Practice Questions

Q85.If the area enclosed between the curves y = kx2 and x = ky2, (k > 0), is 1 sq. unit. Then k is (1) √3 (2) 1 √3 (3) √3 (4) 2 2 √3 JEE Main 2019 (10 Jan Shift 1) JEE Main Previous Year Paper 3 1

Q85.The area (in sq. units) of the region bounded by the curve x2 = 4y and the straight line x = 4y −2 is : (1) 5 (2) 9 4 8 (3) 7 (4) 3 8 4

Q86.Let →a, b and →cbe three unit vectors, out of which vectors b and →care non-parallel. If α and β are the angles → → → b = 21 b, then |α −β| is equal to : which vector →a makes with vectors b and →crespectively and →a× ( ×→c) (1) 90o (2) 60o (3) 45o (4) 30o y−2

Q86.Let 𝑆𝛼= 𝑥, 𝑦: 𝑦2 ≤𝑥, 0 ≤𝑥≤𝛼 and A𝛼 is area of the region 𝑆𝛼. If for a 𝜆, 0 < 𝜆< 4, A𝜆: A4 = 2: 5, then 𝜆 equals: (1) 2 13 (2) 4 13 4 2 5 25 (3) 4 13 (4) 2 13 4 2 25 5

Q86.Consider the differential equation, 𝑦2𝑑𝑥+ 𝑥- 𝑦𝑑𝑦= 0. If value of 𝑦 is 1 when 𝑥= 1, then the value of 𝑥 for which 𝑦= 2, is (1) 3 - 1 (2) 3 - 2 2 √𝑒 √𝑒 1 1 5 1 (3) + (4) + 2 √𝑒 2 √𝑒

Q86.Let 𝑦= 𝑦( 𝑥) be the solution of the differential equation, 𝑥2 + 1 2 𝑑𝑦 2𝑥(𝑥2 + 1)𝑦= 1 such that 𝑦0 = 0 . 𝑑𝑥+ If 𝑦1 = 𝜋 then the value of 𝑎 is √𝑎 32, (1) 1 (2) 1 (3) 1 (4) 1 16 2 4

Q86.If y(x) is the solution of the differential equation dxdy + ( 2x+1x )y = e−2x, x > 0, where y(1) = 21 e−2, then: (1) y (loge 2) = loge 4 (2) y (loge 2) = loge4 2 (3) y(x) is decreasing in ( 12 , 1) (4) y(x) is decreasing in (0,1)

Q86.The solution of the differential equation x y(1) = 1, is dx + 2y = x2, (x ≠0) with (1) y = x35 + 5x21 (2) y = 34 x2 + 4x21 (3) y = x24 + 4x23 (4) y = 45 x3 + 5x21 → → → → → → → → →−−−−

Q86.If dy + y = dx , x ∈(−π3 , π3 ), and y( π4 ) = 34 , then y(−π4 ) equals x x cos2 cos2 (1) 1 (2) 1 3 3 + e3 (3) 3 1 + e6 (4) −43 →

Q86.Let f(x) be a differentiable function such that f ′(x) = 7 −34 f(x)x , (x > 0) and f(1) ≠4. Then lim x→0+ (1) does not exist. (2) exists and equals 4 . (3) exists and equals 4 . (4) exists and equals 0 . 7 → → → → →

Q86.Let √3^i + ^j,^i + √3^j and β^i + (1 −β)^j respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is 3 , √2 then the sum of all possible values of β is: (1) 4 (2) 3 (3) 2 (4) 1

Q86.If y = y(x) is the solution of the differential equation dxdy = (tanx −y)sec2x , y(0) = 0, then y(−π4 ) is equal to: (1) 1 e −2 (2) 2 + 1e (3) e −2 (4) 12 −e

