Differential Equations PYQs Set 9 (PDF)

Set 9 of 11 · Questions 241-270 of 326

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Differential Equations · Set 9 (Q241-270)
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241. $The integrating factor of \frac{dy}{dx} + y = \frac{1 + y}{x} is$ (1) $x e^{x}$ (2) $x e^{1/ x}$ (3) $\frac{e ^{x}}{x}$ (4) $\frac{x}{e ^{x}}$ 242. $The solution of the differential equation \frac{d ^{2} y}{d x ^{2}} + y = 0 is$ (1) $y = 3 sin x + 4 cos x$ (2) $y = x^{2}$ (3) $y = x + 2$ (4) $y = log x$ 243. $The general solution of the differential equation \frac{dy}{dx} = \frac{x + y - 3}{x + y - 7} is$ (1) $(x + y - 5)^{2} = C e^{y + x}$ (2) $(x + y - 5)^{2} = C e^{y - x}$ (3) $2 log (x + y - 5) = 3 x + 2 y + C$ (4) $log (x + y - 3) = 3 (x + y - 2)^{2} + C$ 244. $If y = (\frac{2}{π} x - 1) cosec x is the solution of the differential equation, \frac{dy}{dx} + p (x) y = \frac{2}{π} cosec x; 0 < x < \frac{π}{2}, then the function p (x) is equal to:$ (1) $tan x$ (2) $cot x$ (3) $cosec x$ (4) $sec x$ 245. $The general solution of the differential equation dx = (2 x + 3 y - 4) dy is$ (1) $2 x + 6 y - 3 lo g ∣4 x + 6 y - 5∣ = c$ (2) $6 y - 3 log \| 4 x + 6 y - 5 \| = c$ (3) $2 x + 6 y - 8 - 3 log \| 4 x + 6 y - 5 \| = c$ (4) $6 x + 6 y - 3 lo g ∣4 x + 6 y - 5∣ = c$ 246. $If m and n are respectively the order and degree of the differential equation of the family of parabolas with origin as its focus and X -axis as its axis, then mn - m + n =$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 247. $Solution of the differential equation 2 y sin x \frac{dy}{dx} = 2 sin x cos x - y^{2} cos x satisfying y (π /2) = 1 is given by$ (1) $y^{2} = sin x$ (2) $y = sin^{2} x$ (3) $y^{2} = cos x + 1$ (4) $y^{2} sin x = 4 cos^{2} x$ 248. $Let y = y (x) be the solution curve of the differential equation (1 + x^{2}) dy + (y - tan^{- 1} x) dx = 0, y (0) = 1 . Then the value of y (1) is :$ (1) $\frac{4}{e ^{π /4}} + \frac{π}{2} - 1$ (2) $\frac{2}{e ^{π /4}} + \frac{π}{4} - 1$ (3) $\frac{2}{e ^{π /4}} - \frac{π}{4} - 1$ (4) $\frac{4}{e ^{π /4}} - \frac{π}{2} - 1$ 249. $If the solution curve of the differential equation (2x - 10 y^{3}) dy+ ydx= 0, passes through the points (0,1) and (2, β), then β is a root of the equation:$ (1) $y^{5} - 2y - 2 = 0$ (2) $2 y^{5} - 2y - 1 = 0$ (3) $2 y^{5} - y^{2} - 2 = 0$ (4) $y^{5} - y^{2} - 1 = 0$ 250. $Let the solution curve of the differential equation x d y = (x^{2} + y^{2} + y) d x, x > 0, intersect the line x = 1 at y = 0 and the line x = 2 at y = α . Then the value of α is :$ (1) $\frac{1}{2}$ (2) $\frac{3}{2}$ (3) $- \frac{3}{2}$ (4) $\frac{5}{2}$ 251. $Let x = x(y) be the solution of the differential equation 2 y e^{x / y^{2}} d x + (y^{2} - 4 x e^{x / y^{2}}) d y = 0 such that x(1) = 0. Then, x(e) is