Practice Questions

Q76.The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is (1) 9 (2) 7 (3) 5 (4) 3

Q76.If dx + 2x−1 = 0, x, y > 0, y(1) = 1 , then y(2) is equal to (1) 2 + log2 3 (2) 2 + log2 2 (3) 2 −log−2 3 (4) 2 −log2 3 → →

Q76.If y = y(x) is the solution of the differential equation x dxdy + 2y = xex, y(1) = 0 then the local maximum value of the function z(x) = x2y(x) −ex, x ∈R is (1) 1 −e (2) 0 (3) 1 (4) 4 e −e 2

Q76.The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by x2 . If the curve xy−x2y2−1 passes through the point (1, 1), then e ⋅y(e) is equal to (1) 1−tan(1) (2) tan(1) 1+tan(1) (3) 1 (4) 1+tan(1) 1−tan(1)

Q76.The area bounded by the curve y = x2 −9 and the line y = 3 is (1) 8√6 −16√12 −72 (2) 8√6 + 8√12 −72 (3) 16√6 + 16√12 −72 (4) 16√6 −16√12 −64 → → → → → is b b × b × × (→c×→a) →c

Q76.The general solution of the differential equation 𝑥- 𝑦2𝑑𝑥+ 𝑦5𝑥+ 𝑦2𝑑𝑦= 0 is 4 3 4 3 (1) 𝑦2 + 𝑥 = 𝐶𝑦2 + 2𝑥 (2) 𝑦2 + 2𝑥 = 𝐶𝑦2 + 𝑥 3 4 3 4 (3) 𝑦2 + 𝑥 = 𝐶2𝑦2 + 𝑥 (4) 𝑦2 + 2𝑥 = 𝐶2𝑦2 + 𝑥 → → → → → →

Q76.If dx dy + 2y tan x = sin x, 0 < x < π2 and y( π3 ) = 0 , then the maximum value of y(x) is JEE Main 2022 (26 Jul Shift 1) JEE Main Previous Year Paper (1) 1 (2) 3 8 4 (3) 1 (4) 3 4 8 → →

Q76.Let a smooth curve y = f(x) be such that the slope of the tangent at any point (x, y) on it is directly proportional to ( −yx ). If the curve passes through the points (1, 2) and (8, 1), then y( 81 ) is equal to (1) 2 loge 2 (2) 4 (3) 1 (4) 4 loge 2 → → → →

Q76.Let 𝑦= 𝑦𝑥 be the solution of the differential equation 𝑥+ 1𝑦' - 𝑦= e3𝑥𝑥+ 12, with 𝑦0 = 13. Then, the point 4 𝑥= - for the curve 𝑦= 𝑦𝑥 is 3 (1) not a critical point (2) a point of local minima (3) a point of local maxima (4) a point of inflection

Q76.If 𝑦= 𝑦𝑥, 𝑥∈0, 𝜋 be the solution curve of the differential equation 2 sin22𝑥 𝑑𝑦 8sin22𝑥+ 2sin4𝑥𝑦= 𝑑𝑥+ 2𝑒-4𝑥2sin2𝑥+ cos2𝑥, with 𝑦𝜋 = 𝑒-𝜋, then 𝑦𝜋 is equal to 4 6 2 2𝜋 3 (2) 3 (1) √3𝑒-2𝜋2 √3𝑒 1 2𝜋 3 (4) 3 (3) √3𝑒-2𝜋1 √3𝑒 JEE Main 2022 (28 Jul Shift 1) JEE Main Previous Year Paper

Q77.The area enclosed by y2 = 8x and y = √2x that lies outside the triangle formed by y = √2x, x = 1, y = 2√2 , is equal to (1) 16√2 (2) 11√2 6 6 (3) 13√2 (4) 5√2 6 6

Q77.Let →a = 3ˆi + ˆj and→b = ˆi + 2ˆj + ˆk. Let →cbe a vector satisfying →a× (→ ×→c) parallel, then the value of λ is (1) −5 (2) 5 (3) 1 (4) −1 θ is the angle between the vectors

Q77.Let →a = ˆi −ˆj + 2ˆk and let b be a vector such that →a× b = 2ˆi −ˆk and →a⋅ b = 3 . Then the projection of b on the → vector →a− b is: (1) 2 (2) √21 2√37 (3) 2 (4) 2 3 3 √73

Q77.If 2, 3, 9, 5, 2, 1, 1, 𝜆, 8 and 𝜆, 2, 3 are coplanar, then the product of all possible values of 𝜆 is (1) 21 (2) 59 2 8 57 95 (3) (4) 8 8

