Practice Questions

Q76.Let y = y(x) be the solution curve of the differential equation sec y dydx + 2x sin y = x3 cos y, y(1) = 0. Then y(√3) is equal to : (1) π (2) π 3 6 (3) π (4) π 12 4

Q76.The solution of the differential equation (x2 + y2)dx −5xy dy = 0, y(1) = 0, is : (1) x2 −2y2 6 = x (2) x2 −4y2 6 = x (3) x2 −4y2 5 = x2 (4) x2 −2y2 5 = x2 →

Q76.The area (in square units) of the region enclosed by the ellipse x2 + 3y2 = 18 in the first quadrant below the line y = x is (1) √3π −34 (2) √3π + 1 (3) √3π (4) √3π + 34

Q76.If (a, b) be the orthocentre of the triangle whose vertices are (1, 2), (2, 3) and (3, 1), and I1 = ∫ba xsin(4x −x2) dx, I2 = ∫ba sin(4x −x2) dx , then 36 I1I2 is equal to : (1) 72 (2) 88 (3) 80 (4) 66

Q76.A function y = f(x) satisfies f(x) sin 2x + sin x −(1 + cos2 x)f ′(x) = 0 with condition f(0) = 0. Then f( π2 ) is equal to (1) 1 (2) 0 (3) −1 (4) 2 → → →

Q76.If sin( xy ) = loge x + α2 is the solution of the differential equation x cos( xy ) dxdy = y cos( xy ) + x and y(1) = π3 , then α2 is equal to (1) 3 (2) 12 (3) 4 (4) 9 −−−

Q76.The area enclosed by the curves 𝑥𝑦+ 4𝑦= 16 and 𝑥+ 𝑦= 6 is equal to: (1) 28 −30log𝑒2 (2) 30 −28log𝑒2 (3) 30 −32log𝑒2 (4) 32 −30log𝑒2 2

Q76.Let 𝛼 be a non-zero real number. Suppose 𝑓: 𝑅→𝑅 is a differentiable function such that 𝑓0 = 1 and 𝑥→−∞𝑓𝑥=lim 1. If 𝑓'𝑥= 𝛼𝑓𝑥+ 3, for all 𝑥∈𝑅, then 𝑓−log𝑒2 is equal to ________. JEE Main 2024 (01 Feb Shift 2) JEE Main Previous Year Paper (1) 1 (2) 5 (3) 9 (4) 7

Q76.One of the points of intersection of the curves y = 1 + 3x −2x2 and y = x1 is ( 21 , 2). Let the area of the region enclosed by these curves be 1 (l√5 + m) −n loge(1 + √5), where l, m, n ∈N. Then l + m + n is 24 equal to (1) 29 (2) 31 (3) 30 (4) 32

Q76.The area (in sq. units) of the region described by {(x, y) : y2 ≤2x, and y ≥4x −1} is (1) 11 (2) 8 32 9 (3) 11 (4) 9 12 32

Q76.The integral ∫π/40 3 sin136x+5sincosx x (1) 3π −50 loge 2 + 20 loge 5 (2) 3π −25 loge 2 + 10 loge 5 (3) 3π −10 loge(2√2) + 10 loge 5 (4) 3π −30 loge 2 + 20 loge 5

Q76.The differential equation of the family of circles passing through the origin and having centre at the line y = x is : (1) (x2 −y2 + 2xy)dx = (x2 −y2 −2xy)dy (2) (x2 + y2 + 2xy)dx = (x2 + y2 −2xy)dy (3) (x2 + y2 −2xy)dx = (x2 + y2 + 2xy)dy (4) (x2 −y2 + 2xy)dx = (x2 −y2 + 2xy)dy π

Q76.Let the area of the region enclosed by the curves y = 3x, 2y = 27 −3x and y = 3x −x√x be A . Then 10A is equal to (1) 172 (2) 162 (3) 154 (4) 184

Q76.Suppose the solution of the differential equation (2+α)x−βy+2 represents a circle passing through dx = βx−2αy−(βγ−4α) origin. Then the radius of this circle is : (1) 2 (2) √17 (3) 1 (4) √17 2 2 → →

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.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.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.

Q77.Let →𝑎= ^𝑖+ 𝛼 ^𝑗+ 𝛽 ^𝑘 , 𝛼, 𝛽∈𝑅. Let a vector →𝑏 be such that the angle between →𝑎 and →𝑏 is 𝜋 and →𝑏 = 6, If 4 →𝑎· →𝑏= 3√2, then the value of 𝛼2 + 𝛽2 | →𝑎× →𝑏|2 is equal to (1) 90 (2) 75 (3) 95 (4) 85 2 is equal to

Q77.The position vectors of the vertices A, B and C of a triangle are 2 ^i - 3 ^j + 3 ^k, 2 ^i + 2 ^j + 3 ^k and - ^i + ^j + 3 ^k respectively. Let 𝑙 denotes the length of the angle bisector AD of ∠BAC where D is on the line segment BC, then 2𝑙2 equals : (1) 49 (2) 42 (3) 50 (4) 45

Q77.Let →a, b and→cbe three non-zero vectors such that b and→care non-collinear if →a+ 5b is collinear with →c,→b + 6→cis collinear with →a and →a+ α→b + β→c= →0, then α + β is equal to (1) 35 (2) 30 (3) −30 (4) −25

Q77.Let →𝑎= 3 ^𝑖+ ^𝑗−2 ^𝑘, 𝑏= 4 ^𝑖+ ^𝑗+ 7 ^𝑘 and →𝑐= ^𝑖−3 ^𝑗+ 4 ^𝑘 be three vectors. If a vectors →𝑝 satisfies →𝑝× →𝑏= →𝑐× →𝑏 and →𝑝⋅→𝑎= 0, then →𝑝⋅ ^𝑖− ^𝑗− ^𝑘 is equal to (1) 24 (2) 36 (3) 28 (4) 32

Q77.If y = y(x) is the solution of the differential equation dydx + 2y = sin(2x), y(0) = 43 , then y ( π8 ) is equal to: JEE Main 2024 (05 Apr Shift 1) JEE Main Previous Year Paper (1) eπ/8 (2) eπ/4 (3) e−π/4 (4) e−π/8

Q77.Let →a = 2^i + ^j −^k, b = ((→a× (^i + ^j)) ×^i) ×^i. Then the square of the projection of →a on b is : (1) 1 (2) 2 3 3 (3) 2 (4) 1 5 →

Q77.The set of all α, for which the vectors →a = αt^i + 6^j −3^k and →b = t^i −2^j −2αt^k are inclined at an obtuse angle for all t ∈R, is (1) (−43 , 1) (2) [0, 1) (3) (−43 , 0] (4) (−2, 0] L1 : →r = (2 + λ)^i + (1 −3λ)^j + (3 + 4λ)^k, λ ∈R m

Q77.Let OA→ =→a, OB→ = 12→a+ 4→b and OC→ = →b, where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then area of the areaquadrilateralof S OABC is equal to _____ (1) 6 (2) 10 (3) 7 (4) 8