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Three Dimensional Geometry — JEE Main & Advanced Mathematics PYQs

756 previous year questions from Three Dimensional Geometry with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

756 questionsMathematicsSolutions on every page
1

Let P be the plane such that it contains the straight line x-1 2 = y-3 3 = z+2 1 and is perpendicular to the plane x + 2y + 3z = 4 . Let P₁ be the plane which passes through the po

JEE Advanced 2026 Solution View → 2

Let L be the straight line joining the points P(1, 2, -1) and Q(2, 3, 1) . Let S be the foot of the perpendicular drawn from the point R(4, -1, 5) to the line L . Another line pass

JEE Advanced 2026 Solution View → 3

Let the foot of perpendicular from the point ( , 2, 3) on the line x-4 1 = y-9 2 = z-5 1 be the point (1, , 2) . Then the distance between the lines x-1 2 = y-2 3 = z+4 6 and x- 2

JEE Main 2026 Solution View → 4

Let a line L₁ pass through the origin and be perpendicular to the lines L₂: r = (3+t) i + (2t-1) j + (2t+4) k and L₃: r = (3+2s) i + (3+2s) j + (2+s) k , t, s R . If (a, b, c) , a

JEE Main 2026 Solution View → 5

The shortest distance between the lines x-4 1 = y-3 2 = z-2 -3 and x+2 2 = y-6 4 = z-5 -5 is :

JEE Main 2026 Solution View → 6

Let the image of the point P(0, -5, 0) in the line x-1 2 = y 1 = z+1 -2 be the point R and the image of the point Q (0, -1 2 , 0 ) in the line x-1 -1 = y+9 4 = z+1 1 be the point S

JEE Main 2026 Solution View → 7

Let the image of the point P(1, 6, a) in the line L: x 1 = y-1 2 = z-a+1 b , b > 0 , be ( a 3 , 0, a+c ) . If S( , , ) , > 0 , is the point on L such that the distance of S from th

JEE Main 2026 Solution View → 8

Let a line L be perpendicular to both the lines L₁: x+1 3 = y+3 5 = z+5 7 and L₂: x-2 1 = y-4 4 = z-6 7 . If is the acute angle between the lines L and L₃: x - 8 7 2 = y - 4 7 1 =

JEE Main 2026 Solution View → 9

Let a triangle PQR be such that P and Q lie on the line x+3 8 = y-4 2 = z+1 2 and are at a distance of 6 units from R(1, 2, 3) . If ( , , ) is the centroid of PQR , then + + is equ

JEE Main 2026 Solution View → 10

If the distance of the point (a, 2, 5) from the image of the point (1, 2, 7) in the line x 1 = y-1 1 = z-2 2 is 4 , then the sum of all possible values of a is equal to :

JEE Main 2026 Solution View → 11

The square of the distance of the point P(5, 6, 7) from the line x-2 2 = y-5 3 = z-2 4 is equal to:

JEE Main 2026 Solution View → 12

The square of the distance of the point of intersection of the lines r = ( i + j - k ) + (a i - j ) , a 0 and r = (4 i - k ) + (2 i + a k ) from the origin is:

JEE Main 2026 Solution View → 13

The shortest distance between the lines r = ( 1 3 i +2 j + 8 3 k )+ (2 i -5 j +6 k ) and r = (- 2 3 i - 1 3 k )+ ( j - k ) , , R , is:

JEE Main 2026 Solution View → 14

If (2 +1, ^2-3 , -1 2 ) is the image of ( , 2 , 1) in the line x-2 3 = y-1 2 = z 1 , then the possible value(s) of is (are)

JEE Main 2026 Solution View → 15

A line with direction ratios 1, -1, 2 intersects the lines x 2 = y 3 = z+1 3 and x+1 -1 = y-2 1 = z 4 at the points P and Q , respectively. If the length of the line segment PQ is

JEE Main 2026 Solution View → 16

The square of the distance of the point (-2, -8, 6) from the line x-1 1 = y-1 2 = z -1 along the line x+5 1 = y+5 -1 = z 2 is equal to:

JEE Main 2026 Solution View → 17

Let the point A be the foot of perpendicular drawn from the point P(a, b, 0) on the line x-1 2 = y-2 1 = z- 3 . If the midpoint of the line segment PA is (0, 3 4 , -1 4 ) , then th