Q77.Let →a = αˆi + ˆj −ˆk and b = 2ˆi + ˆj −αˆk, α > 0 . If the projection of →a× b on the vector −ˆi + 2ˆj −2ˆk is 30 , then α is equal to (1) 15 (2) 8 2 (3) 13 (4) 7 2

Q77.Let the vectors →𝑎= 1 + 𝑡 ^𝑖+ 1 - 𝑡 ^𝑗+ ^𝑘, →𝑏= 1 - 𝑡 ^𝑖+ 1 + t ^𝑗+ 2 ^𝑘 and →𝑐= 𝑡 ^𝑖- 𝑡 ^𝑗+ ^𝑘, 𝑡∈𝑅 be such that for 𝛼, 𝛽, 𝛾∈𝑅, 𝛼 →𝑎+ 𝛽 →𝑏+ 𝛾 →𝑐= →0 ⇒𝛼= 𝛽= 𝛾= 0. Then, the set of all values of 𝑡 is (1) a non-empty finite set (2) equal to 𝑁 (3) equal to 𝑅- 0 (4) equal to 𝑅

Q77.If →a⋅ b = 1, b ⋅→c= 2 and →c⋅→a = 3 , then the value of [→a ( ×→c) ( ×→a)] b (1) 0 (2) −6→a⋅(→ ×→c) → −12b ⋅(→c×→a) (3) 12→c⋅(→a×→b) (4)

Q77.The area of the region enclosed between the parabolas 𝑦2 = 2𝑥- 1 and 𝑦2 = 4𝑥- 3 is. 1 1 (1) (2) 3 6 2 3 (3) (4) 3 4

Q77.Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 tan x(cos x −y). if the curve passes π through the point ( π4 , 0), then the value of ∫ 0 2 ydx is equal to (1) (2 −√2) + √2π (2) 2 − √2π (3) (2 + √2) + √2π (4) 2 + √2π →

Q77.If the solution curve 𝑦= 𝑦𝑥 of the differential equation 𝑦2 d𝑥+ 𝑥2 - 𝑥𝑦+ 𝑦2d𝑦= 0, which passes through the point 1, 1 and intersects the line 𝑦= √3𝑥 at the point 𝛼, √3𝛼, then value of log𝑒√3𝛼 is equal to 𝜋 𝜋 (1) (2) 2 4 (3) 𝜋 (4) 𝜋 6 12 JEE Main 2022 (25 Jun Shift 1) JEE Main Previous Year Paper

Q77.Let a and b be two unit vectors such that |(a + b) + 2(a × b)| = 2. If θ ∈(0, π) is the angle between ˆa and ˆb , then among the statements: (S1) : 2 ˆa × ˆb = ˆa −ˆb is 1 + (S2) : The projection of ˆa on 2 (ˆa ˆb) (1) Only (S1) is true. (2) Only (S2) is true. (3) Both (S1) and (S2) are true. (4) Both (S1) and (S2) are false. JEE Main 2022 (24 Jun Shift 2) JEE Main Previous Year Paper

Q77.Let A, B, C be three points whose position vectors respectively are: →a = ˆi + 4ˆj + 3ˆk → b = 2ˆi + αˆj + 4ˆk, α ∈R →c= 3ˆi −2ˆj + 5ˆk → If α is the smallest positive integer for which →a, b, →care non-collinear, then the length of the median, △ABC , through A is: (1) √82 (2) √62 2 2 (3) √69 (4) √66 2 2 y+1

Q77.The area bounded by the curves 𝑦= 𝑥2 - 1 and 𝑦= 1 is (1) 2 + 1 (2) 4 - 1 3√2 3√2 8 (3) 2√2 - 1 (4) 3√2 - 1

Q77.If x = x(y) is the solution of the differential equation y dxdy = 2x + y3(y + 1)ey, x(1) = 0 ; then x(e) is equal to (1) ee(e3 −1) (2) e3(ee −1) (3) ee −1 (4) ee(e2 −1) ×

Q77.Let →a = αˆi + ˆj + βˆk and b = 3ˆi −5ˆj + 4ˆk be two vectors, such that →a× b = −ˆi + 9ˆi + 12ˆk. Then the → → projection of b −2→a on b +→a is equal to (1) 2 (2) 395 (3) 9 (4) 465 → → → 23 × b × 2ˆj is equal to ⋅ˆk = 2 , then