Differential Equations PYQs Set 6 (PDF)

Set 6 of 11 · Questions 151-180 of 326

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Differential Equations · Set 6 (Q151-180)
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151. $The solution of the differential equation (2 x - 4 y + 3) \frac{dy}{dx} + (x - 2 y + 1) = 0 is (C is an arbitrary constant)$ (1) $log [(2 x - 4 y) + 3] = x - 2 y + C$ (2) $log [2 (2 x - 4 y) + 3] = 2 (x - 2 y) + C$ (3) $log [2 (x - 2 y) + 5] = 2 (x + y) + C$ (4) $log [4 (x - 2 y) + 5] = 4 (x + 2 y) + C$ 152. $The order of the differential equation obtained by eliminating arbitrary constants in the family of curves c_{1} y = (c_{2} + c_{3}) e^{x + c_{4}} is$ (1) $2$ (2) $3$ (3) $4$ (4) $1$ 153. $If the solution of \frac{dy}{dx} - y log_{e} 0.5 = 0, y (0) = 1, and y (x) \to k as x \to \infty then k =$ (1) $\infty$ (2) $-1$ (3) $1$ (4) $0$ 154. $Find the solution of the differential equation (e^{y - x}) dy = (e^{x} - e^{y}) dx$ (1) $e^{y} e^{x} = e^{2 x} - e^{x^{2}} + c$ (2) $e^{y} e^{x} = e^{x} e^{e^{x}} - e^{e^{x}} + c$ (3) $e^{y} e^{e^{x}} = e^{x} e^{e^{x}} - e^{e^{x}} + c$ (4) $e^{e^{y}} e^{x} = e^{x} e^{e^{x}} - e^{e^{x}} + c$ 155. $The integrating factor of the differential equation x \cdot \frac{dy}{dx} + 2 y = x^{2} is (x \neq = 0)$ (1) $e^{log x}$ (2) $log \| x \|$ (3) $x$ (4) $x^{2}$ 156. $The solution of the differential equation (x + 2 y^{3}) \frac{dy}{dx} = y is$ (1) $x = y^{3} + c$ (2) $x = y^{3} + cy$ (3) $y = x^{3} + c$ (4) $y = x^{3} + cx + d$ 157. $The general solution of differential equation \frac{dy}{dx} = e^{x + y} + x^{2} e^{x^{3} + y} is (where C is a constant of integration)$ (1) $e^{x} - e^{- y} - \frac{1}{3} e^{x^{3}} = C$ (2) $e^{x} + e^{- y} + \frac{1}{3} e^{x^{3}} = C$ (3) $e^{x} - e^{- x} + \frac{1}{3} e^{x^{3}} = C$ (4) $e^{x} + e^{- x} - \frac{1}{3} e^{x^{3}} = C$ 158. $If sin (\frac{y}{x}) = log_{e} \| x \| + \frac{α}{2} is the solution of the differential equation x cos (\frac{y}{x}) \frac{dy}{dx} = y cos (\frac{y}{x}) + x and y (1) = \frac{π}{3}, then α^{2} is equal to$ (1) $3$ (2) $12$ (3) $4$ (4) $9$ 159. $If y = y (x) is the solution of the differential equation \frac{d y}{d x} + 2 y = sin (2 x), y (0) = \frac{3}{4}, then y (\frac{π}{8}) is equal to:$ (1) $e^{π /8}$ (2) $e^{π /4}$ (3) $e^{- π /4}$ (4) $e^{- π /8}$ 160. $The differential equation of the family of curves y = ax + \frac{1}{a}, where a (\neq = 0) is an arbitrary constant, has the degree$ (1) $4$ (2) $3$ (3) $1$ (4) $2$ 161. $If l and m are respectively the order and the degree of the differential equation f (x) y^{’ ’} + g (x) y^{’} = \frac{4 y}{x} whose general solution is y = a x^{2} + blog x, then f (m) + g (m) =$ (1) $2 l$ (2) $l$ (3) $3 m$ (4) $l + m$ 162. $The solution of the differential equation \frac{dy}{dx} = tan (\frac{y}{x}) + \frac{y}{x} is$ (1) $cos (\frac{y}{x}) = cx$ (2) $sin (\frac{y}{x}) = cx$ (3) $cos (\frac{y}{x}) = cy$ (4) $sin (\frac{y}{x}) = cy$ 163. $If for the solution curve y = f (x) of the differential equation \frac{d y}{d x} + (ta n x) y = \frac{2 + se c x}{( 1 + 2 sec x ) ^{2}}, x (\frac{- π}{2}, \frac{π}{2}), f (\frac{π}{3}) = \frac{3}{10}, then f (\frac{π}{4}) is equal to:$ (1) $\frac{3 + 1}{10 ( 4 + 3 )}$ (2) $\frac{5 - 3}{2 2}$ (3) $\frac{9 3 + 3}{10 ( 4 + 3 )}$ (4) $\frac{4 - 2}{14}$ 164. $If sin x is the integrating factor (I.F.) of the linear differential equation \frac{dy}{dx} + Py = Q, then P is$ (1) $log sin x$ (2) $cos