Practice Questions

Q78.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+ 𝜆2 ^𝑖+ 3 ^𝑗+ 5 ^𝑘, 𝜆∈ℝ and →𝑟= ^𝑖−2 ^𝑗+ ^𝑘+ 𝜇− ^𝑖+ 3 ^𝑗+ 2 ^𝑘, 𝜇∈ℝ is (1) √86 (2) √20 (3) √54 (4) √74

Q79.The shortest distance between the lines x−3 2 = y+15−7 = z−95 and x+12 = y−11 = z−9−3 is (1) 8√3 (2) 4√3 (3) 5√3 (4) 6√3

Q79.The distance, of the point (7, −2, 11) from the line x−61 = y−40 = z−83 along the line x−52 = y−1−3 = z−56 , is : (1) 12 (2) 14 (3) 18 (4) 21

Q79.Let (α, β, γ) be the foot of perpendicular from the point (1, 2, 3) on the line x+35 = y−12 = z+43 . then 19(α + β + γ) is equal to : (1) 102 (2) 101 (3) 99 (4) 100

Q79.Let the image of the point ( 1, 0, 7 ) in the line = = be the point ( α, β, γ ) . Then which one of the 1 2 3 2π 3π following points lies on the line passing through ( α, β, γ ) and making angles and with y - axis and z - 3 4 axis respectively and an acute angle with x - axis? (1) ( 1, - 2, 1 + √2 ) (2) ( 1, 2, 1 - √2 ) (3) ( 3, 4, 3 - 2√2 ) (4) ( 3, - 4, 3 + 2√2 )

Q79.Let the point, on the line passing through the points P(1, −2, 3) and Q(5, −4, 7), farther from the origin and at distance of 9 units from the point P, be (α, β, γ). Then α2 + β2 + γ 2 is equal to : (1) 165 (2) 160 (3) 155 (4) 150

Q79.If the shortest distance between the lines x−λ 2 = y−43 = z−34 and x−24 = y−46 = z−78 is √2913 , then a value of λ is : (1) -1 (2) −1325 (3) 13 (4) 1 25

Q79.Let P(3, 2, 3), Q(4, 6, 2) and R(7, 3, 2) be the vertices of Δ PQR. Then, the angle ∠QPR is (1) π 6 (2) cos−1( 187 ) (3) cos−1( 181 ) (4) π3

Q79.For λ > 0, let θ be the angle between the vectors →a = ^i + λ^j −3^k and →b = 3^i −^j + 2^k. If the vectors →a + →b and →a −→b are mutually perpendicular, then the value of (14 cos θ)2 is equal to (1) 50 (2) 40 (3) 25 (4) 20 JEE Main 2024 (04 Apr Shift 2) JEE Main Previous Year Paper

Q79.Let PQR be a triangle with R(−1, 4, 2). Suppose M(2, 1, 2) is the mid point of PQ . The distance of the centroid of ΔPQR from the point of intersection of the line x−20 = 2y = z+3−1 and x−11 = y+3−3 = z+11 is (1) 69 (2) 9 (3) √69 (4) √99

Q79.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 y−1 x−2 3 = 1 = z−22 . Then which of the following points lies on L? (1) (−13 , 1, −1) (2) (−13 , −1, 1) (3) (−13 , 1, 1) (4) (−13 , −1, −1)

Q79.Let d be the distance of the point of intersection of the lines x+63 = 2y = z+11 and x−74 = y−93 = z−42 from the point (7, 8, 9) . Then d2 + 6 is equal to : (1) 69 (2) 78 (3) 72 (4) 75

Q79.Let P(x, y, z) be a point in the first octant, whose projection in the xy-plane is the point Q. Let OP = γ ; the angle between OQ and the positive x-axis be θ; and the angle between OP and the positive z-axis be ϕ, where O is the origin. Then the distance of P from the x-axis is θ cos2 ϕ (1) γ√1 −sin2 (2) ϕ cos2 θ γ√1 −sin2 θ sin2 ϕ (3) γ√1 + cos2 (4) ϕ sin2 θ γ√1 + cos2

