Practice Questions

Q68.If y = y(x) is the solution of the differential equation 5+ex2+y ⋅dydx + ex = 0 satisfying y(0) = 1 then value of y(loge 13) is (1) 1 (2) −1 (3) 0 (4) 2

Q68.Let a, b, c ∈R be such that a2 + b2 + c2 = 1. If a cos θ = b cos(θ + 2π3 ) = c cos(θ + 4π3 ),where θ = π9 , then the angle between the vectors aˆi + bˆj + cˆk and bˆi + cˆj + aˆk is: (1) 0 (2) 2π3 (3) π (4) π 2 9

Q68.Let →a = ˆi −2ˆj + ˆk and b = ˆi −ˆj + ˆk, be two vectors. If →c, is a vector such that b ×→c= b ×→a and →c⋅→a = 0, → then →c⋅ b, is equal to. (1) −32 (2) 21 (3) −12 (4) −1

Q68.If y = ( 2π −1) then the function p(x) is equal to : (1) cot x (2) cosec x (3) sec x (4) tan x

Q68.Let y = y(x) be the solution curve of the differential equation, (y2 −x) dxdy = 1 , satisfying y(0) = 1 . This curve intersects the X−axis at a point whose abscissa is (1) 2 −e (2) −e (3) 2 (4) 2 + e → → → → →

Q68.If a curve y = f(x) , passing through the point (1, 2), is the solution of the differential equation 2x2dy = (2xy + y2)dx, then f( 21 ) is equal to JEE Main 2020 (02 Sep Shift 2) JEE Main Previous Year Paper (1) 1 (2) 1 1+loge 2 1−loge 2 (3) 1 + loge 2 (4) 1+loge−1 2

Q68.The general solution of the differential equation √1 + x2 + y2 + x2y2 + xy dxdy = 0 (where C is a constant of integration) + C (1) √1 + y2 + √1 + x2 = 12 loge( √1+x2+1√1+x2−1 ) + C (2) √1 + y2 −√1 + x2 = 12 loge( √1+x2+1√1+x2−1 ) + C (3) √1 + y2 + √1 + x2 = 12 loge( √1+x2−1√1+x2+1 ) (4) 1 √1+x2+1 + C √1 + y2 −√1 + x2 = 2 loge( √1+x2−1 )

Q68.The foot of the perpendicular drawn from the point (4, 2, 3) to the line joining the points (1, −2, 3) and (1, 1, 0) lies on the plane (1) 2 x + y −z = 1 (2) x −y −2 z = 1 (3) x −2 y + z = 1 (4) x + 2 y −z = 1 + + +

Q68.If f '(x) = tan−1(sec x + tan x), −π2 < x < π2 and f(0) = 0 , then f(1) is equal to: (1) π+1 (2) 1 4 4 (3) π−1 (4) π+2 4 4

Q69.The mirror image of the point (1, 2, 3), in a plane is (−73 , −43 , −13 ). Which of the following points lies on this plane? (1) (1, 1, 1) (2) (1, −1, 1) (3) (−1, −1, 1) (4) (−1, −1, −1)

Q69.The plane which bisects the line joining the points (4, −2, 3) and (2, 4, −1) at right angles also passes through the point : (1) (0, −1, 1) (2) (4, 0, −1) (3) (4, 0, 1) (4) (0, 1, –1)

Q69.Let →a, b and →c, be three unit vectors such that →a+ b +→c= 0. If λ =→a⋅ b + b ⋅→c+→c⋅→a and → → → → , is equal to. d =→a× b + b ×→c+→c×→a, then the order pair, (λ, d) 3 → , 3→a×→c) (1) ( 2 (2) (−3 2 , 3→c× b) 2 , 3b (3) ( 3 → (4) → ×→c) (−3 2 , 3→a× b)

Q69.The shortest distance between the lines x−3 3 = −1 = z−31 and x+3−3 = y+72 = z−64 is (1) 2√30 (2) 72 √30 (3) 3√30 (4) 3

Q69.A plane P meets the coordinate axes at A, B and C respectively. The centroid of Δ ABC is given to be (1, 1, 2) . Then the equation of the line through this centroid and perpendicular to the plane P is : y−1 (1) x−1 2 = 1 = z−21 (2) x−11 = y−11 = z−22 y−1 (3) x−1 2 = 2 = z−21 (4) x−11 = y−12 = z−22

Q69.Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6) .Then the image of R in the plane P is (1) (6, 5, 2) (2) (6, 5, −2) (3) (4, 3, 2) (4) (3, 4, −2)

Q69.Let y = y(x) be the solution of the differential equation cos x dxdy + 2y sin x = sin 2x, x ∈(0, π2 ) If y(π/3) = 0, then y(π/4) is equal to : (1) 2 −√2 (2) 2 + √2 (3) √2 −2 (4) 1 −1 √2

Q69.Let D be the centroid of the triangle with vertices (3, −1) , (1, 3) and (2, 4) . Let P be the point of intersection of the lines x + 3y −1 = 10 and 3x −y + 1 = 0 . Then, the line passing through the points D and P also passes through the point: (1) (−9, −6) (2) (9,7) (3) (7,6) (4) (−9, −7)

Q69.A plane passing through the point (3, 1, 1) contains two lines whose direction ratios are 1, – 2, 2 and 2, 3, –1 respectively. If, this plane also passes through the point (α, –3, 5), then α is equal to (1) 5 (2) −10 (3) 10 (4) −5

Q69.The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also passes through the point (1) (0, 6, −2) (2) (−2, 0, 1) (3) (0, −6, 2) (4) (2, 0, − 1)

Q69.The lines →r= (ˆi −ˆj) l(2ˆi ˆk) and →r= (2ˆi −ˆj) m(ˆi + ˆj −ˆk) (1) Do not intersect for any values of l and m (2) Intersect for all values of l and m (3) Intersect when l = 2 and m = 21 (4) Intersect when l = 1 and m = 2

Q69.The distance of the point (1, −2, 3) from the plane x −y + z = 5 measured parallel to the line x2 = 3y = −6z is : (1) 7 (2) 1 5 (3) 1 (4) 7 7

Q69.If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of these boxes contain exactly 2 and 3 balls is: (1) 965 (2) 965 211 210 (3) 945 (4) 945 210 211

Q69.Let y = y(x) be the solution of the differential equation, xy′ −y = x2(x cos x + sin x), x > 0. If y(π) = π, then y′′( π2 ) + y( π2 ) is equal to : (1) 2 + π2 (2) 1 + π2 + π24 (3) 2 + π2 + π24 (4) 1 + π2 b where→a = xˆi −2ˆj + 3ˆk, →b = −2ˆi + xˆj −ˆk and

Q69.If the volume of a parallelopiped, whose coterminous edges are given by the vectors →a = ˆi + ˆj + nˆk , → b = 2ˆi + 4ˆj − nˆk and,→c= ˆi + nˆj + 3ˆk (n ≥0) is 158 cubic units, then : → (1) →a⋅→c= 17 (2) b ⋅→c= 10 (3) n = 7 (4) n = 9

Q69.The shortest distance between the lines x−1 0 = y+1−1 = 1z and x + y + z + 1 = 0, 2 x −y + z + 3 = 0 is JEE Main 2020 (06 Sep Shift 1) JEE Main Previous Year Paper (1) 1 (2) 1 √3 (3) 1 (4) 1 √2 2