Differential Equations PYQs Set 3 (PDF)

Set 3 of 11 · Questions 61-90 of 326

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Differential Equations · Set 3 (Q61-90)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
61. $Let y = y (x) be a solution of the differential equation, 1 - x^{2} \frac{dy}{dx} + 1 - y^{2} = 0, (x) < 1 . If y (\frac{1}{2}) = \frac{3}{2}, then y (- \frac{1}{2}) is equal to:$ (1) $- \frac{1}{2}$ (2) $\frac{3}{2}$ (3) $\frac{1}{2}$ (4) $\frac{1}{2}$ 62. $If f (x), f^{'} (x), f^{''} (x) are positive functions and f (0) = 1, f^{'} (0) = 2, then the solution of the differential equation f (x) f^{'} (x) f^{'} (x) f^{''} (x) = 0 is$ (1) $e^{2 x}$ (2) $2 sin x + 1$ (3) $sin^{2} x + 2 x + 1$ (4) $e^{4 x}$ 63. $The general solution of \frac{dy}{dx} = cos^{2} (x - y - 1) is given by x =$ (1) $C - cot (x - y - 1)$ (2) $C - tan (x - y + 1)$ (3) $y + Ccot (x - y - 1)$ (4) $Cy + tan (x - y - 1)$ 64. $If y = y (x) is the solution of the differential equation \frac{dy}{dx} + (tan x) y = sin x, 0 \leq x \leq \frac{π}{3}, with y (0) = 0, then y (\frac{π}{4}) equal to :$ (1) $\frac{1}{4} lo g_{e} 2$ (2) $\frac{1}{2 2} lo g_{e} 2$ (3) $log_{e} 2$ (4) $12 log_{e} 2$ 65. $If l and m are order and degree of a differential equation of all the straight lines at constant distance of P units from the origin, then l m^{2} + l^{2} m =$ (1) $2$ (2) $6$ (3) $12$ (4) $30$ 66. $The general solution of differential equation \frac{dx}{dy} = cos (x + y) is$ (1) $tan (\frac{x + y}{2}) = y + c$ (2) $tan (\frac{x + y}{2}) = x + c$ (3) $cot (\frac{x + y}{2}) = y + c$ (4) $cot (\frac{x + y}{2}) = x + c$ 67. $The particular solution of the differential equation (1 + y^{2}) dx - xydy = 0, y (1) = 0 represents$ (1) $a circle$ (2) $a part of parabola$ (3) $a part of ellipse$ (4) $a part of hyperbola$ 68. $Which one of the following is a linear differential equation?$ (1) $\frac{dx}{dy} + y^{2} = e^{e^{x}}$ (2) $dr + (2 r^{2} cot θ + sin 2 θ) dθ = 0$ (3) $\frac{dy}{dx} = e^{x - y} (e^{x} - e^{- y})$ (4) $x^{2} dy + xydx - 1 = 0$ 69. $The degree and order of the differential equation (1 + (\frac{dy}{dx})^{3})^{7/3} = 7 (\frac{d ^{2} y}{d x ^{2}}) respectively are$ (1) $3 and 7$ (2) $3 and 2$ (3) $7 and 3$ (4) $2 and 3$ 70. $Let the solution curve y = y (x) of the differential equation \frac{d y}{d x} = \frac{3 x ^{5} t a n ^{- 1} ( x ^{3} )}{( 1 + x ^{6} ) ^{3/2}} y + 2 x exp (\frac{x ^{3} - t a n ^{- 1} x ^{3}}{1 + x ^{6}}) pass through the origin. then y (1) is equal to$ (1) $exp (\frac{4 - π}{4 2})$ (2) $exp (\frac{π - 4}{4 2})$ (3) $exp (\frac{1 - π}{4 2})$ (4) $exp (\frac{4 + π}{4 2})$ 71. $The solution of the equation \frac{dy}{dx} + 2 ytan x = sin x satisfying y = 0 when x = \frac{π}{3}, is$ (1) $y = 2 sin^{2} x + cos x - 2$ (2) $y = 2 sin^{2} x - cos x - 2$ (3) $y = 2 cos^{2} x - sin x + 2$ (4) $y = 2 cos x - sin^{2} x - 1$ 72. $The order and degree of the differential equation \frac{dy}{dx} - 4 \frac{dy}{dx} - 7 x = 0 are respectively$ (1) $1 and \frac{1}{2}$ (2) $2 and 1$ (3) $1 and 1$ (4) $1 and 2$ 73. $If c and d are arbitrary constants, then y = e^{2 x} (ccosh 2 x + dsinh 2 x) is the general solution of the differential equation$ (1) $y^{’ ’} + 4 y^{’} + 2 y = 0$ (2) $y^{’ ’} - 4 y^{’} + 2 y = 0$ (3) $y^{’ ’} - 4 y^{’} + 4 y = 0$ (4) $y^{’ ’} - 22 y^{’} + 2 y = 0$ 74. $The differential equation representing the family of circles of constant radius r is$ (1) $r^{2} y^{’ ’} = (1 + (y^{’})^{2})^{2}$ (2) $r^{2} (y^{’ ’})^{2} = (1 + (y^{’})^{2})^{2}$ (3) $r^{2} (y^{’ ’})^{2} = (1 + (y^{’})^{2})^{3}$ (4) $(y^{’ ’})^{2} = r^{2} (1 + (y^{’})^{2})^{2}$ 75. $If a curve y = f (x), passing through the point (1, 2) is the solution of the differential equation, 2 x^{2} dy = (2 xy + y^{2}) dx, then f (\frac{1}{2}) is equal to:$ (1) $\frac{1}{1 + l o g _{e} 2}$ (2) $- \frac{1}{1 + l o g _{e} 2}$ (3) $1 + log_{e} 2$ (4) $\frac{1}{1 - log _{e} 2}$ 76. $Let y = y (x) be the solution curve of the differential equation sec y \frac{d y}{d x} + 2 x sin y = x^{3} cos y, y (1) = 0 . Then y (3) is equal to :$ (1) $\frac{π}{3}$ (2) $\frac{π}{6}$ (3) $\frac{π}{12}$ (4) $\frac{π}{4}$ 77. $The curve amongst the family of curves represented by the differential equation, (x^{2} - y^{2}) dx + 2 xydy = 0 which passes through (1, 1) is :$ (1) $a circle with centre on the x-axis$ (2) $an ellipse with major axis along the y-axis.