Practice Questions

Q75.Let the solution curve of the differential equation 𝑥𝑑𝑦= √𝑥2 + 𝑦2 + 𝑦𝑑𝑥, 𝑥> 0, intersect the line x = 1 at 𝑦= 0 and the line 𝑥= 2 at 𝑦= 𝛼. Then the value of 𝛼 is (1) 1 (2) 3 2 2 3 5 (3) - (4) 2 2

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 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.The area of the region {(x, y) : |x −1| ≤y ≤√5 −x2} (1) 5 2 sin−1( 53 ) −12 (2) 5π4 −32 (3) 3π 4 + 23 (4) 5π4 −12 + = 1 pass through the point

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.

Q75.The slope of the tangent to a curve 𝐶: 𝑦= 𝑦𝑥 at any point [𝑥, 𝑦) on it is 2e2x - 6e-x + 9 . If 𝐶 passes through the 2 + 9e-2x 1 𝜋 1 points 0, + and 𝛼, then 𝑒𝛼 is equal to 2 2√2 2e2𝛼 (1) 3 + √2 (2) 3 3 + √2 3 - √2 √2 3 - √2 (3) 1 √2 + 1 (4) √2 + 1 √2 √2 - 1 √2 - 1

Q75. n→∞(lim (n2+1)(n+1)n2 + (n2+4)(n+2)n2 + (n2+9)(n+3)n2 + … + (n2+n2)(n+n)n2 ) is equal to (1) π 8 + 14 ln 2 (2) π4 + 18 ln 2 (3) π 4 −18 ln 2 (4) π8 + ln √2

Q75.If the solution curve of the differential equation 𝑑𝑦 𝑥+ 𝑦- 2 passes through the point 2, 1 and 𝑘+ 1, 2, k > 0, 𝑑𝑥= 𝑥- 𝑦 then (1) 2tan-11 + 1 𝑘= loge𝑘2 + 1 (2) tan-11𝑘= loge𝑘2 1 𝑘2 + 1 (3) 2tan-1 = loge𝑘2 + 2𝑘+ 2 (4) 2tan-11 𝑘+ 1 𝑘= loge 𝑘2

Q75.The value of ∫0 1 + cos2𝑥ecos𝑥+ e-cos𝑥d𝑥 is equal to (1) 𝜋2 (2) 𝜋 4 4 (3) 𝜋 (4) 𝜋2 6 2

Q75.The sum of absolute maximum and absolute minimum values of the function f(x) = 2x2 + 3x −2 + sin x cos x in the interval [0, 1] is 1 sin(1) cos2( (1) 2 ) (2) 3 + 12 (1 + 2 cos(1)) sin(1) 3 + 2 (3) 5 + 12 (sin(1) + sin(2)) (4) 2 + sin( 21 ) cos( 12 )

Q75.The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = ya is 3643 , equal to: (1) 3 (2) 5 (3) 7 (4) 9

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.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.Let [t] denote the greatest integer less than or equal to t. Then the value of the integral ∫101−3 ([sin(πx)] + e[cos(2πx)])dx is equal to (1) 52(1−e) (2) 52 e e (3) 52(2+e) (4) 104 e e

Q75.If ∫20 (√2x −√2x −x2)dx + I , then I equal to + ∫21 (2 −y22 )dy ∫10 (1 −√1 −y2 −y22 )dy −y2 + + √1 −y2)dy (2) ∫10 ( y22 −√1 1)dy (1) ∫10 (1 + √1 −y2 + 1)dy (3) ∫10 (1 −√1 −y2)dy (4) ∫10 ( y22

Q75.Let y = y(x) be the solution curve of the differential equation dx 1 1 y = ( x−1x+1 ) 2 , x > 1 passing through x2−1 the point . Then √7y(8) is equal to 3 (2, √1 ) (1) 11 + 6 loge 3 (2) 19 (3) 12 −2 loge 3 (4) 19 −6 loge 3

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 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.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.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

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 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.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.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.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 → →