JEE Main 2026 Solution View → 18

If the point of intersection of the lines x+1 3 = y+a 5 = z+b+1 7 and x-2 1 = y-b 4 = z-2a 7 lies on xy -plane, then the value of a + b is :

JEE Main 2026 Solution View → 19

Let a line L passing through the point (1, 1, 1) be perpendicular to both the vectors 2 i + 2 j + k and i + 2 j + 2 k . If P(a, b, c) is the foot of perpendicular from the origin o

JEE Main 2026 Solution View → 20

Let Q ( a , b , c ) be the image of the point P (3,2,1) in the line x-1 1 = y 2 = z-1 1 . Then the distance of Q from the line x-9 3 = y-9 2 = z-5 -2 is

JEE Main 2026 Solution View → 21

If the distance of the point P (43, , ), <0 , from the line r =4 i - k + (2 i +3 k ), R along a line with direction ratios 3,-1,0 is 13 10 , then ²+ ² is equal to _ _ _ _

JEE Main 2026 Solution View → 22

If the distances of the point (1,2, a) from the line x-1 1 = y 2 = z-1 1 along the lines L ₁: x-1 3 = y-2 4 = z-a b and L ₂: x-1 1 = y-2 4 = z-a c are equal, then a+b+c is equal to

JEE Main 2026 Solution View → 23

The sum of all values of , for which the shortest distance between the lines x+1 = y-2 -1 = z-4 - and x = y-1 2 = z-1 2 is 2 , is

JEE Main 2026 Solution View → 24

Let the lines L ₁: r = i +2 j +3 k + (2 i +3 j +4 k ), R and L ₂: r =(4 i + j )+ (5 i +2 j + k ), R , intersect at the point R. Let P and Q be the points lying on lines L ₁ and L ₂

JEE Main 2026 Solution View → 25

Let a line L passing through the point P (1,1,1) be perpendicular to the lines x-4 4 = y-1 1 = z-1 1 and x-17 1 = y-71 1 = z 0 . Let the line L intersect the y z- plane at the poin

JEE Main 2026 Solution View → 26

If the image of the point P (a, 2, a) in the line x 2 = y+a 1 = z 1 is Q and the image of Q in the line x-2 b 2 = y-a 1 = z+2 b -5 is P, then a+b is equal to _ _ _ _ .

JEE Main 2026 Solution View → 27

Let the direction cosines of two lines satisfy the equations : 4 l+m-n=0 and 2 m n+10 n l+3 l m=0 . Then the cosine of the acute angle between these lines is :

JEE Main 2026 Solution View → 28

The vertices B and C of a triangle ABC lie on the line x 1 = 1-y -2 = z -2 3 . The coordinates of A and B are (1,6,3) and (4,9, ) respectively and C is at a distance of 10 units fr

JEE Main 2026 Solution View → 29

Let L be the line x+1 2 = y+1 3 = z+3 6 and let S be the set of all points ( a , b , c ) on L, whose distance from the line x+1 2 = y+1 3 = z-9 0 along the line L is 7. Then _ (a,

JEE Main 2026 Solution View → 30

If the image of the point P (1,2, a) in the line x-6 3 = y-7 2 = 7- z 2 is Q (5, b, c ) , then a²+b²+c² is equal to

JEE Main 2026 Solution View → 31

Let P ( , , ) be the point on the line x-1 2 = y+1 -3 =z at a distance 4 14 from the point (1,-1,0) and nearer to the origin. Then the shortest distance, between the lines x- 1 = y

JEE Main 2026 Solution View → 32

Let the line L pass through the point (-3,5,2) and make equal angles with the positive coordinate axes. If the distance of L from the point (-2, r , 1) is 14 3 , then the sum of al

JEE Main 2026 Solution View → 33

Let the line L ₁ be parallel to the vector -3 i +2 j +4 k and pass through the point (2,6,7), and the line L ₂ be parallel to the vector 2 i + j +3 k and pass through the point (4,

JEE Main 2026 Solution View → 34

Let ( , , ) be the co-ordinates of the foot of the perpendicular drawn from the point (5,4,2) on the line r =(- i +3 j + k )+ (2 i +3 j - k ) . Then the length of the projection of

