Differential Equations PYQs Set 8 (PDF)

Set 8 of 11 · Questions 211-240 of 326

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Differential Equations · Set 8 (Q211-240)
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211. $An integrating factor of the differential equation (x^{2} + 1) \frac{dy}{dx} + xy = x^{3} is$ (1) $\frac{x}{1 + x ^{2}}$ (2) $\frac{1}{2} log (1 + x^{2})$ (3) $1 + x^{2}$ (4) $e^{log (1 + x^{2})}$ 212. $The general solution of \frac{dy}{dx} = x + sin xcos y + xcos y$ (1) $tan \frac{x}{2} = \frac{y ^{2}}{2} - cos y + C$ (2) $tan \frac{y}{2} = \frac{x ^{2}}{2} - cos x + C$ (3) $sec^{2} \frac{y}{2} = \frac{x ^{2}}{2} - cos x + C$ (4) $tan \frac{y}{2} = \frac{x ^{2}}{2} + cos x + C$ 213. $Let y = y (x) be the solution curve of the differential equation \frac{dy}{dx} = \frac{y}{x} (1 + x y^{2} (1 + log_{e} x)) x > 0, y (1) = 3 Then \frac{y ^{2} ( x )}{9} is equal to$ (1) $\frac{x ^{2}}{5 - 2 x ^{3} ( 2 + log _{e} x ^{3} )}$ (2) $\frac{x ^{2}}{2 x ^{3} ( 2 + log _{e} x ^{3} ) - 3}$ (3) $\frac{x ^{2}}{3 x ^{3} ( 1 + log _{e} x ^{2} ) - 2}$ (4) $\frac{x ^{2}}{7 - 3 x ^{3} ( 2 + log _{e} x ^{2} )}$ 214. $The differential equation of the family of straight lines whose slope is equal to y -intercept is$ (1) $(x + 1) \frac{d y}{d x} - y = 0$ (2) $(x + 1) \frac{d y}{d x} + y = 0$ (3) $\frac{d y}{d x} = \frac{x - 1}{y - 1}$ (4) $\frac{d y}{d x} = \frac{x + 1}{y + 1}$ 215. $Let y = y (x) be the solution of the differential equation, \frac{2 + sin x}{y + 1} ∙ \frac{dy}{dx} = - cos x, y > 0, y (0) = 1 . If y (π) = a and \frac{dy}{dx} at x = π is b, then the ordered pair (a, b) is equal to:$ (1) $(2, 1)$ (2) $(1, - 1)$ (3) $(2, \frac{3}{2})$ (4) $(1, 1)$ 216. $Let y = y (x) be the solution of the differentialequation \frac{dy}{dx} = (y + 1) ((y + 1) e^{\frac{x ^{2}}{2}} - x), 0 < x < 2.1, with y (2) = 0 . Then the value of \frac{dy}{dx} at x = 1 is equal to:$ (1) $\frac{- e ^{\frac{3}{2}}}{( e ^{2} + 1 ) ^{2}}$ (2) $- \frac{2 e ^{2}}{( 1 + e ^{2} ) ^{2}}$ (3) $\frac{e ^{\frac{5}{2}}}{( 1 + e ^{2} ) ^{2}}$ (4) $\frac{5 e ^{\frac{1}{2}}}{( e ^{2} + 1 ) ^{2}}$ 217. $The solution of the differential equation y^{’} = \frac{1}{e ^{- y} - x}$ (1) $x = e^{- y} (y + C)$ (2) $y + e^{- y} = x + C$ (3) $x = e^{y} (y + C)$ (4) $x + y = e^{- y} + C$ 218. $Let y = y (x) be the solution of the differential equation x \frac{dy}{dx} - y = x^{2} cotx, x \in (0, π) . If y (\frac{π}{2}) = \frac{π}{2}, then 6 y (\frac{π}{6}) - 8 y (\frac{π}{4}) is equal to :$ (1) $3 π$ (2) $π$ (3) $- π$ (4) $- 3 π$ 219. $Let p \in R, then the differential equation of the family of curves y = (α + βx) e^{px}, where α, β are arbitrary constants, is$ (1) $y^{’ ’} + 4 p y^{’} + p^{2} y = 0$ (2) $y^{’ ’} - 2 p y^{’} + p^{2} y = 0$ (3) $y^{’ ’} + 2 p