Q86.Let 𝑦= 𝑦𝑥 be the solution of the differential equation, 𝑑𝑦 𝑦tan𝑥= 2𝑥+ 𝑥2tan𝑥, 𝑥∈- π π such that 𝑑𝑥+ 2, 2, 𝑦0 = 1 . Then JEE Main 2019 (10 Apr Shift 2) JEE Main Previous Year Paper π π π (1) 𝑦'π - 𝑦'- 4 4 = π - √2 (2) y' 4 + y'- 4 = - √2 π2 (3) 𝑦π - 𝑦-π = √2 (4) y'π + y'- π = + 2 4 4 4 4 2

Q86.The area of the region A = {(x, y) : 0 ≤y ≤x|x| + 1 and −1 ≤x ≤1} in sq. units, is (1) 4 (2) 2 3 (3) 1 (4) 2 3 3 →

Q86.If cosx dxdy −ysinx = 6x, (0 < x < π2 ) and y( π3 ) = 0, then y( π6 ) is equal to (1) −π2 (2) π2 4√3 2√3 (3) −π22 (4) −π22√3 π

Q86.If 𝑦= 𝑦( 𝑥) is the solution of the differential equation, 𝑥 𝑑𝑦 2𝑦= 𝑥2 satisfying 𝑦1 = 1, then 𝑦 1 is equal to 𝑑𝑥+ 2 (1) 7 (2) 1 64 4 13 49 (3) (4) 16 16 2

Q86.Let y = y(x) be the solution of the differential equation, x dxdy + y = x loge x, (x > 1). If 2y(2) = loge 4 −1, then y(e) is equal to (1) −e2 (2) 4e (3) −e22 (4) e24

Q86.Let α ∈R and the three vectors →a = αˆi + ˆj + 3ˆk, b = 2ˆi + ˆj −αˆk and →c= αˆi −2ˆj + 3ˆk. Then the set S = { → α :→a, b and →care coplanar} (1) is singleton (2) contains exactly two positive numbers (3) is empty (4) contains exactly two numbers only one of which is positive

Q87.Let α = (λ −2) →a+ b and β = (4λ −2) →a+ 3 b, be two given vectors where vectors →a and b are non-collinear. → → The value of λ for which vectors α and β are collinear, is: (1) −4 (2) −3 (3) 4 (4) 3

Q87.The distance of the point having position vector -^𝑖+ 2^𝑗+ 6^𝑘 from the straight line passing through the point 2, 3, - 4 and parallel to the vector, 6^𝑖+ 3^𝑗- 4^𝑘 is (1) 4√3 (2) 6 (3) 2√13 (4) 7

Q87.If a unit vector →a makes angles θ ∈(0, π) with ˆk, then a value of θ is: 3 with ˆi, π4 with ˆj and (1) 5π (2) 5π 6 12 (3) π (4) 2π 4 3

Q87.Let →𝑎= 3^𝑖+ 2^𝑗+ 𝑥^𝑘 and →𝑏= ^𝑖- ^𝑗+ ^𝑘, for some real 𝑥. Then the condition for →𝑎× →𝑏 = 𝑟 to follow (1) 0 < 𝑟≤ 3 (2) 𝑟≥ 3 √ 2 5√ 2 (3) 3 < 3 (4) 3 3 < r < 3 √ 2 𝑟≤3√ 2 √ 2 5√ 2

Q87.If an angle between the line, x+1 , then a value 2 = 1 = z−3−2 and the plane, x −2y −kz = 3 is cos−1( 2√23 ) of k is (1) √53 (2) √35 (3) −35 (4) −53

Q87.Let →a = ˆi +ˆj + √2ˆk, b = b1ˆi + b2ˆj + √2ˆk and →c= 5ˆi +ˆj + √2ˆk be three vectors such that the projection → → → vector of b on →a is →a . If →a+ b is perpendicular to →c, then b is equal to: (1) √22 (2) √32 (3) 6 (4) 4

Q87.The magnitude of the projection of the vector 2^𝑖+ 3^𝑗+ ^𝑘 on the vector perpendicular to the plane containing the vectors ^𝑖+ ^𝑗+ ^𝑘 and ^𝑖+ 2^𝑗+ 3^𝑘, is: (1) 3√6 (2) √ 32 (3) √6 (4) √32