Practice Questions

Q88.Let rk = , k ∈N. Then the value of ∑10k=1 7(rk−1)1 is equal to________ (1−x7)k+1dx ∫1 0

Q88.Let y = y(x) be the solution of the differential equation (x + y + 2)2dx = dy, y(0) = −2. Let the maximum and minimum values of the function y = y(x) in [0, π3 ] be α and β , respectively. If (3α + π)2 + β2 = γ + δ√3, γ, δ ∈Z , then γ + δ equals ______

Q88.If ∫−𝜋/𝜋/ 2 2 1 +8√2cos𝑥𝑑𝑥𝑒sin𝑥1 + sin4𝑥=

Q88.The area (in sq. units) of the part of circle x2 + y2 = 169 which is below the line 5x −y = 13 is πα 2β −652 + αβ sin−1( 1312 ) where α, β are coprime numbers. Then α + β is equal to

Q88.If S = {a ∈R : |2a −1| = 3[a] + 2{a}} , where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t , then 72 ∑a∈S a is equal to ______

Q88.If the solution y(x) of the given differential equation (ey + 1) cos x dx + ey sin x dy = 0 passes through the 6 ) is equal to_________ point ( π2 , 0), then the value of ey( π

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 f(t) = ∫π0 1−cos22x dxt sin2 x , 0 < t < π, then the value of ∫ 0 2 π2dtf(t) equals_________ 1 dy 2x (1+x2) y = xe ; y(0) = 0.

Q88.If ∫ B 1 dx = A( αx−1βx+3 ) 5√(x−1)4(x+3)6 α + β + 20AB is__________

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 = 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 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 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.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.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 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 __________.

Q90.A line with direction ratio 2, 1, 2 meets the lines x = y + 2 = z and x + 2 = 2y = 2z respectively at the point P and Q. if the length of the perpendicular from the point (1, 2, 12) to the line PQ is l, then l2 is JEE Main 2024 (29 Jan Shift 1) JEE Main Previous Year Paper

Q90.Let the point (−1, α, β) lie on the line of the shortest distance between the lines x+2−3 = y−24 = z−52 and y+6 x+2 −1 = 2 = z−10 . Then (α −β)2 is equal to___________ JEE Main 2024 (05 Apr Shift 2) JEE Main Previous Year Paper

Q90.If the shortest distance between the lines x+2 2 = y+33 = z−54 and x−31 = y−2−3 = z+42 is 3√538 k , and α −√α, where [x] denotes the greatest integer function, then 6α3 is equal to________ ∫k0 [x2]dx = JEE Main 2024 (04 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let the line of the shortest distance between the lines 𝐿1: →𝑟= ^𝑖+ 2 ^𝑗+ 3 ^𝑘+ 𝜆 ^𝑖− ^𝑗+ ^𝑘 and 𝐿2: →𝑟= 4 ^𝑖+ 5 ^𝑗+ 6 ^𝑘+ 𝜇 ^𝑖+ ^𝑗− ^𝑘 intersect 𝐿1 and 𝐿2 at 𝑃 and 𝑄 respectively. If 𝛼, 𝛽, 𝛾 is the midpoint of the line segment 𝑃𝑄, then 2𝛼+ 𝛽+ 𝛾 is equal to___________ JEE Main 2024 (01 Feb Shift 1) JEE Main Previous Year Paper

Q90.A line passes through 𝐴4, −6, −2 and 𝐵16, −2, 4. The point 𝑃𝑎, 𝑏, 𝑐 where 𝑎, 𝑏, 𝑐 are non-negative integers, on the line 𝐴𝐵 lies at a distance of 21 units, from the point 𝐴. The distance between the points 𝑃𝑎, 𝑏, 𝑐 and 𝑄4, −12, 3 is equal to ______. JEE Main 2024 (31 Jan 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

Q61.Let α1, α2, … , α7α1, α2, … , α7 be the roots of the equation x7 + 3x5 −13x3 −15x = 0 and |α1| ≥|α2| ≥… ≥|α7|. Then, α1α2 −α3α4 + α5α6 is equal to _______ ¯

Q61.Let S = {α : log2(92α−4 + 13) −log2( 25 ⋅32α−4 + 1) = 2}. Then the maximum value of β for which the equation x2 −2(∑α∈s α) 2x + ∑a∈s (α + 1)2β = 0 has real roots, is _____ .

Q61.The number of real roots of the equation √𝑥2 - 4𝑥+ 3 + √𝑥2 - 9 = √4𝑥2 - 14𝑥+ 6, is: (1) 0 (2) 1 (3) 3 (4) 2