Practice Questions

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 (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

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 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.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.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.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 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.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.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.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.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.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.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.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.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 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.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

Q80.An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is : (1) 5 (2) 5 256 715 3 3 (3) (4) 715 256 1

Q80.If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is: JEE Main 2024 (06 Apr Shift 2) JEE Main Previous Year Paper (1) 18 (2) 12 25 25 (3) 6 (4) 4 25 25

Q80.Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is : (1) 5 (2) 5 18 16 (3) 4 (4) 7 17 18 1C0+1C1 2C0+2C1+2C2 3C0+3C1+3C2+3C3 , b = 1 +

Q80.A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is (1) 5 (2) 1 6 6 (3) 5 (4) 6 11 11

Q80.The shortest distance between the lines x−34 = −11y+7 = z−15 and x−53 = y−9−6 = z+21 is: (1) 178 (2) 187 √563 √563 (3) 185 (4) 179 √563 √563

Q80.The coefficients a, b, c in the quadratic equation ax2 + bx + c = 0 are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8} . The probability of this equation having repeated roots is : (1) 1 (2) 1 128 64 (3) 3 (4) 3 256 128