JEE Main 2026 Solution View → 35

Let L₁ be the line of intersection of the planes given by the equation- 2 x+3 y+z=4 and x+2 y+z=5 . Let L₂ be the line passing through the point P(2,-1,3) and parallel to L₁ . Let

JEE Advanced 2025 Solution View → 36

Let the values of for which the shortest distance between the lines x-1 2 = y-2 3 = z-3 4 and x- 3 = y-4 4 = z-5 5 is 1 6 be ₁ and ₂ . Then the radius of the circle passing through

JEE Main 2025 Solution View → 37

Let the area of the triangle formed by the lines x+2=y-1=z, x-3 5 = y -1 = z-1 1 and x -3 = y-3 3 = z-2 1 be A . Then A^2 is equal to ________

JEE Main 2025 Solution View → 38

Consider the lines L ₁: x -1= y -2= z and L ₂: x -2= y = z -1 . Let the feet of the perpendiculars from the point P (5,1,-3) on the lines L ₁ and L ₂ be Q and R respectively. If th

JEE Main 2025 Solution View → 39

If the equation of the line passing through the point (0,- 1 2 , 0 ) and perpendicular to the lines r = ( i +a j +b k ) and r =( i - j -6 k )+ (-b i + a j +5 k ) is x -1 -2 = y +4

JEE Main 2025 Solution View → 40

If the shortest distance between the lines x-1 2 = y-2 3 = z-3 4 and x 1 = y = z-5 1 is 5 6 , then the sum of all possible values of is

JEE Main 2025 Solution View → 41

Let the line L pass through (1,1,1) and intersect the lines x-1 2 = y+1 3 = z-1 4 and x-3 1 = y-4 2 = z 1 . Then, which of the following points lies on the line L ?

JEE Main 2025 Solution View → 42

Let the values of p , for which the shortest distance between the lines x+1 3 = y 4 = z 5 and r =( p i +2 j + k )+ (2 i +3 j +4 k ) is 1 6 , be a , b , (a b) . Then the length of t

JEE Main 2025 Solution View → 43

Let A be the point of intersection of the lines L₁: x-7 1 = y-5 0 = z-3 -1 and L₂: x-1 3 = y+3 4 = z+7 5 . Let B and C be the point on the lines L₁ and L₂ respectively such that AB

JEE Main 2025 Solution View → 44

Let the shortest distance between the lines x-3 3 = y- -1 = z-3 1 and x+3 -3 = y+7 2 = z- 4 be 3 30 . Then the positive value of 5 + is

JEE Main 2025 Solution View → 45

Let A and B be two distinct points on the line L : x -6 3 = y -7 2 = z -7 -2 . Both A and B are at a distance 2 17 from the foot of perpendicular drawn from the point (1,2,3) on th

JEE Main 2025 Solution View → 46

Each of the angles and that a given line makes with the positive y - and z -axes, respectively, is half of the angle that this line makes with the positive x -axes. Then the sum of

JEE Main 2025 Solution View → 47

The distance of the point (7,10,11) from the line x-4 1 = y-4 0 = z-2 3 along the line x-9 2 = y-13 3 = z-17 6 is

JEE Main 2025 Solution View → 48

Let a line passing through the point (4,1,0) intersect the line L₁ ; x-1 2 = y-2 3 = z-3 4 at the point A ( , , ) and the line L₂: x-6=y=-z+4 at the point B(a, b, c) . Then | array

JEE Main 2025 Solution View → 49

Line L₁ passes through the point (1,2,3) and is parallel to Z -axis. Line L ₂ passes through the point ( , 5,6) and is parallel to y -axis. Let for = ₁, ₂, ₂ ₁ , the shortest dista

JEE Main 2025 Solution View → 50

The line L₁ is parallel to the vector a =-3 i +2 j +4 k and passes through the point (7,6,2) and the line L₂ is parallel to the vector b =2 i + j +3 k and passes through the point

JEE Main 2025 Solution View → 51

If the image of the point P (1,0,3) in the line joining the points A (4,7,1) and B (3,5,3) is Q ( , , ) , then + + is equal to

JEE Main 2025 Solution View → 52

Let the vertices Q and R of the triangle P Q R lie on the line x+3 5 = y-1 2 = z+4 3 , Q R=5 and the coordinates of the point P be (0,2,3) . If the area of the triangle P Q R is m