y^{’} - p^{2} y = 0$ (4) $y^{’ ’} + 2 p y^{’} + p^{2} y = 0$ 220. $The order and degree of the differential equation y = x \frac{dy}{dx} + \frac{2}{\frac{dy}{dx}} is$ (1) $1,2$ (2) $1,3$ (3) $2,1$ (4) $1,1$ 221. $Let y = y (x) be a differentiable function in the interval (0, \infty) such that y (1) = 2, and lim t \to x (\frac{t ^{2} y ( x ) - x ^{2} y ( t )}{x - t}) = 3 for each x > 0 . Then 2 y (2) is equal to$ (1) $23$ (2) $27$ (3) $12$ (4) $18$ 222. $The solution of the differential equation x y^{’} = 2 x e^{- y / x} + y is$ (1) $e^{y / x} + log \| Cx \| = 0$ (2) $e^{- y / x} = x + C$ (3) $e^{y / x} = log \| Cx \|$ (4) $e^{y / x} = 2 log \| Cx \|$ 223. $The order and degree of the differential equation 3 x^{2} \frac{d ^{2} y}{d x ^{2}} - sin (\frac{d ^{3} y}{d x ^{3}}) + cos (xy) = 0 are$ (1) $Order can't be defined and degree is 3$ (2) $Order is 3 and degree can't be defined$ (3) $Order is 3 and degree is 1$ (4) $Order is 1 and degree is 3$ 224. $The solution curve of the differential equation, (1 + e^{- x}) (1 + y^{2}) \frac{dy}{dx} = y^{2}, which passes the point (0, 1), is:$ (1) $y^{2} + 1 = y (log_{e} (\frac{1 + e ^{- x}}{2}) + 2)$ (2) $y^{2} + 1 = y (log_{e} (\frac{1 + e ^{x}}{2}) + 2)$ (3) $y^{2} = 1 + y log_{e} (\frac{1 + e ^{- x}}{2})$ (4) $y^{2} = 1 + y log_{e} (\frac{1 + e ^{x}}{2})$ 225. $Let f : (0, \infty) \to R be a function which is differentiable at all points of its domain and satisfies the condition x^{2} f^{’} (x) = 2 xf (x) + 3, with f (1) = 4 . Then 2 f (2) is equal to:$ (1) $39$ (2) $19$ (3) $29$ (4) $23$ 226. $The temperature T (t) of a body at time t = 0 is 160^{\circ} F and it decreases continuously as per the differential equation \frac{dT}{dt} = - K (T - 80), where K is positive constant. If T (15) = 120^{\circ} F, then T (45) is equal to$ (1) $85^{\circ} F$ (2) $95^{\circ} F$ (3) $90^{\circ} F$ (4) $80^{\circ} F$ 227. $Let y = y (x) be the solution curve of the differential equation x (x^{2} + e^{x}) dy + (e^{x} (x - 2) y - x^{3}) dx = 0, x, passing through the point (1, 0) . Then y (2) is equal to:$ (1) $\frac{2}{2 + e ^{2}}$ (2) $\frac{4}{4 - e ^{2}}$ (3) $\frac{4}{4 + e ^{2}}$ (4) $\frac{2}{2 - e ^{2}}$ 228. $By multiplying with e^{\int Pdx} on both sides of the equation \frac{dy}{dx} + P (x) y = Q (x), the left side of the equation takes the form \frac{d}{dx} (yf (x)), then f (x) =$ (1) $\int y e^{\int Pdx} dx$ (2) $yP (x)$ (3) $e^{\int Pdx}$ (4) $P (x) e^{\int Pdx}$ 229. $The population P = P (t) at time^{’} t^{’} of a certain species follows the differential equation \frac{dP}{dt} = 0.5 P - 450 . If P (0) = 850, then the time at which population becomes zero is$ (1) $\frac{1}{2} log_{e} 18$ (2) $2 log_{e} 18$ (3) $log_{e} 9$ (4) $log_{e} 18$ 230. $The differential equation for which 