Differential Equations PYQs Set 2 (PDF)

Set 2 of 11 · Questions 31-60 of 326

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Differential Equations · Set 2 (Q31-60)
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31. $The particular solution of the differential equation y (1 + log x) \frac{dx}{dy} - x log x = 0 when x = e, y = e^{2} is$ (1) $y = ex log x$ (2) $ey = x log x$ (3) $xy = e log x$ (4) $y log x = ex$ 32. $Let x = x (t) and y = y (t) be solutions of the differential equations \frac{dx}{dt} + ax = 0 and \frac{dy}{dt} + by = 0 respectively, a, b \in R . Given that x (0) = 2; y (0) = 1 and 3 y (1) = 2 x (1), the value of t, for which x (t) = y (t), is :$ (1) $log_{\frac{2}{3}} 2$ (2) $log_{4} 3$ (3) $log_{3} 4$ (4) $log_{\frac{4}{3}} 2$ 33. $The differential equation of the family of circles with fixed radius r units and centre on the line y = 3, is$ (1) $1 + (\frac{dy}{dx})^{2} = \frac{r ^{2}}{( y - 3 ) ^{2}}$ (2) $1 + (\frac{dy}{dx})^{2} = \frac{r ^{2}}{y - 3}$ (3) $(\frac{dy}{dx})^{2} = \frac{r ^{2}}{( y - 3 ) ^{2}}$ (4) $(\frac{dy}{dx})^{2} = \frac{r ^{2}}{y - 3}$ 34. $If y_{1} (x) is a solution of the differential equation dy / dx - f (x) y = 0, then a solution of the differential equation \frac{dy}{dx} + f (x) y = r (x) is$ (1) $\frac{1}{y _{1} ( x )} \int r (x) y_{1} (x) d x$ (2) $y1(x) \int r (x) y_{1} (x) dx$ (3) $\int r (x) y_{1} (x) dx$ (4) $None of these$ 35. $Let y = y (x) be the solution curve of the differential equation, (y^{2} - x) \frac{dy}{dx} = 1, satisfying y (0) = 1 . This curve intersects the x -axis at a point whose abscissa is:$ (1) $2 - e$ (2) $2 + e$ (3) $2$ (4) $e$ 36. $If y = y (x) is the solution of the differential equation \frac{dy}{dx} = (tan x - y) sec^{2} x, x \in (- \frac{π}{2}, \frac{π}{2}), such that y (0) = 0, then y (- \frac{π}{4}) is equal to :$ (1) $e - 2$ (2) $\frac{1}{2} - e$ (3) $2 + \frac{1}{e}$ (4) $1 e - 2$ 37. $The solution of the equation (x - 4 y^{3}) \frac{dy}{dx} - y = 0, (y > 0) is$ (1) $x = y^{3} + cy$ (2) $x + 2 y^{3} = cy$ (3) $y = x^{3} + cx$ (4) $y + 2 x^{3} = cx$ 38. $If \frac{dy}{dx} + \frac{3}{cos ^{2} x} y = \frac{1}{cos ^{2} x}, x \in (- \frac{π}{3}, \frac{π}{3}), and y (\frac{π}{4}) = \frac{4}{3}, then y (- \frac{π}{4}) equals:$ (1) $\frac{1}{3} + e^{6}$ (2) $\frac{1}{3}$ (3) $- \frac{4}{3}$ (4) $\frac{1}{3} + e^{3}$ 39. $The order of the differential equation of all parabolas, whose latus rectum is 4 a and axis parallel to the x -axis, is$ (1) $one$ (2) $four$ (3) $three$ (4) $two$ 40. $The solution of the differential equation \frac{dy}{dx} = (x + y)^{2} is$ (1) $tan^{- 1} (x + y) = x + c$ (2) $cot^{- 1} (x + y) = c$ (3) $tan^{- 1} (x + y) = 0$ (4) $cot^{- 1} (x + y) = x + c$ 41. $The general solution of the differential equation \frac{dy}{dx} + \frac{y}{x} = 3 x is$ (1) $y = x - \frac{c}{x}$ (2) $y = x + \frac{c}{x}$ (3) $y = x^{2} - \frac{c}{x}$ (4) $none of the above$ 42. $Integrating factor of x \frac{dy}{dx} - y = x^{4} - 3 x is$ (1) $x$ (2) $log x$ (3) $\frac{1}{x}$ (4) $- x$ 43. $The general solution of the differential equation (sec x + tan x) \frac{dy}{dx} + (sec^{2} x + sec xtan x) y = 1 is$ (1) $(1 + sin x) y = ncos x + c$ (2) $(1 + cos x) y = xsin x + c$ (3) $(sec x + tan x) y = xsec x + c$ (4) $(sec x + tan x) y = x + c$ 44. $Let f be a twice differentiable function defined on R such that f (0) = 1, f^{'} (0) = 2 and f^{'} (x) \neq = 0 for all x \in R . If f (x) f^{'} (x) f^{'} (x) f^{''} (x) = 0, for all x \in R then the value of f (1) lies in the interval:$ (1) $(9, 12)$ (2) $(6, 9)$ (3) $(3, 6)$ (4) $(0, 3)$ 45. $If \frac{dy}{dx} = f (x, y) is a homogeneous differential equation, then the general form of f (x, y) is$ (1) $x^{n} φ (\frac{y}{x}), n \neq = 1$ (2) $y^{n} φ (\frac{x}{y}), n \neq = 1$ (3) $φ (\frac{y}{x})$ (4) $K^{n} f (x, y), n \neq = 1$ 46. $Let y = y (x) be the solution of the differential equation \frac{dy}{dx} = 2 x (x + y)^{3} - x (x + y) - 1, y (0) = 1 . Then, (\frac{1}{2} + y (\frac{1}{2}))^{2} equals :$ (1) $\frac{4}{4 + e}$ (2) $\frac{3}{3 - e}$ (3) $\frac{2}{1 + e}$ (4) $\frac{1}{2 - e}$ 47. $A function y = f (x) satisfies f (x) sin 2 x + sin x - (1 + cos^{2} x) f^{’} (x) = 0 with condition f (0) = 0 . Then f (\frac{π}{2}) is equal to$ (1) $1$ (2) $0$ (3) $-1$ (4) $2$ 48. $If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is \frac{x ^{2} - 4 x + y + 8}{x - 2}, then thiscurve also passes through the point:$ (1) $(4, 5)$ (2) $(5, 4)$ (3) $(4, 4)$ (4) $(5, 5)$ 49. $If log y is an integrating factor of \frac{dx}{dy} + P (y) x = Q (y), then P (y) =$ (1) $\frac{1}{y + log y}$ (2) $\frac{y}{log y}$ (3) $\frac{log y}{y}$ (4) $\frac{1}{ylog y}$ 50. $The differential equation of all parabolas, whose axes are parallel to Y -axis, is$ (1) $y_{3} = 1$ (2) $y_{3} = - 1$ (3) $y_{3} = 0$ (4) $y_{3} + y_{1} = 0$ 51. $Let αx = exp (x^{β} y^{γ}) be the solution of the differential equation 2 x^{2} y dy - (1 - x y^{2}) dx = 0, x > 0, y (2) = log_{e} 2 . Then α + β - γ equals$ (1) $1$ (2) $- 1$ (3) $0$ (4) $3$ 52. $y = A e^{x} + B e^{- 2 x} satisfies which of the following differential equations?$ (1) $\frac{d ^{2} y}{d x ^{2}} - \frac{dy}{dx} + 2 y = 0$ (2) $\frac{d ^{2} y}{d x ^{2}} - 2 \frac{dy}{dx} - y = 0$ (3) $\frac{d ^{2} y}{dx} - 2 \frac{dy}{dx} + y = 0$ (4) $\frac{d ^{2} y}{d x ^{2}} + \frac{dy}{dx} - 2 y = 0$ 53. $The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteriacount is 1000 at initial time t = 0 . The number of bacteria is increased by 20 % in 2 hours. If the population ofbacteria is 2000 after \frac{k}{log _{e} ( \frac{6}{5} )} hours, then (\frac{k}{log _{e} 2})^{2} is equal to$ (1) $4$ (2) $2$ (3) $16$ (4) $8$ 54. $The equation of the curve passing through the point (1, 1) such that the slope of the tangent at any point (x, y) is equal to the product of its coordinates is$ (1) $2 log x = y^{2} - 1$ (2) $2 log y = x^{2} - 1$ (3) $2 log y = x^{2} + 1$ (4) $2 log x = y^{2} + 1$ 55. $The difference between degree and order of differential equation that represents the family of curves given by y^{2} = a (x + \frac{a}{2}), a > 0 is$ 56. $Let y = f (x) be the solution of the differential equation \frac{d y}{d x} + \frac{xy}{x ^{2} - 1} = \frac{x ^{6} + 4 x}{1 - x ^{2}}, - 1 < x < 1 such that f (0) = 0 . If 6 \sum f (x) d x = 2 π - α then α^{2} is equal to .$ 57. $If y = y (x) is the solution of the differential equation, 4 - x^{2} \frac{d y}{d x} = ((sin^{- 1} \frac{x}{2})^{2} - y) sin^{- 1} \frac{x}{2}, - 2 \leq x \leq 2, y (2) = \frac{π ^{2} - 8}{4}, then y^{2} (0) is equal to$ 58. $Let y = y (x) be the solution of the differential equation \frac{dy}{dx} + 2 y sec^{2} x = 2 sec^{2} x + 3 tan x \cdot sec^{2} x such that y (0) = \frac{5}{4} . Then 12 (y (\frac{π}{4}) - e^{- 2}) is equal to ________.$ 59. $If the solution of the differential equation (2 x + 3 y - 2) dx + (4 x + 6 y - 7) dy = 0, y (0) = 3, is αx + βy + 3 log_{e} \| 2 x + 3 y - γ \| = 6, then α + 2 β + 3 γ is equal to_______$ 60. $Let y = y (x) be the solution of the differential equation (1 - x^{2}) dy = (xy + (x^{3} + 2) 3 (1 - x^{2})) dx, - 1 < x < 1, y (0) = 0 . If y (\frac{1}{2}) = \frac{m}{n}, m and n are coprime numbers, then m + n is equal to______$