JEE Main 2025 Solution View → 53

Let P be the foot of the perpendicular from the point (1,2,2) on the line L : x-1 1 = y+1 -1 = z-2 2 . Let the line r =(- i + j -2 k )+ ( i - j + k ), R , intersect the line L at Q

JEE Main 2025 Solution View → 54

Let a straight line L pass through the point P(2,-1,3) and be perpendicular to the lines x-1 2 = y+1 1 = z-3 -2 and x-3 1 = y-2 3 = z+2 4 . If the line L intersects the y z -plane

JEE Main 2025 Solution View → 55

Let ( a = i +2 j + k ) and ( b =2 i +7 j +3 k . ) Let ( L ₁: r =(- i +2 j + k )+ a , R ) and ( L ₂: r =( j + k )+ b , R ) be two lines. If the line ( L ₃ ) passes through the point

JEE Main 2025 Solution View → 56

Let ( L ₁: x-1 1 = y-2 -1 = z-1 2 ) and ( L ₂: x+1 -1 = y-2 2 = z 1 ) be two lines. Let (L₃ ) be a line passing through the point (( , , ) ) and be perpendicular to both (L₁ ) and

JEE Main 2025 Solution View → 57

The square of the distance of the point ( 15 7 , 32 7 , 7 ) from the line x+1 3 = y+3 5 = z+5 7 in the direction of the vector i +4 j +7 k is :

JEE Main 2025 Solution View → 58

If the image of the point (4,4,3) in the line x-1 2 = y-2 1 = z-1 3 is ( , , ) , then + + is equal to

JEE Main 2025 Solution View → 59

Let A (x, y, z) be a point in x y -plane, which is equidistant from three points (0,3,2),(2,0,3) and ( 0,0,1 ). Let B =(1,4,-1) and C =(2,0,-2) . Then among the statements (S1) : A

JEE Main 2025 Solution View → 60

Let P be the image of the point Q (7,-2,5) in the line L : x-1 2 = y+1 3 = z 4 and R (5, p , q ) be a point on L . Then the square of the area of P Q R is ________.

JEE Main 2025 Solution View → 61

Let in a A B C , the length of the side A C be 6 , the vertex B be (1,2,3) and the vertices A, C lie on the line x-6 3 = y-7 2 = z-7 -2 . Then the area (in sq. units) of ABC is:

JEE Main 2025 Solution View → 62

Let the line passing through the points (-1,2,1) and parallel to the line x-1 2 = y+1 3 = z 4 intersect the line x+2 3 = y-3 2 = z-4 1 at the point P . Then the distance of P from

JEE Main 2025 Solution View → 63

The distance of the line x-2 2 = y-6 3 = z-3 4 from the point (1,4,0) along the line x 1 = y-2 2 = z+3 3 is :

JEE Main 2025 Solution View → 64

If the square of the shortest distance between the lines x-2 1 = y-1 2 = z+3 -3 and x+1 2 = y+3 4 = z+5 -5 is m n , where m , n are coprime numbers, then m + n is equal to :

JEE Main 2025 Solution View → 65

Let the point A divide the line segment joining the points P(-1,-1,2) and Q(5,5,10) internally in the ratio r : 1( r 0) . If O is the origin and ( OQ OA )- 1 5 | OP OA |^2=10 , the

JEE Main 2025 Solution View → 66

Let P be the foot of the perpendicular from the point Q(10,-3,-1) on the line x-3 7 = y-2 -1 = z+1 -2 . Then the area of the right angled triangle P Q R , where R is the point (3,-

JEE Main 2025 Solution View → 67

Let a line pass through two distinct points P(-2,-1,3) and Q , and be parallel to the vector 3 i +2 j +2 k . If the distance of the point Q from the point R (1,3,3) is 5 , then the

JEE Main 2025 Solution View → 68

The perpendicular distance, of the line x-1 2 = y+2 -1 = z+3 2 from the point P (2,-10,1) , is :

JEE Main 2025 Solution View → 69

Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle P Q R is formed such that Q

JEE Main 2025 Solution View → 70

Let L ₁: x-1 2 = y-2 3 = z-3 4 and L ₂: x-2 3 = y-4 4 = z-5 5 be two lines. Then which of the following points lies on the line of the shortest distance between L₁ and L₂ ?

