019, 11 January Shift 1, Q79] The direction ratios of the normal to the plane through the points (0,−1,0)(0,-1,0) and (0,0,1)(0,0,1) and making an angle π4\dfrac\pi4 with the plane y−z+5=0y-z+5=0 are:
Not attempted
A
23, 1, −12\sqrt3,\,1,\,-1
B
2, 2, −22,\,\sqrt2,\,-\sqrt2
C
2, −1, 12,\,-1,\,1
D
2, 1, −1\sqrt2,\,1,\,-1
236
[JEE Main 2019, 11 January Shift 1, Q88] The plane containing the line x−32=y+2−1=z−13\dfrac{x-3}2=\dfrac{y+2}{-1}=\dfrac{z-1}3 and also containing its projection on the plane 2x+3y−z=52x+3y-z=5, contains which one of the following points?
Not attempted
A
(2,0,−2)(2,0,-2)
B
(−2,2,2)(-2,2,2)
C
(0,−2,2)(0,-2,2)
D
(2,2,0)(2,2,0)
237
[JEE Main 2019, 10 January Shift 2, Q81] The plane which bisects the line segment joining the points (−3,−3,4)(-3,-3,4) and (3,7,6)(3,7,6) at right angles, passes through which one of the following points?
Not attempted
A
(−2,3,5)(-2,3,5)
B
(4,−1,7)(4,-1,7)
C
(2,1,3)(2,1,3)
D
(4,1,−2)(4,1,-2)
238
[JEE Main 2019, 10 January Shift 2, Q87] On which of the following lines lies the point of intersection of the line x−42=y−52=z−31\dfrac{x-4}{2}=\dfrac{y-5}{2}=\dfrac{z-3}{1} and the plane x+y+z=2x+y+z=2?
Not attempted
A
x+33=4−y3=z+1−2\dfrac{x+3}{3}=\dfrac{4-y}{3}=\dfrac{z+1}{-2}
B
x−41=y−51=z−5−1\dfrac{x-4}{1}=\dfrac{y-5}{1}=\dfrac{z-5}{-1}
C
x−11=y−32=z+4−5\dfrac{x-1}{1}=\dfrac{y-3}{2}=\dfrac{z+4}{-5}
D
x−22=y−32=z+33\dfrac{x-2}{2}=\dfrac{y-3}{2}=\dfrac{z+3}{3}
239
[JEE Main 2019, 10 January Shift 1, Q83] The plane passing through the point (4,−1,2)(4,-1,2) and parallel to the lines x+23=y−2−1=z+12\dfrac{x+2}{3}=\dfrac{y-2}{-1}=\dfrac{z+1}{2} and x−21=y−32=z−43\dfrac{x-2}{1}=\dfrac{y-3}{2}=\dfrac{z-4}{3} also passes through the point:
Not attempted
A
(−1,−1,−1)(-1,-1,-1)
B
(−1,−1,1)(-1,-1,1)
C
(1,1,−1)(1,1,-1)
D
(1,1,1)(1,1,1)
240
[JEE Main 2019, 10 January Shift 1, Q87] Let AA be a point on the line r⃗=(1−3μ)i^+(μ−1)j^+(2+5μ)k^\vec r=(1-3\mu)\hat i+(\mu-1)\hat j+(2+5\mu)\hat k and B(3,2,6)B(3,2,6) be a point in the space. Then the value of μ\mu for which the vector AB→\overrightarrow{AB} is parallel to the plane x−4y+3z=1x-4y+3z=1 is:
Not attempted
A
12\dfrac12
B
−14-\dfrac14
C
14\dfrac14
D
18\dfrac18
241
[JEE Main 2019, 9 January Shift 2, Q74] The equation of the plane containing the straight line x2=y3=z4\dfrac x2=\dfrac y3=\dfrac z4 and perpendicular to the plane containing the straight lines x3=y4=z2\dfrac x3=\dfrac y4=\dfrac z2 and x4=y2=z3\dfrac x4=\dfrac y2=\dfrac z3 is:
Not attempted
A
x+2y−2z=0x+2y-2z=0
B
x−2y+z=0x-2y+z=0
C
5x+2y−4z=05x+2y-4z=0
D
3x+2y−3z=03x+2y-3z=0
242