x$ (3) $tan x$ (4) $cot x$ 165. $The particular solution of the differential equation (1 + y^{2}) dx - xydy = 0 at x = 1, y = 0, represents$ (1) $circle$ (2) $pair of straight lines$ (3) $hyperbola$ (4) $ellipse$ 166. $If the order and degree of the differential equation corresponding to the family of curves y^{2} = 4 a (x + a) (a is parameter) are m and n respectively, then m + n^{2} =$ (1) $3$ (2) $4$ (3) $5$ (4) $2$ 167. $Let a curve y = f (x) pass through the points (0, 5) and (log_{e} 2, k) . If the curve satisfies the differential equation 2 (3 + y) e^{2 x} dx - (7 + e^{2 x}) dy = 0, then k is equal to$ (1) $4$ (2) $32$ (3) $8$ (4) $16$ 168. $If y = y (x) satisfies the differential equation 16 (x + 9 x) (4 + 9 + x) cosy dy = (1 + 2 siny) dx, x > 0 and y (256) = \frac{π}{2}, y (49) = α, then 2 sinα is equal to :$ (1) $2 (2 - 1)$ (2) $3 (2 - 1)$ (3) $22 - 1$ (4) $2 - 1$ 169. $If the slope of the tangent of the curve at any point is equal to - y + e^{- x}, then the equation of the curve passing through origin is$ (1) $y + x e^{x} = 0$ (2) $y e^{x} + x = 0$ (3) $y e^{x} - x = 0$ (4) $y - x e^{x} = 0$ 170. $Let y = f (x) is the solution of the differential equation e^{y} (\frac{dy}{dx} - 1) = e^{x} such that y (0) = 0, then y (1) is equal to:$ (1) $ln 2$ (2) $2 + ln 2$ (3) $1 + ln 2$ (4) $3 + ln 2$ 171. $If y = y (x) is the solution of the differential equation \frac{5 + e ^{x}}{2 + y} ∙ \frac{dy}{dx} + e^{x} = 0 satisfying y (0) = 1 then a value of y (log_{e} 13) is:$ (1) $- 1$ (2) $2$ (3) $0$ (4) $1$ 172. $Let the solution curve of the differential equation xdy - ydx = x^{2} + y^{2} dx, x > 0, y (1) = 0, be y = y (x) . Then y (3) is equal to$ (1) $1$ (2) $4$ (3) $2$ (4) $6$ 173. $If x = f (y) is the solution of the differential equation (1 + y^{2}) + (x - 2 e^{tan^{- 1} y}) \frac{d y}{d x} = 0, y (- \frac{π}{2}, \frac{π}{2}) with f (0) = 1, then f (\frac{1}{3}) is equal to:$ (1) $e^{π /12}$ (2) $e^{π /4}$ (3) $e^{π /3}$ (4) $e^{π /6}$ 174. $The solution of the differential equation cos (x + y) dy = dx given that y (0) = 0 is$ (1) $y = tan (\frac{x + y}{2})$ (2) $y = sin (\frac{x + y}{2})$ (3) $y = tan (\frac{y}{2})$ (4) $y = tan (\frac{x}{2})$ 175. $If a curve passes through (1, 2) and has the slope of its tangent 1 - \frac{1}{x ^{2}} at a point (x, y), then the equation of that curve is$ (1) $y = 3 x - \frac{1}{x}$ (2) $y = x + \frac{1}{x}$ (3) $y = 2 x + \frac{1}{x} - 1$ (4) $y = x + \frac{2}{x} - 1$ 176. $The order and degree of the differential equation (1 + (\frac{dy}{dx})^{5})^{\frac{1}{3}} = \frac{d ^{2} y}{d x ^{2}} are respectively$ (1) $1,5$ (2) $2,1$ (3) $2,5$ (4) $2,3$ 177. $The order of differential equation of all circles of given radius ' a' is$ (1) $2$ (2) $3$ (3) $4$ (4) $1$ 178. $By eliminating the arbitrary constants from y = (a + b) sin (x + c) - d e^{x + e + f}, the differential equation obtained is of order$ (1) $6$ (2) $4$ (3) $3$ (4) $5$ 179. $Let slope of the tangent line to a curve at any point P (x, y) be given by \frac{x y ^{2} + y}{x} . If the curve intersects the line x + 2 y = 4 at x = - 2, then the value of y, for which the point (3, y) lies on the curve, is:$ (1) $- \frac{18}{11}$ (2) $- \frac{18}{19}$ (3) $- \frac{4}{3}$ (4) $\frac{18}{35}$ 180. $If y = y (x) is the solution of the equation e^{sin y} cos y \frac{dy}{dx} + e^{sin y} cos x = cos x, y (0) = 0; then 1 + y (\frac{π}{6}) + \frac{3}{2} y (\frac{π}{3}) + \frac{1}{2} y (\frac{π}{4}) is equal to$