JEE Main 2025 Solution View → 71

Let L ₁: x-1 3 = y-1 -1 = z+1 0 and L ₂: x-2 2 = y 0 = z+4 , R , be two lines, which intersect at the point B . If P is the foot of perpendicular from the point A(1,1,-1) on L₂ , t

JEE Main 2025 Solution View → 72

A straight line drawn from the point P(1,3,2) , parallel to the line x-2 1 = y-4 2 = z-6 1 , intersects the plane L₁: x-y+3 z=6 at the point Q . Another straight line which passes

JEE Advanced 2024 Solution View → 73

Let R ^3 denote the three-dimensional space. Take two points P=(1,2,3) and Q=(4,2,7) . Let dist (X, Y) denote the distance between two points X and Y in R ^3 . Let gathered S= X R

JEE Advanced 2024 Solution View → 74

Let R be such that the lines L ₁: x+11 1 = y +21 2 = z +29 3 and L₂: x+16 3 = y+11 2 = z+4 intersect. Let R ₁ be the point of intersection of L ₁ and L ₂ . Let O =(0,0,0) , and n d

JEE Advanced 2024 Solution View → 75

Consider the line L passing through the points (1,2,3) and (2,3,5) . The distance of the point ( 11 3 , 11 3 , 19 3 ) from the line L along the line 3 x-11 2 = 3 y-11 1 = 3 z-19 2

JEE Main 2024 Solution View → 76

The square of the distance of the image of the point (6,1,5) in the line x-1 3 = y 2 = z-2 4 , from the origin is _________

JEE Main 2024 Solution View → 77

Let the line L intersect the lines x-2=-y=z-1,2(x+1)=2(y-1)=z+1 and be parallel to the line x-2 3 = y-1 1 = z-2 2 . Then which of the following points lies on L?

JEE Main 2024 Solution View → 78

The shortest distance between the lines x-3 4 = y+7 -11 = z-1 5 and x-5 3 = y-9 -6 = z+2 1 is:

JEE Main 2024 Solution View → 79

If the shortest distance between the lines x- 2 = y-4 3 = z-3 4 and x-2 4 = y-4 6 = z-7 8 is 13 29 , then a value of is :

JEE Main 2024 Solution View → 80

Let P ( , , ) be the image of the point Q (1,6,4) in the line x 1 = y-1 2 = z-2 3 . Then 2 + + is equal to_______

JEE Main 2024 Solution View → 81

If the shortest distance between the lines array ll L₁: r =(2+ ) i +(1-3 ) j +(3+4 ) k , & R L₂: r =2(1+ ) i +3(1+ ) j +(5+ ) k , R array is m n , where gcd (m, n)=1 , then the val

JEE Main 2024 Solution View → 82

Let P(x, y, z) be a point in the first octant, whose projection in the x y -plane is the point Q . Let O P= ; the angle between O Q and the positive x -axis be ; and the angle betw

JEE Main 2024 Solution View → 83

Let P ( , , ) be the image of the point Q (3,-3,1) in the line x-0 1 = y-3 1 = z-1 -1 and R be the point (2,5,-1) . If the area of the triangle P Q R is and ^2=14 K , then K is equ

JEE Main 2024 Solution View → 84

If the shortest distance between the lines x- 3 = y-2 -1 = z-1 1 and x+2 -3 = y+5 2 = z-4 4 is 44 30 , then the largest possible value of | | is equal to _________

JEE Main 2024 Solution View → 85

The shortest distance between the lines x-3 2 = y+15 -7 = z-9 5 and x+1 2 = y-1 1 = z-9 -3 is

JEE Main 2024 Solution View → 86

Let P be the point (10,-2,-1) and Q be the foot of the perpendicular drawn from the point R(1,7,6) on the line passing through the points (2,-5,11) and (-6,7,-5) . Then the length

JEE Main 2024 Solution View → 87

Let ( , , ) be the image of the point (8,5,7) in the line x-1 2 = y+1 3 = z-2 5 . Then + + is equal to :

JEE Main 2024 Solution View → 88

Let the point (-1, , ) lie on the line of the shortest distance between the lines x+2 -3 = y-2 4 = z-5 2 and x+2 -1 = y+6 2 = z-1 0 . Then ( - )^2 is equal to___________