Q79.Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is 2 4 (1) (2) 25 25 (3) 2 (4) 4 3 75

Q79.If the shortest distance between the lines 𝑥−𝜆 = 𝑦−2 = 𝑧−1 and 𝑥−√3 = 𝑦−1 = 𝑧−2 is 1, then the sum of all −2 1 1 1 −2 1 possible values of 𝜆 is (1) 0 (2) 2√3 (3) 3√3 (4) −2√3

Q79.Consider the line L passing through the points (1, 2, 3) and (2, 3, 5). The distance of the point ( 113 , 113 , 193 ) from the line L along the line 3x−112 = 3y−111 = 3z−192 is equal to (1) 6 (2) 5 (3) 4 (4) 3

Q79.Let 𝐿1: →𝑟= ^𝑖- ^𝑗+ 2 ^𝑘+ 𝜆 ^𝑖- ^𝑗+ 2 ^𝑘, 𝜆∈𝑅, 𝐿2: →𝑟= ^𝑗- ^𝑘+ 𝜇3 ^𝑖+ ^𝑗+ 𝑝 ^𝑘, 𝜇∈𝑅 and 𝐿3: →𝑟= 𝛿(𝑙 ^𝑖+ 𝑚 ^𝑗+ 𝑛 ^𝑘), 𝛿∈𝑅 be three lines such that 𝐿1 is perpendicular to 𝐿2 and 𝐿3 is perpendicular to both 𝐿1 and 𝐿2. Then the point which lies on 𝐿3 is (1) ( - 1, 7, 4 ) (2) ( - 1, - 7, 4 ) (3) ( 1, 7, - 4 ) (4) ( 1, - 7, 4 )

Q79.Let P(α, β, γ) be the image of the point Q(3, −3, 1) in the line x−01 = y−31 = z−1−1 and R be the point (2, 5, −1). If the area of the triangle PQR is λ and λ2 = 14K , then K is equal to : (1) 36 (2) 81 (3) 72 (4) 18

Q79.Let 𝑃 and 𝑄 be the points on the line = = which are at a distance of 6 units from the point 8 2 2 𝑅( 1, 2, 3 ) . If the centroid of the triangle 𝑃𝑄𝑅 is 𝛼, 𝛽, 𝛾, then 𝛼2 + 𝛽2 + 𝛾2 is: (1) 26 (2) 36 (3) 18 (4) 24

Q79.The shortest distance between lines 𝐿1 and 𝐿2, where 𝐿1: 2 = −3 = 2 and 𝐿2 is the line passing through 𝑥−3 𝑦 𝑧−1 the points 𝐴−4, 4, 3, 𝐵−1, 6, 3 and perpendicular to the line = = , is −2 3 1 (1) 121 (2) 24 √221 √117 (3) 141 (4) 42 √221 √117

Q79.Let (α, β, γ) be the image of the point (8, 5, 7) in the line x−12 = y+13 = z−25 . Then α + β + γ is equal to : (1) 16 (2) 20 (3) 14 (4) 18

Q80.A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is: (1) 2 (2) 2 5 7 1 1 (3) (4) 7 5

Q80.An integer is chosen at random from the integers 1 , 2, 3, . . . . . , 50. The probability that the chosen integer is a multiple of atleast one of 4, 6 and 7 is (1) 8 (2) 21 25 50 (3) 9 (4) 14 50 25 is equal to _______. +

Q80.Let P be the point of intersection of the lines x−21 = y−45 = z−21 and x−32 = y−23 = z−32 . Then, the shortest distance of P from the line 4x = 2y = z is (1) 5√14 (2) 3√14 7 7 (3) √14 (4) 6√14 7 7

Q80.A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is- 2 1 (1) (2) 9 9 (3) 2 (4) 1 27 27