[JEE Main 2019, 9 January Shift 2, Q86] If the lines x=ay+bx=ay+b, z=cy+dz=cy+d and x=a′z+b′x=a'z+b', y=c′z+d′y=c'z+d' are perpendicular, then:
Not attempted
A
cc′+a+a′=0cc'+a+a'=0
B
aa′+c+c′=0aa'+c+c'=0
C
ab′+bc′+1=0ab'+bc'+1=0
D
bb′+cc′+1=0bb'+cc'+1=0
243
[JEE Main 2019, 9 January Shift 1, Q74] The equation of the line passing through (−4,3,1)(-4,3,1), parallel to the plane x+2y−z−5=0x+2y-z-5=0 and intersecting the line x+1−3=y−32=z−2−1\dfrac{x+1}{-3}=\dfrac{y-3}{2}=\dfrac{z-2}{-1} is:
Not attempted
A
x+42=y+31=z+14\dfrac{x+4}{2}=\dfrac{y+3}{1}=\dfrac{z+1}{4}
B
x+41=y−31=z−13\dfrac{x+4}{1}=\dfrac{y-3}{1}=\dfrac{z-1}{3}
C
x+43=y−3−1=z−11\dfrac{x+4}{3}=\dfrac{y-3}{-1}=\dfrac{z-1}{1}
D
x+4−1=y−31=z−11\dfrac{x+4}{-1}=\dfrac{y-3}{1}=\dfrac{z-1}{1}
244
[JEE Main 2019, 9 January Shift 1, Q77] The plane through the intersection of the planes x+y+z=1x+y+z=1 and 2x+3y−z+4=02x+3y-z+4=0 and parallel to yy-axis also passes through the point:
Not attempted
A
(−3,0,−1)(-3,0,-1)
B
(−3,1,1)(-3,1,1)
C
(3,3,−1)(3,3,-1)
D
(3,2,1)(3,2,1)
245
[JEE Main 2018, 8 April, Q4] If L1L_1 is the line of intersection of the planes 2x−2y+3z−2=02x-2y+3z-2=0, x−y+z+1=0x-y+z+1=0 and L2L_2 is the line of intersection of the planes x+2y−z−3=0x+2y-z-3=0, 3x−y+2z−1=03x-y+2z-1=0, then the distance of the origin from the plane, containing the lines L1L_1 and L2L_2, is:
Not attempted
A
12\dfrac{1}{\sqrt2}
B
142\dfrac{1}{4\sqrt2}
C
132\dfrac{1}{3\sqrt2}
D
122\dfrac{1}{2\sqrt2}
246
[JEE Main 2018, 8 April, Q28] The length of the projection of the line segment joining the points (5,−1,4)(5,-1,4) and (4,−1,3)(4,-1,3) on the plane, x+y+z=7x+y+z=7 is:
Not attempted
A
23\sqrt{\dfrac23}
B
23\dfrac{2}{\sqrt3}
C
23\dfrac23
D
13\dfrac13
247
[JEE Main 2017, 2 April, Q9] If the image of the point P(1,−2,3)P(1,-2,3) in the plane 2x+3y−4z+22=02x+3y-4z+22=0 measured parallel to the line x1=y4=z5\dfrac x1=\dfrac y4=\dfrac z5 is QQ, then PQPQ is equal to:
Not attempted
A
353\sqrt5
B
2422\sqrt{42}
C
42\sqrt{42}
D
656\sqrt5
248
[JEE Main 2017, 2 April, Q15] The distance of the point (1,3,−7)(1,3,-7) from the plane passing through the point (1,−1,−1)(1,-1,-1), having normal perpendicular to both the lines x−11=y+2−2=z−43\dfrac{x-1}1=\dfrac{y+2}{-2}=\dfrac{z-4}3 and x−22=y+1−1=z+7−1\dfrac{x-2}2=\dfrac{y+1}{-1}=\dfrac{z+7}{-1}, is:
Not attempted
A
2074\dfrac{20}{74}
B
1083\dfrac{10}{\sqrt{83}}
C
583\dfrac5{\sqrt{83}}
D
1074\dfrac{10}{\sqrt{74}}
249
[JEE Main 2016, 3 April, Q83] The distance of the point (1,−5,9)(1,-5,9) from the plane x−y+z=5x-y+z=5 measured along the line x=y=zx=y=z is :
Not attempted
A
3103\sqrt{10}
B
10310\sqrt3
C
103\dfrac{10}{\sqrt3}
D
203\dfrac{20}3
250
