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 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 area of the region enclosed by the parabola 𝑦= 4𝑥−𝑥2 and 3𝑦= 𝑥−42 is equal to 32 (1) (2) 4 9 14 (3) 6 (4) 3

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.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.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.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.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 → →

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 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 (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

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.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 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.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 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= 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 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 →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 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 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

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 = 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.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.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 →