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.Let y = y(x) be the solution of the differential equation (1 + y2)etan xdx + cos2 x (1 + e2 tan x)dy = 0, y(0) = 1. Then y ( π4 ) is equal to (1) 2 (2) 2 e e2 (3) 1 (4) 1 e e2

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 𝑦= 𝑦( 𝑥) be the solution of the differential equation 𝑑𝑦 tan𝑥+ 𝑦 𝜋 𝑑𝑥= sin𝑥sec𝑥- sin𝑥tan𝑥, 𝑥∈0, 2 satisfying the 𝜋 𝜋 condition 𝑦 = 2. Then, 𝑦 is 4 3 2 + log𝑒3 (1) √32 + log𝑒√3 (2) √32 (3) √31 + 2log𝑒3 (4) √32 + log𝑒3 →

Q76.Let y = y(x) be the solution of the differential equation sec xdy + {2(1 −x) tan x + x(2 −x)}dx = 0 such that y(0) = 2. Then y(2) is equal to : (1) 2 (2) 2{1 −sin(2)} (3) 2{sin(2) + 1} (4) 1

Q76.Let 𝑓: 𝑅→𝑅 be defined 𝑓𝑥= 𝑎𝑒2𝑥+ 𝑏𝑒𝑥+ 𝑐𝑥. If 𝑓(0) = - 1, 𝑓'log𝑒2 = 21 and ∫0log4 2 the value of |𝑎+ 𝑏+ 𝑐| equals: (1) 16 (2) 10 (3) 12 (4) 8 2

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.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.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.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 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

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 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 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.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 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.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.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 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 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, 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.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= ^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