1 + y^{2} = Cx e^{tan - 1} x is the general solution of$ (1) $xy (1 + x^{2}) dy - e^{tan^{- 1} x} (1 + x + x^{2}) dx = 0$ (2) $xy (1 + y^{2}) dy - (1 + x^{2}) (1 + y + y^{2}) dx = 0$ (3) $(1 + y^{2}) tan^{- 1} x \frac{dy}{dx} = \frac{1 + x ^{2}}{xy}$ (4) $xy (1 + x^{2}) dy - (1 + y^{2}) (1 + x + x^{2}) dx = 0$ 231. $Let y = y (x) be the solution of the differential equation \frac{dy}{dx} = \frac{( tan x ) + y}{sin x ( sec x - sin x tan x )}, x \in (0, \frac{π}{2}) satisfying the condition y (\frac{π}{4}) = 2 . Then, y (\frac{π}{3}) is$ (1) $3 (2 + log_{e} 3)$ (2) $\frac{3}{2} (2 + log_{e}) 3)$ (3) $3 (1 + 2 log_{e} 3)$ (4) $3 (2 + log_{e} 3)$ 232. $The solution of the differential equation (2 x - 3 y + 5) dx + (9 y - 6 x - 7) dy = 0, is$ (1) $3 x - 3 y + 8 log \| 6 x - 9 y - 1 \| = c$ (2) $3 x - 9 y + 8 log \| 6 x - 9 y - 1 \| = c$ (3) $3 x - 9 y + 8 log \| 2 x - 3 y - 1 \| = c$ (4) $3 x - 9 y + 4 log \| 2 x - 3 y - 1 \| = c$ 233. $The value of ' k' for which the function f (x) = k (x + sin x) + k is increasing, is equal to$ (1) $k < 0$ (2) $k > 0$ (3) $k = 0$ (4) $Data Insufficient$ 234. $Let y = y (x) be the solution of the differential equation \frac{dy}{dx} + 3 (tan^{2} x) y + 3 y = sec^{2} x y (0) = \frac{1}{3} + e^{3} . Then y (\frac{π}{4}) is equal to$ (1) $\frac{2}{3}$ (2) $\frac{4}{3}$ (3) $\frac{4}{3} + e^{3}$ (4) $\frac{2}{3} + e^{3}$ 235. $Let y = y (x) be the solution of the differential equation (1 + x^{2}) \frac{dy}{dx} + y = e^{tan^{- 1} x}, y (1) = 0 . Then y (0) is$ (1) $\frac{1}{2} (e^{π /2} - 1)$ (2) $\frac{1}{2} (1 - e^{π /2})$ (3) $\frac{1}{4} (1 - e^{π /2})$ (4) $\frac{1}{4} (e^{π /2} - 1)$ 236. $Find the order and degree of the differential equation (1 + (\frac{dy}{dx})^{2} + sin (\frac{dy}{dx}))^{3/4} = \frac{d ^{2} y}{d x ^{2}} .$ (1) $order =2, degree = 3$ (2) $order =2, degree = 4$ (3) $order =2, degree = 3 4$ (4) $order = 2, degree = not defined$ 237. $If the solution curve y = f (x) of the differential equation (x^{2} - 4) y^{’} - 2 xy + 2 x (4 - x^{2})^{2} = 0, x > 2, passes through the point (3, 15), then the local maximum value of f is____.$ 238. $If the solution curve, of the differential equation \frac{dy}{dx} = \frac{x + y - 2}{x - y} passing through the point (2, 1) is tan^{- 1} (\frac{y - 1}{x - 1}) - \frac{1}{β} log_{e} (α + (\frac{y - 1}{x - 1})^{2}) = log_{e} \| x - 1 \|, then 5 β + α is equal to$ 239. $For a differentiable function f : R \to R, suppose f^{’} (x) = 3 f (x) + α, where α \in R, f (0) = 1 and lim_{x \to - \infty} f (x) = 7 . Then 9 f (- log_{e} 3) is equal to_______$ 240. $Let y = y (x) be the solution of the differential equation sec^{2} xdx + (e^{2 y} tan^{2} x + tan x) dy = 00 < x < \frac{π}{2}, y (\frac{π}{4}) = 0. If y (\frac{π}{6}) = α, Then e^{8 α} is equal to$