$ (3) $a circle with centre on the y -axis.$ (4) $a hyperbola with transverse axis along the x -axis.$ 78. $The particular solution of the differential equation xdy + 2 ydx = 0, when x = 2, y = 1 is$ (1) $xy = 4$ (2) $x^{2} y = 4$ (3) $x y^{2} = 4$ (4) $x^{2} y^{2} = 4$ 79. $f (x, y, c_{1}, c_{2}) = 0 is an equation containing two arbitrary constants c_{1} and c_{2} . If the differential equation having f (x, y, c_{1}, c_{2}) = 0 as its general solution is of k^{hh} order, then the differential equation corresponding to x^{k} + y^{k} = c^{2} (c is an arbitrary constant) is$ (1) $\frac{dy}{dx} + \frac{x}{y} = 0$ (2) $\frac{dy}{dx} + \frac{y}{x} = 0$ (3) $\frac{dy}{dx} - \frac{x}{y} = 0$ (4) $\frac{dy}{dx} - \frac{y}{x} = 0$ 80. $The solution of the differential equation, \frac{dy}{dx} = (x - y)^{2}, when y (1) = 1, is :$ (1) $log_{e} (\frac{2 - x}{2 - y}) = x - y$ (2) $- log_{e} (\frac{1 - x + y}{1 + x - y}) = 2 (x - 1)$ (3) $- log_{e} (\frac{1 + x - y}{1 - x + y}) = x + y - 2$ (4) $log_{e} (\frac{2 - y}{2 - x}) = 2 (y - 1)$ 81. $The general solution of the differential equation \frac{dy}{dx} = \frac{xy + x - 2 y - 2}{xy - 2 x + y - 2} is$ (1) $x + y + 3 log (\frac{x + 1}{y + 1}) = c$ (2) $x + y + 3 log (\frac{y + 1}{x + 1}) = c$ (3) $x - y + 3 log (\frac{x + 1}{y + 1}) = c$ (4) $x - y + 3 log (\frac{y + 1}{x + 1}) = c$ 82. $Solution of the differential equation cosx dy = y(sinx - y) dx, 0 < x < \frac{π}{2} is$ (1) $y sec x = tan x + c$ (2) $y tan x = sec x + c$ (3) $tanx = (sec x + c)y$ (4) $secx = (tanx + c) y$ 83. $The solution of the differential equation \frac{dθ}{dt} = - k (θ - θ_{0}) where k is constant, is$ (1) $θ = θ_{0} + a e^{- kt}$ (2) $θ = θ_{0} + a e^{kt}$ (3) $θ = 2 θ_{0} - a e^{kt}$ (4) $θ = 2 θ_{0} - a e^{- kt}$ 84. $If x^{α} \frac{dy}{dx} = y^{β} (γlog x + δlog y + 1) is a homogeneous differential equation, then$ (1) $α = β and γ = - δ$ (2) $α = β and γ = δ$ (3) $α \neq = β and γ = δ$ (4) $α \neq = β and γ \neq = δ 80 .$ 85. $The differential equation of the family of all straight lines passing through the origin is$ (1) $x = y \frac{dy}{dx}$ (2) $\frac{dy}{dx} = 0$ (3) $y = x \frac{dy}{dx}$ (4) $\frac{d ^{2} y}{d x ^{2}} = \frac{y}{x}$ 86. $Consider the differential equation, y^{2} dx + (x - \frac{1}{y}) dy = 0 . If y is 1 when x = 1, then value of x for which y = 2, is:$ (1) $\frac{5}{2} + \frac{1}{e}$ (2) $\frac{3}{2} - \frac{1}{e}$ (3) $\frac{1}{2} + \frac{1}{e}$ (4) $\frac{3}{2} - e$ 87. $The particular solution of the differential equation log (\frac{dy}{dx}) = x, when x = 0, y = 1 is$ (1) $y = e^{x} + 2$ (2) $y = - e^{x}$ (3) $y = - e^{x} + 2$ (4) $y = e^{x}$ 88. $General solution of the differential equation cos x (1 + cos y) dx - sin y (1 + sin x) dy = 0 is$ (1) $(1 + cos x) (1 + sin y) = c, where c is a constant of integration.$ (2) $1 + sin x + cos y = c, where c is a constant of integration.$ (3) $(1 + sin x) (1 + cos y) = c, where c is a constant of integration.$ (4) $1 + sin x cos y = c, where c is a constant of integration.$ 89. $The differential equation representing the family of curves y^{2} = 2 c (x + c), where c is a positive parameter, is of$ (1) $order 1, degree 4$ (2) $order 2, degree 3$ (3) $order 2, degree 4$ (4) $order 1 , degree 3$ 90. $If \frac{dx}{dy} = \frac{1 + x - y ^{2}}{y}, x (1) = 1, then 5 x (2) is equal to :$