Practice Questions

Q75.Let the solution curve y = y(x) of the differential equation, [ √x2−y2x [ √x2−y2x through the points (1, 0) and (2α, α), α > 0 . Then α is equal to (1) 2 1 exp( π6 + √e −1) (2) 12 exp( π3 + √e −1) (3) exp( π6 + √e + 1) (4) 2 exp( π3 + √e −1)

Q75.If the angle made by the tangent at the point 𝑥0, 𝑦0 on the curve 𝑥= 12𝑡+ sin𝑡cos𝑡, 𝜋 𝜋 𝑦= 121 + sin𝑡2, 0 < 𝑡< 2, with the positive 𝑥-axis is 3, then 𝑦0 is equal to (1) 63 + 2√2 (2) 37 + 4√3 (3) 27 (4) 48 𝜋 𝑛∈ℕ, then

Q75.The integral ∫10 [11x ] 7 (1) 1 −6 ln( 76 ) (2) 1 + 6 ln( 76 ) (3) 1 −7 ln( 76 ) (4) 1 + 7 ln( 76 )

Q75.Let = , where a, b, c are constants. represent a circle passing through the point (2, 5). Then the dx bx+cy+a shortest distance of the point (11, 6) from this circle is (1) 10 (2) 8 (3) 7 (4) 5 dy 2x−y(2y−1)

Q75.The area of the bounded region enclosed by the curve y = 3 −x −12 −|x + 1| and the x-axis is (1) 9 (2) 45 4 16 (3) 278 (4) 1663 x x −4xe y2 = 0 such that x(1) = 0.

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.Consider a curve y = y(x) in the first quadrant as shown in the figure. Let the area A1 is twice the area A2 . Then the normal to the curve perpendicular to the line 2x −12y = 15 does NOT pass through the point __ JEE Main 2022 (27 Jul Shift 2) JEE Main Previous Year Paper ​ (1) (6, 21) (2) (8, 9) (3) (10, −4) (4) (12, −15)

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 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.Let the solution curve y = y(x) of the differential equation (1 + e2x)( dxdy y) (0, π2 ). Then, x→∞exy(x)lim is equal to JEE Main 2022 (29 Jul Shift 1) JEE Main Previous Year Paper (1) π (2) 3π 4 4 (3) π (4) 3π 2 2 → b = b + λ→c. If→b and →care non-

Q76.If the solution curve of the differential equation ((tan−1 y) −x)dy = (1 + y2)dx passes through the point (1, 0) then the abscissa of the point on the curve whose ordinate is tan(1) is (1) 2 (2) 2e (3) 3 (4) 2e e →

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.Let 𝑦= 𝑦𝑥 be the solution curve of the differential equation 𝑑𝑦 2𝑥2 + 11𝑥+ 13 𝑥+ 3 𝑥> - 1, which 𝑑𝑥+ 𝑥3 + 6𝑥2 + 11𝑥+ 6𝑦= 𝑥+ 1, passes through the point 0, 1. Then 𝑦1 is equal to 1 3 (1) (2) 2 2 5 7 (3) (4) 2 2

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 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.Let y = y1(x) and y = y2(x) be two distinct solutions of the differential equation dxdy = x + y, with y1(0) = 0 and y2(0) = 1 respectively. Then, the number of points of intersection of y = y1(x) and y = y2(x) is (1) 0 (2) 1 (3) 2 (4) 3 → →

Q76.Let x = x(y) be the solution of the differential equation 2ye y2 dx + (y2 )dy Then, x(e) is equal to (1) e loge(2) (2) −e loge(2) (3) e2 loge(2) (4) −e2 loge(2)

Q76.Let y = y(x) be the solution of the differential equation x(1 −x2) dxdy + (3x2y −y −4x3) = 0, x > 1 with y(2) = −2. Then y(3) is equal to (1) −18 (2) −12 (3) −6 (4) −3

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.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

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.If y = y(x) is the solution of the differential equation (1 + e2x) dxdy + 2(1 + y2)ex = 0 and y(0) = 0, then 2 + (y(logc √3)) is equal to: 6(y′(0) ) (1) 2 (2) −2 (3) −4 (4) −1

Q76.The differential equation of the family of circles passing through the points (0, 2) and (0, −2) is (1) 2xy dxdy + (x2 −y2 + 4) = 0 (2) 2xy dxdy + (x2 + y2 −4) = 0 (3) 2xy dxdy + (y2 −x2 + 4) = 0 (4) 2xy dxdy −(x2 −y2 + 4) = 0 →

Q76.If 𝑏𝑛= ∫02 cos2𝑛𝑥sin𝑥𝑑𝑥, 1 1 1 (1) 𝑏3 - 𝑏2, 𝑏4 - 𝑏3, 𝑏5 - 𝑏4 are in an A.P. with (2) 𝑏3 - 𝑏2, 𝑏4 - 𝑏3, 𝑏5 - 𝑏4 are in an A.P. with common common difference-2 difference 2 (3) 𝑏3 - 𝑏2, 𝑏4 - 𝑏3, 𝑏5 - 𝑏4 are in a G.P. (4) 1 1 1 are in an A.P. with common 𝑏3 - 𝑏2, 𝑏4 - 𝑏3, 𝑏5 - 𝑏4 difference -2