equal to$ (1) $e l o g_{e} (2)$ (2) $- e l o g_{e} (2)$ (3) $e^{2} l o g_{e} (2)$ (4) $- e^{2} l o g_{e} (2)$ 252. $Let y = y(x) be the solution of the differential equation cose c^{2} xdy + 2dx = (1 + y cos 2x) cose c^{2} xdx, with y (\frac{π}{4}) = 0 . Then, the value of (y (0) + 1)^{2} is equal to:$ (1) $e^{1/2}$ (2) $e^{- 1/2}$ (3) $e^{- 1}$ (4) $e$ 253. $A differential equation representing the family of parabolas with axis parallel to y -axis and whose length of latus rectum is the distance of the point (2, - 3) form the line 3 x + 4 y = 5, is given by :$ (1) $10 \frac{d ^{2} y}{d x ^{2}} = 11$ (2) $11 \frac{d ^{2} x}{d y ^{2}} = 10$ (3) $10 \frac{d ^{2} x}{d y ^{2}} = 11$ (4) $11 \frac{d ^{2} y}{d x ^{2}} = 10$ 254. $Let y = y(x) be the solution of the differential equation xdy = (y + x^{3} cos x) dx with y(π) = 0, then y (\frac{π}{2}) is equal to:$ (1) $\frac{π ^{2}}{4} + \frac{π}{2}$ (2) $\frac{π ^{2}}{2} + \frac{π}{4}$ (3) $\frac{π ^{2}}{2} - \frac{π}{4}$ (4) $\frac{π ^{2}}{4} - \frac{π}{2}$ 255. $Consider the family of all circles whose centers lie on the straight line y = x. If this family of circles is represented by the differential equation Py" + Qy' + 1 = 0, where P, Q are functions of x, y and y' (here y' = , y" = ), then which of the following statements is (are) true?$ (1) $P = y + x$ (2) $P = y - x$ (3) $P + Q = 1 - x + y + y' + (y')2$ (4) $P - Q = x + y - y' - (y')2$ 256. $Let y = y(x), y > 0, be a solution curve of the differential equation (1 + x^{2})dy = y(x - y)dx. If y(0)= 1 and y (22) = β then$ (1) $e^{3 β - 1} = e (3 + 22)$ (2) $e^{3 β - 1} = e (5 + 2)$ (3) $e^{β - 1} = e^{- 2} (3 + 22)$ (4) $e^{β - 1} = e^{- 2} (5 + 2)$ 257. $Let \frac{d y}{d x} = \frac{a x - b y + a}{b x + cy + a}, where a, b, c are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the point (11,6) from this circle is :$ (1) $10$ (2) $8$ (3) $7$ (4) $5$ 258. $If the solution curve of the differential equation \frac{d y}{d x} = \frac{x + y - 2}{x - y} passes through the point (2,1) and (k+ 1, 2), k>0, then$ (1) $2 tan^{- 1} (\frac{1}{k}) = lo g_{e} (k^{2} + 1)$ (2) $tan^{- 1} (\frac{1}{k}) = lo g_{e} (k^{2} + 1)$ (3) $2 tan^{- 1} (\frac{1}{k + 1}) = lo g_{e} (k^{2} + 2 k + 2)$ (4) $2 tan^{- 1} (\frac{1}{k}) = lo g_{e} (\frac{k ^{2} + 1}{k ^{2}})$ 259. $The differential equation of the family of circles passing through the points (0, 2) and (0, - 2) is$ (1) $2 x y \frac{d y}{d x} + (x^{2} - y^{2} + 4) = 0$ (2) $2 x y \frac{d y}{d x} + (x^{2} + y^{2} - 4) = 0$ (3) $2 x y \frac{d y}{d x} + (y^{2} - x^{2} + 4) = 0$ (4) $2 x y \frac{d y}{d x} - (x^{2} - y^{2} + 4) = 0$ 260. $Let us consider a curve, y = f(x) passing through the point (-2, 2) and the slope of the tangent to the curve at any point (x, f(x)) is given by f(x) + xf’(x) = x^{2} .Then :$ (1) $x^{2} + 2xf(x) - 12 = 0$ (2) $x^{3} + xf(x) + 12 = 0$ (3) $x^{3} - 3xf(x) - 4 = 0$ (4) $x^{2} + 2xf(x) + 4 = 0$ 261. $Let y = y(x) be a solution curve of the differential equation, (1- x^{2} y^{2})dx = ydx + xdy, If the line x = 1 intersects the curve y = y(x) at y = 2 and the line x = 2 intersects the curve y = y(x) at y = α, then a value of α is$ (1) $\frac{1 - 3 e ^{2}}{2 ( 3 e ^{2} + 1 )}$ (2) $\frac{1 + 3 e ^{2}}{2 ( 3 e ^{2} - 1 )}$ (3) $\frac{3 e ^{2}}{2 ( 3 e ^{2} - 1 )}$ (4) $\frac{3 e ^{2}}{2 ( 3 e ^{2} + 1 )}$ 262. $Let y = y_{1} (x) and y = y_{2} (x) be the solution curves the differential equation \frac{d y}{d x} = y + 7 with initial conditions y_{1} (0)= 0 and y_{2} (0)= 1 respectively. Then the curves y = y_{1} (x) and y = y_{2} (x) intersect at$ (1) $no point$ (2) $two points$ (3) $one point$ (4) $infinite number of points$ 263. $Let y = y(x) be the solution of the differential equation (x - x^{3}) dy = (y + y x^{2} - 3 x^{4}) dx, x > 2. If y(3) = 3, then y(4) is equal to:$ (1) $4$ (2) $12$ (3) $8$ (4) $16$ 264. $Let Ι be the purchase value of an equipment and V(t) be the value after it has been used for t years. The value V(t) depreciates at a rate given by differential equation \frac{d V ( t )}{d t} = - k(T - t), where k > 0 is a constant and T is the total life in years of the equipment. Then the scrap value V(T) of the equipment is :$ (1) $T2 - \frac{1}{k}$ (2) $Ι - \frac{k T ^{2}}{2}$ (3) $Ι - \frac{k ( T - t ) ^{2}}{2}$ (4) $e-kT$ 265. $The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by \frac{x ^{2}}{x y - x ^{2} y ^{2} - 1},If the curve passes through the point (1, 1), then e.y(e) is equal to$ (1) $\frac{1 - t a n ( 1 )}{1 + t a n ( 1 )}$ (2) $tan (1)$ (3) $1$ (4) $\frac{1 + t a n ( 1 )}{1 - t a n ( 1 )}$ 266. $Let y(x) be the solution of the differential equation 2 x^{2} dy + (e^{y} - 2x) dx = 0, x > 0. If y(e) = 1, then y(1) is equal to :$ (1) $0$ (2) $2$ (3) $l o g_{e} 2$ (4) $l o g_{e} (2e)$ 267. $Suppose y = y(x) be the solution curve to the differential equation \frac{d y}{d x} - y = 2 - e^{- x} such that lim_{x \to \infty} y_{(} x_{)} is finite. If a and b are respectively the x- and y- intercepts of the tangent to the curve at x=0, then the value of a-4b is equal to_________.$ 268. $Let y = y (x) be the solution of the differential equation dy = e^{ax + y} dx; a \in N . If y (log_{e} 2) = log_{e} 2 and y (0) = log_{e} (\frac{1}{2}), then the value of α is equal to -$ 269. $Let f : R \to R be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential equation \frac{d y}{d x} = (2 + 5y)(5y - 2), then the value of lim_{x \to -\infty} f(x) is _______ .$ 270. $Let the solution curve y = y(x) of the differential equation (4 + x^{2})dy - 2x(x^{2} + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to________.$