Differential Equations PYQs Set 7 (PDF)

Set 7 of 11 · Questions 181-210 of 326

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Differential Equations · Set 7 (Q181-210)
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181. $Let y = y (x) be the solution of the differential equation, \frac{dy}{dx} + y tan x = 2 x + x^{2} tan x, x \in (- \frac{π}{2}, \frac{π}{2}), such that y (0) = 1 . Then:$ (1) $y (\frac{π}{4}) + y (- \frac{π}{4}) = \frac{π ^{2}}{2} + 2$ (2) $y^{’} (\frac{π}{4}) + y^{’} (\frac{π}{4}) = - 2$ (3) $y (\frac{π}{4}) - y (- \frac{π}{4}) = 2$ (4) $y^{’} (\frac{π}{4}) + y^{’} (- \frac{π}{4}) = π - 2$ 182. $Let f : (0, 1) \to R be such that f (xy) == f (x) . f (y), for all x, y \in (0, 1), and f (0) \neq = 0 . If y = y (x) satisfies the differential equation, \frac{d y}{dx} = f (x) with y (0) = 1, then y (\frac{1}{4}) + y (\frac{3}{4}) is equal to :$ (1) $3$ (2) $4$ (3) $2$ (4) $5$ 183. $The degree and order of the differential equation of the family of parabolas whose axis is the X-axis, are respectively$ (1) $2,2$ (2) $2,1$ (3) $1,2$ (4) $3,2$ 184. $If the curve y = y (x) is the solution of the differential equation 2 (x^{2} + x^{\frac{5}{4}}) dy - y (x + x^{\frac{1}{4}}) dx = 2 x^{\frac{9}{4}} dx, x > 0 which passes through the point (1, 1 - \frac{4}{3} log_{e} 2), then the value of y(16) is equal to :$ (1) $4 (\frac{31}{3} + \frac{8}{3} log_{e} 3)$ (2) $(\frac{31}{3} + \frac{8}{3} log_{e} 3)$ (3) $4 (\frac{31}{3} - \frac{8}{3} log_{e} 3)$ (4) $(\frac{31}{3} - \frac{8}{3} log_{e} 3)$ 185. $Let x = x (y) be the solution of the differential equation y = (x - y \frac{d x}{d y}) sin (\frac{x}{y}), y > 0 and x (1) = \frac{π}{2} . Then cos (x (2)) is equal to:$ (1) $1 - 2 (log_{e} 2)^{2}$ (2) $1 - 2 (log_{e} 2)$ (3) $2 (log_{e} 2) - 1$ (4) $2 (log_{e} 2)^{2} - 1$ 186. $If y = y (x) is the solution of the differential equation x \frac{d y}{d x} + 2 y = x^{2} satisfying y (1) = 1, then y (\frac{1}{2}) is equal to :$ (1) $\frac{7}{64}$ (2) $\frac{1}{4}$ (3) $\frac{49}{16}$ (4) $\frac{13}{16}$ 187. $The integrating factor of the differential equation (2 x + 3 y^{2}) dy = ydx (y > 0) is$ (1) $e^{\frac{1}{y}}$ (2) $\frac{1}{x}$ (3) $- \frac{1}{y ^{2}}$ (4) $\frac{1}{y ^{2}}$ 188. $p and q are positive integers and n < r < m . If the order and degree of the differential equation (\frac{d ^{m} y}{d x ^{m}} + \frac{d ^{n} y}{d x ^{n}})^{p / q} = 5 \frac{d ^{r} y}{d x ^{r}} are respectively 4 and 3 , then$ (1) $n = 4, q = 3$ (2) $m = 4, q = 3$ (3) $r = 4, q = 3$ (4) $m = 4, p = 3$ 189. $The solution of the differential equation (x^{2} + y^{2}) d x - 5 xy