Practice Questions

Q76.If y = y ( x ) is the solution curve of the differential equation x2 - 4dy - y2 - 3ydx = 0, x > 2, y(4) = 3 and 2 the slope of the curve is never zero, then the value of y ( 10 ) equals : 3 3 (1) 1 (2) 1 + 2√2 1 + ( 8 ) 4 3 3 (3) (4) 1 1 - 2√2 1 - ( 8 ) 4

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

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.Let y = y(x) be the solution of the differential equation (1 + x2) dxdy + y = etan−1 x , y(1) = 0. Then y(0) is (1) 2 1 (eπ/2 −1) (2) 21 (1 −eπ/2) (3) 4 1 (1 −eπ/2) (4) 14 (eπ/2 −1)

Q77.Consider three vectors →a,→b, →c. Let |→a| = 2, |→b| = 3 and →a = →b × →c. If α ∈[0, 3 ] is the angle between the vectors →b and →c, then the minimum value of 27|→c −→a|2 is equal to: (1) 110 (2) 124 (3) 121 (4) 105

Q77.Let three vectors →a = α^i + 4^j + 2^k, b = 5^i + 3^j + 4^k,→c= x^i + y^j + z^k form a triangle such that →c = →a −→b and the area of the triangle is 5√6. If α is a positive real number, then |→c|2 is equal to: (1) 16 (2) 14 (3) 12 (4) 10 → →−−−

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.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.Let →a = 4^i −^j + ^k,→b = 11^i −^j + ^k and →c be a vector such that (→a + →b) × →c = →c × (−2→a + 3→b). If (2→a + 3→b) ⋅→c = 1670, then |→c|2 is equal to : (1) 1609 (2) 1618 (3) 1600 (4) 1627 →

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

Q77.Let x = x(t) and y = y(t) be solutions of the differential equations dxdt + ax = 0 and dydt + by = 0 respectively, a, b ∈R. Given that x(0) = 2 ; y(0) = 1 and 3 y(1) = 2 x(1), the value of t, for which x(t) = y(t), is : (1) log 2 2 (2) log4 3 3 4 2 (3) log3 4 (4) log 3 → → and→cbe the vector such that →a×→c= b and →a⋅→c= 3, then

Q77.The temperature 𝑇𝑡 of a body at time 𝑡= 0 is 160° 𝐹 and it decreases continuously as per the differential 𝑑𝑇 equation 𝑑𝑡= −𝐾𝑇−80, where 𝐾 is positive constant. If 𝑇15 = 120° 𝐹, then 𝑇45 is equal to (1) 85° 𝐹 (2) 95° 𝐹 (3) 90° 𝐹 (4) 80° 𝐹

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.Consider a 𝛥𝐴𝐵𝐶 where 𝐴1, 3, 2, 𝐵−2, 8, 0 and 𝐶3, 6, 7. If the angle bisector of ∠𝐵𝐴𝐶 meets the line 𝐵𝐶 at 𝐷, then the length of the projection of the vector →𝐴𝐷 on the vector →𝐴𝐶 is: (1) 37 (2) √38 2√38 2 39 (3) (4) √19 2√38

Q77.Let A(2, 3, 5) and C(−3, 4, −2) be opposite vertices of a parallelogram ABCD if the diagonal −→ BD = ˆi + 2ˆj + 3ˆk then the area of the parallelogram is equal to (1) 1 2 √410 (2) 21 √474 (3) 1 2 √586 (4) 21 √306 → → →

Q77.Let 𝑦= 𝑦𝑥 be the solution of the differential equation 𝑑𝑦 2𝑥𝑥+ 𝑦3 −𝑥𝑥+ 𝑦−1, 𝑦0 = 1. Then, 1 + 𝑦1 𝑑𝑥= √2 √2 equals: (1) 4 (2) 3 4 + √𝑒 3 −√𝑒 2 1 (3) (4) 1 + √𝑒 2 −√𝑒