Practice Questions

Q78.Let →𝑎= −5 ^𝑖+ ^𝑗−3 ^𝑘, →𝑏= ^𝑖+ 2 ^𝑗−4 ^𝑘 and →𝑐= →𝑎× →𝑏× ^𝑖× ^𝑖× ^𝑖. Then →𝑐⋅− ^𝑖+ ^𝑗+ ^𝑘 is equal to (1) -12 (2) -10 (3) -13 (4) -15

Q78.Let OA→ = 2→a, OB = 6→a + 5→b and OC = 3→b, where O is the origin. If the area of the parallelogram with −−→ → adjacent sides OA and OC is 15 sq. units, then the area (in sq. units) of the quadrilateral OABC is equal to : (1) 32 (2) 40 (3) 38 (4) 35

Q78.Let y = y(x) be the solution of the differential equation (2x loge x) dxdy + 2y = x3 loge x, x > 0 and y (e−1) = 0. Then, y(e) is equal to (1) −3e (2) −32e (3) −23e (4) −2e

Q78.Let a unit vector ˆu = xˆi + yˆj + zˆk make angles π2 , π3 and 2π3 with the vectors √2ˆi1 + √21 ˆk, √21 ˆj + √21 ˆk and 1 + 1 ˆj respectively. If →v= 1 + 1 ˆj + 1 ˆk, then |^u −→v|2 is equal to √2ˆi √2 √2ˆi √2 √2 (1) 11 (2) 5 2 2 (3) 9 (4) 7

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 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 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 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.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.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 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 (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 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 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.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 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.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 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(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.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 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 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 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

Q80.Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable 𝑥 to be the number of rotten apples in a draw of two apples, the variance of 𝑥 is 37 57 (1) (2) 153 153 47 40 (3) (4) 153 153