Practice Questions

Q78.Let →a = ^i + ^j + ^k,→b = 2^i + 4^j −5^k and →c = x^i + 2^j + 3^k, x ∈R. If →d is the unit vector in the direction of →b + →c such that →a ⋅→d = 1, then (→a × →b) ⋅→c is equal to (1) 11 (2) 3 (3) 9 (4) 6

Q78.Let 𝛼, 𝛽, 𝛾 be mirror image of the point 2, 3, 5 in the line 𝑥−1 = 𝑦−2 = 𝑧−3 . Then 2𝛼+ 3𝛽+ 4𝛾 is equal to 2 3 4 (1) 32 (2) 33 (3) 31 (4) 34 𝑥−1 𝑦+ 1 𝑧+ 4

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 = 2^i + α^j + ^k,→b = −^i + ^k, →c = β^j −^k, where α and β are integers and αβ = −6. Let the values of the √21 ordered pair (α, β), for which the area of the parallelogram of diagonals →a + →b and →b + →c is , be (α1, β1) 2 and (α2, β2). Then α21 + β21 −α2β2 is equal to (1) 19 (2) 17 (3) 24 (4) 21

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 →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 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 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 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 (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 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 (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.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 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.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 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 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(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.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 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.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.Two integers x and y are chosen with replacement from the set {0, 1, 2, 3, … . . , 10}. Then the probability that |x −y| > 5 is : (1) 30 (2) 62 121 121 (3) 60 (4) 31 121 121

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.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