JEE Main 2024 Solution View → 89

If the line 2-x 3 = 3 y-2 4 +1 =4-z makes a right angle with the line x+3 3 = 1-2 y 6 = 5-z 7 , then 4 +9 is equal to :

JEE Main 2024 Solution View → 90

Let d be the distance of the point of intersection of the lines x+6 3 = y 2 = z+1 1 and x-7 4 = y-9 3 = z-4 2 from the point (7,8,9) . Then d ^2+6 is equal to :

JEE Main 2024 Solution View → 91

Let P be the point of intersection of the lines x-2 1 = y-4 5 = z-2 1 and x-3 2 = y-2 3 = z-3 2 . Then, the shortest distance of P from the line 4 x=2 y=z is

JEE Main 2024 Solution View → 92

Consider a line L passing through the points P (1,2,1) and Q (2,1,-1) . If the mirror image of the point A (2,2,2) in the line L is ( , , ) , then + +6 is equal to _______

JEE Main 2024 Solution View → 93

Let the point, on the line passing through the points P(1,-2,3) and Q(5,-4,7) , farther from the origin and at distance of 9 units from the point P , be ( , , ) . Then ^2+ ^2+ ^2 i

JEE Main 2024 Solution View → 94

If the shortest distance between the lines x+2 2 = y+3 3 = z-5 4 and x-3 1 = y-2 -3 = z+4 2 is 38 3 5 k , and ₀^ k [x^2 ] d x= - , where [x] denotes the greatest integer function,

JEE Main 2024 Solution View → 95

If the mirror image of the point P ( 3 , 4 , 9 ) in the line x − 1 3 = y + 1 2 = z − 2 1 is α , β , γ , then 14 α + β + γ is:

JEE Main 2024 Solution View → 96

Let P and Q be the points on the line x + 3 8 = y − 4 2 = z + 1 2 which are at a distance of 6 units from the point R ( 1 , 2 , 3 ) . If the centroid of the triangle P Q R is α , β

JEE Main 2024 Solution View → 97

If the shortest distance between the lines x − λ − 2 = y − 2 1 = z − 1 1 and x − 3 1 = y − 1 − 2 = z − 2 1 is 1 , then the sum of all possible values of λ is

JEE Main 2024 Solution View → 98

Let the line of the shortest distance between the lines L 1 : r → = i ^ + 2 j ^ + 3 k ^ + λ i ^ − j ^ + k ^ and L 2 : r → = 4 i ^ + 5 j ^ + 6 k ^ + μ i ^ + j ^ − k ^ intersect L 1

JEE Main 2024 Solution View → 99

Let α , β , γ be mirror image of the point 2 , 3 , 5 in the line x − 1 2 = y − 2 3 = z − 3 4 . Then 2 α + 3 β + 4 γ is equal to

JEE Main 2024 Solution View → 100

The shortest distance between lines L 1 and L 2 , where L 1 : x − 1 2 = y + 1 − 3 = z + 4 2 and L 2 is the line passing through the points A − 4 , 4 , 3 , B − 1 , 6 , 3 and perpend

JEE Main 2024 Solution View → 101

A line passes through A 4 , − 6 , − 2 and B 16 , − 2 , 4 . The point P a , b , c where a , b , c are non-negative integers, on the line A B lies at a distance of 21 units, from the

JEE Main 2024 Solution View → 102

The distance of the point Q ( 0 , 2 , – 2 ) form the line passing through the point P ( 5 , – 4 , 3 ) and perpendicular to the lines r → = − 3 i ^ + 2 k ^ + λ 2 i ^ + 3 j ^ + 5 k ^

JEE Main 2024 Solution View → 103

Let Q and R be the feet of perpendiculars from the point P a , a , a on the lines x = y , z = 1 and x = − y , z = − 1 respectively. If ∠ Q P R is a right angle, then 12 a 2 is equa

JEE Main 2024 Solution View → 104

Let L 1 : r → = i ^ - j ^ + 2 k ^ + λ i ^ - j ^ + 2 k ^ , λ ∈ R , L 2 : r → = j ^ - k ^ + μ 3 i ^ + j ^ + p k ^ , μ ∈ R and L 3 : r → = δ ( l i ^ + m j ^ + n k ^ ) , δ ∈ R be three

