Practice Questions

Q88.If the solution of the differential equation (2x + 3y −2)dx + (4x + 6y −7)dy = 0, y(0) = 3, is αx + βy + 3 loge |2x + 3y −γ| = 6, then α + 2β + 3γ is equal to ______.

Q88.The sum of squares of all possible values of 𝑘, for which area of the region bounded by the parabolas 2𝑦2 = 𝑘𝑥 and 𝑘𝑦2 = 2𝑦−𝑥 is maximum, is equal to:

Q88.If ∫ π3 √1 −sin 2xdx = α + β√2 + γ√3, where α, β and γ are rational numbers, then 3α + 4β −γ is equal 6 to _____.

Q89.Let y = y(x) be the solution of the differential equation dx + (1+x2)2 − 1 Then the area enclosed by the curve f(x) = y(x)e (1+x2) and the line y −x = 4 is__________

Q89.The least positive integral value of α, for which the angle between the vectors αˆi −2ˆj + 2ˆk and αˆi + 2αˆj −2ˆk is acute, is _____.

Q89.Let the set of all positive values of λ , for which the point of local minimum of the function (1 + x (λ2 −x2)) satisfies x2+x+2 < 0, be (α, β). Then α2 + β2 is equal to _________ x2+5x+6

Q89.The area of the region enclosed by the parabolas y = x2 −5x and y = 7x −x2 is → →

Q89.Let α|x| = |y|exy−β, α, β ∈N be the solution of the differential equation x dy −y dx + xy(x dy + y dx) = 0, y(1) = 2. Then α + β is equal to ________ β + γ is equal

Q89.Let the set of all values of p, for which f(x) = (p2 −6p + 8) (sin2 2x −cos2 2x) + 2(2 −p)x + 7 does not have any critical point, be the interval (a, b). Then 16ab is equal to _______

Q89.Let y = y(x) be the solution of the differential equation (1 −x2)dy = [xy + (x3 + 2)√3(1 −x2)]dx −1 < x < 1, y(0) = 0. If y( 21 ) = mn , m and n are coprime numbers, then m + n is equal to __________.

Q89.If the solution curve y = y(x) of the differential equation (1 + y2)(1 + loge x)dx + xdy = 0, x > 0 passes α−tan( 23 ) through the point (1, 1) and y(e) = 3 , then α + 2β is β+tan( 2 )

Q89.Consider a line L passing through the points P(1, 2, 1) and Q(2, 1, −1). If the mirror image of the point A(2, 2, 2) in the line L is (α, β, γ), then α + β + 6γ is equal to _______

Q89.Let →a = 9^i −13^j + 25^k,→b = 3^i + 7^j −13^k and →c = 17^i −2^j + ^k be three given vectors. If →r is a vector such |593→r+67→a|2 is equal to___________ that →r × →a = (→b + →c) × →a and →r ⋅(→b −→c) = 0 , then (593)2

Q89.Let 𝑌= 𝑌( 𝑋) be a curve lying in the first quadrant such that the area enclosed by the line 𝑌- 𝑦= 𝑌' (𝑥) (𝑋- 𝑥) and the co-ordinate axes, where ( 𝑥, 𝑦) is any point on the curve, is always -𝑦2 + 1, 𝑌'𝑥≠0. If 𝑌( 1 ) = 1, then 12 𝑌( 2 ) equals ________. 2𝑌' (𝑥)

Q89.Let →𝑎= 3 ^𝑖+ 2 ^𝑗+ ^𝑘, →𝑏= 2 ^𝑖− ^𝑗+ 3 ^𝑘 and →𝑐 be a vector such that →𝑎+ →𝑏× →𝑐= 2→𝑎× →𝑏+ 24 ^𝑗−6 ^𝑘 and → 2 →𝑎− 𝑏+ ^𝑖. →𝑐= −3. Then →𝑐 is equal to _______.

Q89.If the shortest distance between the lines x−λ 3 = y−2−1 = z−11 and x+2−3 = y+52 = z−44 is √3044 , then the largest possible value of |λ| is equal to _________

Q89.If 𝑑𝑥 1 + 𝑥−𝑦2 , 𝑥1 = 1, then 5𝑥2 is equal to: 𝑑𝑦= 𝑦

Q89.Let →𝑎 and 𝑏 be two vectors such that →𝑎= 1, 𝑏= 4 and →𝑎⋅ 𝑏= 2. If →𝑐= 2 →𝑎× 𝑏−3 𝑏 and the angle between →𝑏 and→𝑐 is 𝛼, then 192sin2𝛼 is equal to _________

Q89.Let →a = 2^i −3^j + 4^k,→b = 3^i + 4^j −5^k and a vector →c be such that →a × (→b + →c) + →b × →c = ^i + 8^j + 13^k . If →a ⋅→c = 13 , then (24 −→b ⋅→c) is equal to_______

Q89.If the solution curve, of the differential equation 𝑑𝑦 𝑥+ 𝑦- 2 𝑑𝑥= 𝑥- 𝑦 passing through the point ( 2, 1 ) is tan-1𝑦- 1 - 1 𝑦- 1 2 = 1, then 5𝛽+ 𝛼 is equal to 𝑥- 1 𝛽log𝑒𝛼+ 𝑥- 1 log𝑒𝑥- x - 2 y z - 7 x + 3 y + 2 z + 2

Q89.Let ABC be a triangle of area 15√2 and the vectors AB→ = ^i + 2^j −7^k, BC→ = a^i + b^j + ck and −→ AC = 6^i + d^j −2^k, d > 0. Then the square of the length of the largest side of the triangle ABC is _______

Q89.Let the area of the region {(x, y) : 0 ≤x ≤3, 0 ≤y ≤min{x2 + 2, 2x + 2}} be A . Then 12A is equal to ______.

Q89.If 𝑥= 𝑥𝑡 is the solution of the differential equation 𝑡+ 1𝑑𝑥= 2𝑥+ 𝑡+ 14𝑑𝑡, 𝑥0 = 2, then 𝑥1 equals ________

Q90.Let P(α, β, γ) be the image of the point Q(1, 6, 4) in the line x1 = y−12 = z−23 . Then 2α + to_______ JEE Main 2024 (08 Apr Shift 2) JEE Main Previous Year Paper

Q90.The lines = = and = = intersect at the point P. If the distance of P from the line 2 -2 16 4 3 1 x + 1 y - 1 = = z - 1 is 𝑙, then 14𝑙2 is equal to _____. 2 3 1 JEE Main 2024 (27 Jan Shift 2) JEE Main Previous Year Paper