[JEE Main 2016, 3 April, Q84] If the line, x−32=y+2−1=z+43\dfrac{x-3}2=\dfrac{y+2}{-1}=\dfrac{z+4}3 lies in the plane, lx+my−z=9lx+my-z=9, then l2+m2l^2+m^2 is equal to :
Not attempted
A
2626
B
1818
C
55
D
22
251
[JEE Main 2015, 4 April, Q41] The distance of the point (1,0,2)(1,0,2) from the point of intersection of the line x−23=y+14=z−212\dfrac{x-2}{3}=\dfrac{y+1}{4}=\dfrac{z-2}{12} and the plane x−y+z=16x-y+z=16, is:
Not attempted
A
88
B
3213\sqrt{21}
C
1313
D
2142\sqrt{14}
252
[JEE Main 2015, 4 April, Q42] The equation of the plane containing the line 2x−5y+z=32x-5y+z=3; x+y+4z=5x+y+4z=5, and parallel to the plane, x+3y+6z=1x+3y+6z=1, is:
Not attempted
A
x+3y+6z=−7x+3y+6z=-7
B
x+3y+6z=7x+3y+6z=7
C
2x+6y+12z=−132x+6y+12z=-13
D
2x+6y+12z=132x+6y+12z=13
253
[JEE Main 2014, 6 April, Q31] The image of the line x−13=y−31=z−4−5\dfrac{x-1}{3}=\dfrac{y-3}{1}=\dfrac{z-4}{-5} in the plane 2x−y+z+3=02x-y+z+3=0 is the line
Not attempted
A
x+33=y−51=z−2−5\dfrac{x+3}{3}=\dfrac{y-5}{1}=\dfrac{z-2}{-5}
B
x+3−3=y−5−1=z+25\dfrac{x+3}{-3}=\dfrac{y-5}{-1}=\dfrac{z+2}{5}
C
x−33=y+51=z−2−5\dfrac{x-3}{3}=\dfrac{y+5}{1}=\dfrac{z-2}{-5}
D
x−3−3=y+5−1=z−25\dfrac{x-3}{-3}=\dfrac{y+5}{-1}=\dfrac{z-2}{5}
254
[JEE Main 2014, 6 April, Q58] The angle between the lines whose direction cosines satisfy the equations l+m+n=0l+m+n=0 and l2=m2+n2l^2=m^2+n^2 is
Not attempted
A
π3\dfrac\pi3
B
π4\dfrac\pi4
C
π6\dfrac\pi6
D
π2\dfrac\pi2
255
If the lines x−21=y−31=z−4−k\dfrac{x - 2}{1} = \dfrac{y - 3}{1} = \dfrac{z - 4}{-k} and x−1k=y−42=z−51\dfrac{x - 1}{k} = \dfrac{y - 4}{2} = \dfrac{z - 5}{1} are coplanar, then kk can have
Not attempted
A
exactly one value
B
exactly two values
C
exactly three values
D
any value
256
Distance between two parallel planes 2x+y+2z=82x + y + 2z = 8 and 4x+2y+4z+5=04x + 2y + 4z + 5 = 0 is
Not attempted
A
52\frac52
B
72\frac72
C
92\frac92
D
32\frac32
Make the normal coefficients identical first, then use ∣d1−d2∣a2+b2+c2\dfrac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}}.
Deconstruction\textbf{Deconstruction}: Distance between two parallel planes. Strategy\textbf{Strategy}: Normalise both equations to the same (a,b,c)(a, b, c). Execution\textbf{Execution}: - 2x+y+2z−8=02x + y + 2z - 8 = 0 and 2x+y+2z+52=02x + y + 2z + \frac52 = 0. - d=∣−8−52∣4+1+4=21/23=72d = \dfrac{|-8 - \frac52|}{\sqrt{4 + 1 + 4}} = \dfrac{21/2}{3} = \dfrac72. Verification\textbf{Verification}: Point (4,0,0)(4, 0, 0) lies on the first plane; its distance to 4x+2y+4z+5=04x + 2y + 4z + 5 = 0 is ∣16+5∣36=216=72\dfrac{|16 + 5|}{\sqrt{36}} = \dfrac{21}{6} = \dfrac72. Final Answer\textbf{Final Answer}: Option (2): 72\dfrac72.