JEE Main 2024 Solution View → 105

Let a line passing through the point ( - 1 , 2 , 3 ) intersect the lines L 1 : x - 1 3 = y - 2 2 = z + 1 - 2 at M ( α , β , γ ) and L 2 : x + 2 - 3 = y - 2 - 2 = z - 1 4 at N ( a ,

JEE Main 2024 Solution View → 106

Let ( α , β , γ ) be the foot of perpendicular from the point ( 1 , 2 , 3 ) on the line x + 3 5 = y - 1 2 = z + 4 3 . then 19 ( α + β + γ ) is equal to :

JEE Main 2024 Solution View → 107

If d 1 is the shortest distance between the lines x + 1 = 2 y = - 12 z , x = y + 2 = 6 z - 6 and d 2 is the shortest distance between the lines x - 1 2 = y + 8 - 7 = z - 4 5 , x -

JEE Main 2024 Solution View → 108

Let P ( 3 , 2 , 3 ) , Q ( 4 , 6 , 2 ) and R ( 7 , 3 , 2 ) be the vertices of ∆ PQR . Then, the angle ∠ QPR is

JEE Main 2024 Solution View → 109

Let O be the origin, and M and N be the points on the lines x - 5 4 = y - 4 1 = z - 5 3 and x + 8 12 = y + 2 5 = z + 11 9 respectively such that M N is the shortest distance betwee

JEE Main 2024 Solution View → 110

Let O be the origin and the position vector of A and B be 2 i ^ + 2 j ^ + k ^ and 2 i ^ + 4 j ^ + 4 k ^ respectively. If the internal bisector of ∠ A O B meets the line A B at C ,

JEE Main 2024 Solution View → 111

Let P Q R be a triangle with R ( - 1 , 4 , 2 ) . Suppose M ( 2 , 1 , 2 ) is the mid point of P Q . The distance of the centroid of ∆ P Q R from the point of intersection of the lin

JEE Main 2024 Solution View → 112

A line with direction ratio 2 , 1 , 2 meets the lines x = y + 2 = z and x + 2 = 2 y = 2 z respectively at the point P and Q . if the length of the perpendicular from the point ( 1

JEE Main 2024 Solution View → 113

The position vectors of the vertices A , B and C of a triangle are 2 i ^ - 3 j ^ + 3 k ^ , 2 i ^ + 2 j ^ + 3 k ^ and - i ^ + j ^ + 3 k ^ respectively. Let l denotes the length of t

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Let the position vectors of the vertices A , B and C of a triangle be 2 i ^ + 2 j ^ + k ^ , i ^ + 2 j ^ + 2 k ^ and 2 i ^ + j ^ + 2 k ^ respectively. Let l 1 , l 2 and l 3 be the l

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The lines x - 2 2 = y - 2 = z - 7 16 and x + 3 4 = y + 2 3 = z + 2 1 intersect at the point P . If the distance of P from the line x + 1 2 = y - 1 3 = z - 1 1 is l , then 14 l 2 is

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Let the image of the point ( 1 , 0 , 7 ) in the line x 1 = y - 1 2 = z - 2 3 be the point ( α , β , γ ) . Then which one of the following points lies on the line passing through (

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The distance, of the point ( 7 , - 2 , 11 ) from the line x - 6 1 = y - 4 0 = z - 8 3 along the line x - 5 2 = y - 1 - 3 = z - 5 6 , is :

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If the shortest distance between the lines x - 4 1 = y + 1 2 = z - 3 and x - λ 2 = y + 1 4 = z - 2 - 5 is 6 5 , then the sum of all possible values of λ is :

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Let Q be the cube with the set of vertices x 1 , x 2 , x 3 &#8712; &#8477; 3 : x 1 , x 2 , x 3 &#8712; 0 , 1 . Let F be the set of all twelve lines containing the diagonals of the

JEE Advanced 2023 Solution View → 120

Let &#8467; 1 and &#8467; 2 be the lines r &#8594; 1 = &#955; i ^ + j ^ + k ^ and r &#8594; 2 = j ^ - k ^ + &#956; i ^ + k ^ , respectively, Let X be the set of all the planes H th

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