Differential Equations PYQs Set 4 (PDF)

Set 4 of 11 · Questions 91-120 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 4 (Q91-120)
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.
91. $Let y = y (x) be the solution of the differential equation (3 y^{2} - 5 x^{2}) ydx + 2 x (x^{2} - y^{2}) dy = 0 Such that y (1) = 1, then ((y (2))^{3} - 12 y (2)) is equal to$ (1) $322$ (2) $64$ (3) $162$ (4) $32$ 92. $Let y = y (x) be the solution of the differential equation (2 x log_{e} x) \frac{dy}{dx} + 2 y = \frac{3}{x} log_{e} x, x > 0 and y (e^{- 1}) = 0 . Then, y (e) is equal to$ (1) $- \frac{3}{e}$ (2) $- \frac{3}{2 e}$ (3) $- \frac{2}{3 e}$ (4) $- \frac{2}{e}$ 93. $Which one of the following is a homogeneous differential equation?$ (1) $\frac{dy}{dx} = x^{3} + (sin x) y$ (2) $\frac{dy}{dx} = (x^{3} + y^{3}) e^{\frac{x}{y}} + x y$ (3) $(x^{2} + y^{2}) dx = 2 xydy$ (4) $x \frac{dy}{dx} = y + e^{\frac{x}{y}}$ 94. $Let y = y (x) be the solution of the differential equation (x y - 5 x^{2} 1 + x^{2}) d x + (1 + x^{2}) d y = 0, y (0) = 0 . Then y (3) is equal to$ (1) $\frac{15}{2}$ (2) $\frac{5 3}{2}$ (3) $22$ (4) $\frac{14}{3}$ 95. $The solution of the differential equation y_{1} y_{3} = 3 y_{2}^{2} is$ (1) $x = A_{1} y^{2} + A_{2} y + A_{3}$ (2) $x = A_{1} y + A_{2}$ (3) $x = A_{1} y^{2} + A_{2} y$ (4) $None of these$ 96. $The equation of any member of the family of all the ellipses whose axes are along the coordinate axes satisfies the differential equation$ (1) $xy y^{’ ’} + x (y^{’})^{2} - y y^{’} = 0$ (2) $xy y^{’ ’} + x (y^{’})^{2} - y = y^{’}$ (3) $y^{’ ’} + \frac{( y ^{’} ) ^{2}}{y} - \frac{y}{x} = 0$ (4) $y^{’ ’} + (y^{’})^{2} + x^{2} y^{2} = 0$ 97. $If a curve y = y (x) passes through the point (1, \frac{π}{2}) and satisfies the differential equation (7 x^{4} cot y - e^{x} cosecy) \frac{dx}{dy} = x^{5}, x \geq 1, then at x = 2, the value of cos y is :$ (1) $\frac{2 e ^{2} + e}{64}$ (2) $\frac{2 e ^{2} - e}{64}$ (3) $\frac{2 e ^{2} - e}{128}$ (4) $\frac{2 e ^{2} + e}{128}$ 98. $If y = f (x) is the solution of the differential equation x \frac{dy}{dx} = x^{2} + 3 y, x > 0, y (2) = 4, then f (4) =$ (1) $48$ (2) $260$ (3) $80$ (4) $36$ 99. $If the solution y = y (x) of the differential equation (x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 2) d y - (2 x^{2} + 2 x + 3) d x = 0 satisfies y (- 1) = - \frac{π}{4}, then y (0) is equal to :_______$ (1) $\frac{π}{2}$ (2) $- \frac{π}{2}$ (3) $0$ (4) $\frac{π}{4}$ 100. $Find the degree of the differential equation y_{3}^{2/3} + 2 + 3 y_{2} + y_{1} = 0 .$ (1) $4$ (2) $2$ (3) $3$ (4) $1$ 101. $The general solution of the differential equation (1 + y^{2}) dx = (tan^{- 1} y - x) dy is$ (1) $x= tan^{- 1} y-1+Ce- tan^{- 1} y$ (2) $x= tan^{- 1} y-1+Cetan-1y$ (3) $x= tan^{- 1} y-1+C$ (4) $x= tan^{- 1} y+Ce- tan^{- 1} y$ 102. $The degree of the differential equation log (\frac{dy}{dx}) = (2 x + 3 \frac{dy}{dx})^{2} is$ (1) $1$ (2) $2$ (3) $3$ (4) $not defined$ 103. $If x^{3} dy + xydx = x^{2} dy + 2 ydx; y (2) = e and x > 1, then y (4) is equal to:$ (1) $\frac{e}{2}$ (2) $\frac{1}{2} + e$ (3) $\frac{3}{2} + e$ (4) $\frac{3}{2} e$ 104. $The integrating factor of the linear directional equation \frac{dy}{dx} + P (x) y = Q (x) is a solution of the differential equation$ (1) $\frac{dy}{dx} - P (x) y = 0$ (2) $\frac{dy}{dx} + P (x) y = 0$ (3) $\frac{dy}{dx} - \frac{y}{x} = P (x)$ (4) $\frac{dy}{dx} + \frac{x}{y} = P (x)$ 105. $The order of the differential equation y = C_{1} e^{C_{2} + x} + C_{3} e^{C_{4} + x} is$ (1) $1$ (2) $3$ (3) $2$ (4) $4$ 106. $Let y = y (x) be the solution of the differential equation x^{3} dy + (xy - 1) dx = 0, x > 0, y (\frac{1}{2}) = 3 - e Then y (1) is equal to$ (1) $1$ (2) $e$ (3) $2-e$ (4) $3$ 107. $If y = y (x) be the solution of the differential equation cos x \frac{dy}{dx} + 2 y sin x = sin 2 x, x \in (0, \frac{π}{2}) . If y (\frac{π}{3}) = 0, then y (\frac{π}{4}) is equal to:$ (1) $\frac{1}{2} - 1$ (2) $2 - 2$ (3) $2 - 2$ (4) $2 + 2$ 108. $The differential equation of the family of circles passing through the origin and having centre at the line y = x is:$ (1) $(x^{2} - y^{2} + 2 xy) d x = (x^{2} - y^{2} - 2 xy) d y$ (2) $(x^{2} + y^{2} + 2 xy) d x = (x^{2} + y^{2} - 2 xy) d y$ (3) $(x^{2} + y^{2} - 2 xy) d x = (x^{2} + y^{2} + 2 xy) d y$ (4) $(x^{2} - y^{2} + 2 xy) d x = (x^{2} - y^{2} + 2 xy) d y$ 109. $The degree of the differential equation (\frac{d ^{2} y}{d x ^{2}})^{\frac{4}{3}} + x (\frac{dy}{dx})^{2} - ycos (\frac{dy}{dx}) = 0 is$ (1) $4$ (2) $3$ (3) $6$ (4) $not defined$ 110. $The differential equation \frac{dy}{dx} = \frac{1}{ax + by + c}, where a, b, c are all non-zero real numbers, is$ (1) $linear in y$ (2) $linear in x$ (3) $linear in both x and y$ (4) $homogeneous equation$ 111. $The general solution of the differential equation x^{2} dy - 2 xydx = x^{4} cos x dx is$ (1) $y = x^{2} sin x + c$ (2) $y = cos x + c x^{2}$ (3) $y = sin x + c x^{2}$ (4) $y = x^{2} sin x + c x^{2}$ 112. $The differential equation of the family of circles passing through the origin and having their centres on the x -axis is$ (1) $x^{2} = y^{2} + xy \frac{dy}{dx}$ (2) $x^{2} = y^{2} + 3 xy \frac{dy}{dx}$ (3) $y^{2} = x^{2} + 2 xy \frac{dy}{dx}$ (4) $y^{2} = x^{2} - 2 xy \frac{dy}{dx}$ 113. $If \frac{dy}{dx} = \frac{xy}{x ^{2} + y ^{2}}; y (1) = 1 : then a value x satisfying y (x) = e is:$ (1) $3 e$ (2) $\frac{e}{2}$ (3) $2 e$ (4) $\frac{1}{2} 3 e$ 114. $The general solution of the differential equation x^{2} dy - (xy - y^{2}) dx = 0 is$ (1) $y^{2} = 3 x^{2} log (cx)$ (2) $y^{2} = log x + c$ (3) $ylog x = x + cy$ (4) $ylog x = x^{2} + c$ 115. $The degree of the differential equation 1 + (\frac{dy}{dx})^{2} + (\frac{d ^{2} y}{d x ^{2}})^{2} = 3 \frac{d ^{2} y}{d x ^{2}} + 1 is$ (1) $1$ (2) $6$ (3) $2$ (4) $3$ 116. $The differential equation of all parabolas whose axes are parallel to Y -axis is$ (1) $\frac{d ^{3} y}{d x ^{3}} = 0$ (2) $\frac{d ^{2} y}{d x ^{2}} = 0$ (3) $\frac{d ^{2} y}{d x ^{2}} + \frac{dy}{dx} = 0$ (4) $y \frac{d ^{2} y}{d x ^{2}} + (\frac{dy}{dx})^{2} = 0$ 117. $The solution for the differential equation \frac{dy}{y} + \frac{dx}{x} = 0 is$ (1) $\frac{1}{y} + \frac{1}{x} = c$ (2) $log x \cdot log y = c$ (3) $xy = c$ (4) $x + y = c$ 118. $The degree and order of the differential equation \frac{d ^{2} y}{d x ^{2}} = 3 1 + (\frac{dy}{dx})^{2} respectively are$ (1) $2 and 3$ (2) $3 and 2$ (3) $2 and 2$ (4) $3 and 3$ 119. $If the general solution of the differential equation (y - x + 1) dy - (y + x + 2) dx = 0 is f (x, y, c) = 0, then the value of c such that f (1, 1 c) = 0 is$ (1) $4$ (2) $-4$ (3) $2$ (4) $1$ 120. $If the solution curve y = y (x) of the differential equation (1 + y^{2}) (1 + log_{e} x) dx + xdy = 0, x > 0 passes through the point (1, 1) and y (e) = \frac{α - tan ( \frac{3}{2} )}{β + tan ( \frac{3}{2} )}, then α + 2 β is$