d y = 0, y (1) = 0, is :$ (1) $(x^{2} - 2 y^{2})^{6} = x$ (2) $(x^{2} - 4 y^{2})^{6} = x$ (3) $(x^{2} - 4 y^{2})^{5} = x^{2}$ (4) $(x^{2} - 2 y^{2})^{5} = x^{2}$ 190. $The solution of the differential equation 1 - y^{2} dx + xdy - sin^{- 1} ydy = 0, is$ (1) $x= sin^{- 1} y-1+ce- sin^{- 1} y$ (2) $y=x1-y2+ sin^{- 1} y+c$ (3) $x=1+ sin^{- 1} y+cesin-1y$ (4) $y= sin^{- 1} y-1+x1-y2+c$ 191. $Let y = y (x) be the solution of the differential equation (x^{2} + 1) y^{’} - 2 xy = (x^{4} + 2 x^{2} + 1) cos x, y (0) = 1 . Then \int_{- 3}^{3} y (x) dx is:$ (1) $36$ (2) $24$ (3) $18$ (4) $30$ 192. $The general solution of the differential equation 2 x \frac{dy}{dx} - y = 3 is a family of$ (1) $parabolas$ (2) $hyperbolas$ (3) $circles$ (4) $straight lines$ 193. $If y = y (x) is the solution curve of the differential equation (x^{2} - 4) d y - (y^{2} - 3 y) d x = 0, x > 2, y (4) = \frac{3}{2} and the slope of the curve is never zero, then the value of y (10) equals :$ (1) $\frac{3}{1 + ( 8 ) ^{1/4}}$ (2) $\frac{3}{1 + 2 2}$ (3) $\frac{3}{1 - 2 2}$ (4) $\frac{3}{1 - ( 8 ) ^{1/4}}$ 194. $If y (x) is the solution of the differential equation \frac{dy}{dx} + (\frac{2 x + 1}{x}) y = e^{- 2 x}, x > 0, where y (1) = \frac{1}{2} e^{- 2}, then$ (1) $y (log_{e} 2) = log_{e} 4$ (2) $y (log_{e} 2) = \frac{log _{e} 2}{4}$ (3) $yxis decreasing in (\frac{1}{2}, 1)$ (4) $y (x) is decreasing in (0, 1)$ 195. $If m and n are degree and order of (1 + y_{1}^{2})^{2/3} = y_{2}, then the value of \frac{m + n}{m - n} is$ (1) $3$ (2) $4$ (3) $5$ (4) $2$ 196. $Let y = y (t) be a solution of the differential equation \frac{dy}{dt} + αy = γ e^{- βt} Where, α > 0, β > 0 and γ > 0 . then lim t \to \infty y (t)$ (1) $is 0$ (2) $does not exist$ (3) $is 1$ (4) $is - 1$ 197. $The solution of differential equation x \frac{dy}{dx} + 2 y = x^{2} is$ (1) $y = \frac{x ^{2}}{4} + C$ (2) $y = \frac{x ^{4} + C}{4 x ^{2}}$ (3) $y = \frac{x ^{2} + C}{4 x ^{2}}$ (4) $y = \frac{x ^{4} + C}{x ^{2}}$ 198. $Let y = y (x) be the solution of the differential equation (x^{2} + 4)^{2} dy + (2 x^{3} y + 8 xy - 2) dx = 0 . If y (0) = 0, then y (2) is equal to$ (1) $\frac{π}{32}$ (2) $2 π$ (3) $\frac{π}{8}$ (4) $\frac{π}{16}$ 199. $The differential equation of the family of circles touching y - axis at the origin is$ (1) $(x^{2} + y^{2}) \frac{dy}{dx} - 2 xy = 0$ (2) $(x^{2} - y^{2}) + 2 xy \frac{dy}{dx} = 0$ (3) $(x^{2} - y^{2}) \frac{dy}{dx} - 2 xy = 0$ (4) $(x^{2} + y^{2}) \frac{dy}{dx} + 2 xy = 0$ 200. $Let x = x (y) be the solution of the differential equation y^{2} d x + (x - \frac{1}{y}) d y = 0 . If x (1) 1, then x (\frac{1}{2}) is:$ (1) $\frac{1}{2} + e$ (2) $3 + e$ (3) $3 - e$ (4) $\frac{3}{2} + e$ 201. $The general solution of \frac{dy}{dx} = 1 - x^{2} - y^{2} + x^{2} y^{2} is$ (1) $2 sin^{- 1} y = x 1 - x^{2} + sin^{- 1} x + c$ (2) $cos^{- 1} y = x cos^{- 1} x + c$ (3) $sin^{- 1} y = \frac{1}{2} sin^{- 1} x + c$ (4) $2 sin^{- 1} y = x 1 - y^{2} + c$ 202. $The solution of the differential equation \frac{dy}{dx} = - (\frac{x ^{2} + 3 y ^{2}}{3 x ^{2} + y ^{2}}) . y (1) = 0$ (1) $log_{e} (x + y) - \frac{xy}{( x + y ) ^{2}} = 0$ (2) $log_{e} (x + y) + \frac{xy}{( x + y ) ^{2}} = 0$ (3) $log_{e} (x + y) + \frac{2 xy}{( x + y ) ^{2}} = 0$ (4) $log_{e} (x + y) - \frac{2 xy}{( x + y ) ^{2}} = 0$ 203. $For the differential equation \frac{d ^{2} y}{d x ^{2}} = 3 (y \frac{dy}{dx} + xsin (\frac{dy}{dx}))^{2}$ (1) $Order is 2 and degree is 3$ (2) $Order is 3 and degree is 3$ (3) $Order is 3 and degree is 2$ (4) $Order is 2 and degree is not defined$ 204. $Which of the following is true for y (x) that satisfies the differential equation \frac{dy}{dx} = xy - 1 + x - y; y (0) = 0$ (1) $y (1) = e^{- \frac{1}{2}} - 1$ (2) $y (1) = e^{\frac{1}{2}} - e^{- \frac{1}{2}}$ (3) $y (1) = 1$ (4) $y (1) = e^{\frac{1}{2}} - 1$ 205. $If cos x \frac{dy}{dx} - y sin x = 6 x, (0 < x < \frac{π}{2}) and y (\frac{π}{3}) = 0, then y (\frac{π}{6}) is equal to :$ (1) $\frac{π ^{2}}{2 3}$ (2) $- \frac{π ^{2}}{2}$ (3) $- \frac{π ^{2}}{2 3}$ (4) $- \frac{π ^{2}}{4 3}$ 206. $General solution of differential equation \frac{dy}{dx} + y = 1 (y \neq = 1) is$ (1) $log (\frac{1}{1 - y}) = x + C$ (2) $log \| 1 - y \| = x + C$ (3) $log \| 1 + y \| = x + C$ (4) $log (\frac{1}{1 - y}) = - x + C$ 207. $The degree of the differential equation x (\frac{d ^{2} y}{d x ^{2}})^{1/3} + 2 x^{2} (\frac{d ^{2} y}{d x ^{2}})^{5/3} + 7 \frac{dy}{dx} + y = 0$ (1) $15$ (2) $5$ (3) $12$ (4) $3$ 208. $The solution of the differential equation e^{- x} (y + 1) dy + (cos^{2} x - sin 2 x) y (dx) = 0 subjected to the condition that y = 1 when x = 0 is$ (1) $y + log y + e^{x} cos^{2} x = 2$ (2) $log (y + 1) + e^{x} cos^{2} x = 1$ (3) $y + log y = e^{x} cos^{2} x$ (4) $(y + 1) + e^{x} cos^{2} x = 2 .$ 209. $The solution curve of the differential equation y \frac{dx}{dy} = x (log_{e} x - log_{e} y + 1), x > 0, y > 0 passing through the point (e, 1) is$ (1) $(log_{e} \frac{y}{x}) = x$ (2) $(log_{e} \frac{y}{x}) = y^{2}$ (3) $(log_{e} \frac{x}{y}) = y$ (4) $2 (log_{e} \frac{x}{y}) = y + 1$ 210. $Let the solution y = y (x) of the differential equation \frac{d y}{d x} - y = 1 + 4 sin x satisfy y (π) = 1 . Then y (\frac{π}{2}) + 10 is equal to_______$