Practice Questions

Q77.Let y = y(x) be the solution of the differential equation (x2 + 4)2dy + (2x3y + 8xy −2)dx = 0. If y(0) = 0, then y(2) is equal to (1) π (2) 2π 32 (3) π (4) π 8 16

Q77.If the solution y = y(x) of the differential equation (x4 + 2x3 + 3x2 + 2x + 2)dy −(2x2 + 2x + 3)dx = 0 satisfies y(−1) = −π4 , then y(0) is equal to : (1) π 2 (2) −π2 (3) 0 (4) π 4 →

Q77.Between the following two statements: Statement I : Let →a = ^i + 2^j −3^k and →b = 2^i + ^j −^k. Then the vector →r satisfying →a × →r = →a × →b and →a ⋅→r = 0 is of magnitude √10. Statement II : In a triangle ABC, cos 2A + cos 2B + cos 2C ≥−32 . (1) Statement I is incorrect but Statement II is (2) Both Statement I and Statement II are correct. correct. (3) Statement I is correct but Statement II is (4) Both Statement I and Statement II are incorrect. incorrect.

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 →a = 6^i + ^j −^k and b = ^i + ^j. If→cis a is vector such that |→c| ≥6,→a⋅→c= 6|→c|, |→c−→a| = 2√2 and the angle between →a × →b and →c is 60∘ , then |(→a × →b) × →c| is equal to: (1) 9 2 (6 −√6) (2) 23 √6 (3) 9 2 (6 + √6) (4) 23 √3

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

Q78.Let the position vectors of the vertices A, B and C of a triangle be 2 ^i + 2 ^j + ^k, ^i + 2 ^j + 2 ^k and 2 ^i + ^j + 2 ^k respectively. Let l1, l2 and l3 be the lengths of perpendiculars drawn from the ortho centre of the triangle on the sides AB, BC and CA respectively, then l12 + l22 + l32 equals : 1 1 (1) (2) 5 2 (3) 1 (4) 1 4 3 x y - 1 z - 2

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.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 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

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.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.If an unbiased dice is rolled thrice, then the probability of getting a greater number in the ith roll than the number obtained in the (i −1)th roll, i = 2, 3, is equal to (1) 3/54 (2) 2/54 (3) 1/54 (4) 5/54

Q81.Let the set C = {(x, y) ∣x2 −2y = 2023, x, y ∈N}. Then ∑(x,y)∈C(x y)

Q81.Let α, β ∈ be roots of equation x2 −70x + λ = 0, where λ2 , λ3 ∉ . If λ assumes the minimum possible value, (√α−1+√β−1)(λ+35) then is equal to : |α−β|

Q81.Let α, β be the roots of the equation x2 −x + 2 = 0 with Im (α) >Im (β). Then α6 + α4 + β4 −5α2 is equal to

Q81.If 𝛼 denotes the number of solutions of 1 −𝑖𝑥= 2𝑥 and 𝛽= 𝑧 where 𝑧= 𝜋 + 𝑖41 −√𝜋· 𝑖 √𝜋−𝑖 arg𝑧, 41 𝑖+ 1 + 𝑖, √𝜋+ √𝜋· 𝑖= √−1, then the distance of the point 𝛼, 𝛽 from the line 4𝑥−3𝑦= 7 is ______ JEE Main 2024 (31 Jan Shift 1) JEE Main Previous Year Paper

Q81.Let x1, x2, x3, x4 be the solution of the equation 4x4 + 8x3 −17x2 −12x + 9 = 0 and (4 + x21) (4 + x22) (4 + x23) (4 + x24) = 12516 m. Then the value of m is

Q81.The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7 , when the repetition of digits is not allowed, and which are not divisible by 3 , is equal to__________ JEE Main 2024 (08 Apr Shift 1) JEE Main Previous Year Paper

Q81.Let 𝑃= 𝑧∈ℂ: 𝑧+ 2 −3𝑖≤1 and 𝑄= 𝑧∈ℂ: 𝑧1 + 𝑖+ ¯𝑧1 −𝑖≤−8. Let in 𝑃∩𝑄, 𝑧−3 + 2𝑖 be maximum and minimum at 𝑧1 and 𝑧2 respectively. If 𝑧12 + 2𝑧2 = 𝛼+ 𝛽√2, where 𝛼, 𝛽 are integers, then 𝛼+ 𝛽 equals __________

Q81.Let a = 1 + 2C23! + 3C24! + 4C25! + … 1! + 2! + 3! + … Then 2b is equal to a2

Q82.Let the first term of a series be T1 = 6 and its rth term Tr = 3Tr−1 + 6r, r = 2, 3, n. If the sum of the first n terms of this series is 1 (n2 −12n + 39) (4 ⋅6n −5 ⋅3n + 1), then n is equal to______ 5 JEE Main 2024 (06 Apr Shift 1) JEE Main Previous Year Paper

Q82.Let S = {sin2 2θ : (sin4 θ + cos4 θ)x2 + (sin 2θ)x + (sin6 θ + cos6 θ) = 0 has real roots }. If α and β be the smallest and largest elements of the set S , respectively, then 3 ((α −2)2 + (β −1)2) equals _________

Q82.Let α = 12 + 42 + 82 + 132 + 192 + 262 + … … . upto 10 terms and β = ∑10n=1 n4 . If 4α −β = 55k + 40, then k is equal to _______. 6

Q82.Let a1, a2, a3, … be in an arithmetic progression of positive terms. Let Ak = a21 −a22 + a23 −a24 + … + a22k−1 −a22k . If A3 = −153, A5 = −435 and a21 + a22 + a23 = 66 , then a17 −A7 is equal to______ is p , then 108p is equal to