Practice Questions

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.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.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.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

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 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.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.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

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 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.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

Q87.Let is parallel to α and α = 3ˆi + ˆj and β = 2ˆi −ˆj + 3ˆk. If β = α, β1 −β2, β2 is perpendicular to where β1 −−→ → then β1 × β2 is equal to: (1) 1 2 (−3ˆi + 9ˆj + 5ˆk) (2) 3ˆi −9ˆj −5ˆk (3) −3ˆi + 9ˆj + 5ˆk (4) 1 + 2 (3ˆi −9ˆj 5ˆk)

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.Let →a = 2ˆi + λ1ˆj + 3ˆk, b = 4ˆi + (3 −λ2)ˆj + 6ˆk and →c= 3ˆi + 6ˆj + (λ3 −1)ˆk be three vectors such that → b = 2→a and →a is perpendicular to →c. Then a possible value of (λ1, λ2, λ3) is (1) (−12 , 4, 0) (2) (1, 5, 1) (3) ( 12 , 4, −2) (4) (1, 3, 1)

Q87.Let →𝑎= ^𝑖- ^𝑗, →𝑏= ^𝑖+ ^𝑗+ ^𝑘 and →𝑐 be a vector such that →𝑎× →𝑐+ →𝑏= →0 and →𝑎. →𝑐= 4, then |→𝑐| is equal to: 19 (1) (2) 9 2 (3) 17 (4) 8 2

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.If the volume of parallelepiped formed by the vectors ^𝑖+ 𝜆^𝑗+ ^𝑘, ^𝑗+ 𝜆^𝑘 and 𝜆^𝑖+ ^𝑘 is minimum, then 𝜆 is equal to: 1 (1) - (2) -√3 √3 1 (3) √3 (4) √3

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.Let A(3, 0, −1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the mid-point of AC . If G divides BM in the ratio, 2 : 1 , then cos(∠GOA) ( O being the origin) is equal to (1) 1 (2) 1 √30 6√10 (3) 1 (4) 1 √15 2√15 , then β

Q87.A plane which bisects the angle between the two given planes 2x −y + 2z −4 = 0 and x + 2y + 2z −2 = 0, passes through the point (1) (2, 4, 1) (2) (1, −4, 1) (3) (1, 4, −1) (4) (2, −4, 1)

Q87.The sum of the distinct real values of μ for which the vectors μˆi + ˆj + ˆk, ˆi + μˆj + ˆk, ˆi + ˆj + μˆk are co- planar, is (1) 0 (2) −1 (3) 1 (4) 2

Q87.Two lines x−3 1 = y+13 = z−6−1 and x+57 = y−2−6 = z−34 intersect at the point R. The reflection of R in the xy - plane has coordinates: (1) (2,-4,-7) (2) (2,4,7) (3) (2,-4,7) (4) (-2,4,7)

Q87.Let →a = ^i + 2^j + 4^k,→b = ^i + λ^j + 4^k and →c = 2^i + 4^j + (λ2 −1)^k be coplanar vectors. Then the non-zero vector →a × →c is: (1) −10^i −5^j (2) −14^i −5^j (3) −14^i + 5^j (4) −10^i + 5^j

Q88.If the plane 2𝑥- 𝑦+ 2𝑧+ 3 = 0 has the distances 1 and 2 units from the planes 4𝑥- 2𝑦+ 4𝑧+ 𝜆= 0 and 3 3 2𝑥- 𝑦+ 2𝑧+ 𝜇= 0 , respectively, then the maximum value of 𝜆+ 𝜇 is equal to: (1) 9 (2) 15 (3) 13 (4) 5 𝑥- 1 𝑦+ 1 𝑧