Practice Questions

Q78.Let O be the origin and the position vector of A and B be 2ˆi + 2ˆj + ˆk and 2ˆi + 4ˆj + 4ˆk respectively. If the internal bisector of ∠AOB meets the line AB at C , then the length of OC is (1) 3 2 √31 (2) 32 √34 (3) 3 4 √34 (4) 23 √31

Q78.Let →a = ^i + 2^j + 3^k, b = 2^i + 3^j −5^k and→c= 3^i −^j + λ^k be three vectors. Let→rbe anit vector along →b + →c. If →r ⋅→a = 3, then 3λ is equal to: (1) 21 (2) 30 (3) 25 (4) 27

Q78.If the line 2−x 3 = 4λ+13y−2 = 4 −z makes a right angle with the line x+33μ = 1−2y6 = 5−z7 , then 4λ + 9μ is equal to : (1) 4 (2) 13 (3) 5 (4) 6

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 = 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.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.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.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 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 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.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.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 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.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 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(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.The coefficients a, b, c in the quadratic equation ax2 + bx + c = 0 are from the set {1, 2, 3, 4, 5, 6}. If the probability of this equation having one real root bigger than the other is p, then 216 p equals : (1) 57 (2) 76 (3) 38 (4) 19

Q80.There are three bags X, Y and Z . Bag X contains 5 one-rupee coins and 4 five-rupee coins; Bag Y contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y, is : (1) 1 (2) 1 4 2 (3) 5 (4) 1 12 3

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

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.Let the sum of two positive integers be 24 . If the probability, that their product is not less than 3 times their 4 greatest possible product, is m , where gcd(m, n) = 1, then n −m equals n (1) 10 (2) 9 (3) 11 (4) 8

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.A company has two plants A and B to manufacture motorcycles. 60% motorcycles are manufactured at plant A and the remaining are manufactured at plant B.80% of the motorcycles manufactured at plant A are rated of the standard quality, while 90% of the motorcycles manufactured at plant B are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If p is the probability that it was manufactured at plant B, then 126p is (1) 54 (2) 66 (3) 64 (4) 56