Differential Equations PYQs Set 10 (PDF)

Set 10 of 11 · Questions 271-300 of 326

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Differential Equations · Set 10 (Q271-300)
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271. $Let y = y(x) be the solution of the differential equation (x + 1)y' - y = e^{3 x} (x + 1)^{2}, with y(0) = \frac{1}{3} . Then, the point x = - \frac{4}{3} for the curve y = y(x) is:$ (1) $not a critical point$ (2) $a point of local minima$ (3) $a point of local maxima$ (4) $a point of inflection$ 272. $Let y = y(x) be the solution of the differential equation x(1 - x^{2}) \frac{d y}{d x} + (3 x^{2} y - y - 4 x^{3}) = 0,x > 1, with y(2) = -2. Then y(3) is equal to$ (1) $-18$ (2) $-12$ (3) $-6$ (4) $-3$ 273. $Let y(x) be the solution of the differential equal (x log x) \frac{d y}{d x} + y = 2x log x, (x \geq 1). Then y(e) is equal to$ (1) $e$ (2) $0$ (3) $2$ (4) $2e$ 274. $Let f : R \to R and g : R \to R be two non-constant differentiable functions. If f'(x) = (e^{(f (x) - g (x)}) g'(x) for all x \in R, and f(a) = g(b) = 1, then which of the following statement(s) is (are) TRUE?$ (1) $f(b) < 1 - l o g_{e} 2$ (2) $f(b) > 1 - l o g_{e} 2$ (3) $g(a) > 1 - l o g_{e} 2$ (4) $g(a) < 1 - l o g_{e} 2$ 275. $The differential equation of all parabolas whose axis is y -axis is$ (1) $x \frac{d ^{2} y}{d x ^{2}} - \frac{dy}{dx} = 0$ (2) $x \frac{d ^{2} y}{d x ^{2}} + \frac{dy}{dx} = 0$ (3) $\frac{d ^{2} y}{d x ^{2}} - y = 0$ (4) $\frac{d ^{2} y}{d x ^{2}} - \frac{dy}{dx} = 0$ 276. $Suppose the solution of the differential equation \frac{dy}{dx} = \frac{( 2 + α ) x - βy + 2}{βx - 2 αy - ( βγ - 4 α )} represents a circle passing through origin. Then the radius of this circle is :$ (1) $2$ (2) $17$ (3) $\frac{1}{2}$ (4) $\frac{17}{2}$ 277. $Let x = x(y) be the solution of the differential equation 2(y + 2) l o g_{e} (y + 2)dx +(x + 4-2 l o g_{e} (y + 2))dy = 0, y > -1 with x(e^{4} - 2)= 1. Then x(e^{9} - 2) is equal to$ (1) $3$ (2) $\frac{4}{9}$ (3) $\frac{32}{9}$ (4) $\frac{10}{3}$ 278. $Let f : [0, 1] \to R be such that f(xy) = f(x).f(y), for all x, y \in [0, 1], and f(0) \neq 0. if y = y(x) satisfies the differential equation, \frac{d y}{d x} = f(x) with y(0) = 1, then y is equal to :$ (1) $5$ (2) $2$ (3) $3$ (4) $4$ 279. $If y = y(x) is the solution of the differential equation 2 x^{2} \frac{d y}{d x} - 2 x y + 3 y^{2} = 0 such that y (e) = \frac{e}{3}, then y(1) is equal to$ (1) $\frac{1}{3}$ (2) $\frac{2}{3}$ (3) $\frac{3}{2}$ (4) $3$ 280. $If \frac{d y}{d x} + 2y tan x = sin x, 0 < x < \frac{π}{2} and y (\frac{π}{3}) =0, then the maximum value of y(x) is$ (1) $\frac{1}{8}$ (2) $\frac{3}{4}$ (3) $\frac{1}{4}$ (4) $\frac{3}{8}$ 281. $Let y = y(x) be the solution of the differential equation \frac{d y}{d x} + \frac{5}{x ( x ^{5} + 1 )} y = \frac{( x ^{5} + 1 ) ^{2}}{x ^{7}}, x > 0 . If y(a)= 2, then y(b) is equal to$ (1) $\frac{637}{128}$ (2) $\frac{679}{128}$ (3) $\frac{693}{128}$ (4) $\frac{697}{128}$ 282. $If the solution of the differential equation \frac{d y}{d x} + e^{x} (x^{2} - 2) y = (x^{2} - 2 x) (x^{2} - 2) e^{2 x} satisfies y(0) = 0, then the value of y(2) is_______.$ (1) $-1$ (2) $1$ (3) $0$ (4) $e$ 283. $Let y = y (x) be the solution of the differential equation, x y^{'} - y = x^{2} (x cos x + sin x), x > 0 . If y (π) = π, then y^{''} (\frac{π}{2}) + y (\frac{π}{2}) is equal to:$ (1) $1 + \frac{π}{2}$ (2) $1 + \frac{π}{2} + \frac{x ^{2}}{4}$ (3) $2 + \frac{π}{2}$ (4) $2 + \frac{π}{2} + \frac{π ^{2}}{4}$ 284. $The order and degree of the differential equation y = \frac{d p}{d x} x + a^{2} p^{2} + b^{2} where p = \frac{d y}{d x} (here a and b are arbitrary constants) respectively are$ (1) $1,1$ (2) $2,2$ (3) $2,1$ (4) $1,2$ 285. $Let y = f (x) be the solution of the differential equation (x + 1) y d x - x^{2} d y = 0, y (1) = e . Then lim_{x \to 0^{+}} f (x) is equal to$ (1) $0$ (2) $\frac{1}{e}$ (3) $e^{2}$ (4) $1 e^{2}$ 286. $Let y = y (x) be the solution of the differential equation x lo g_{x} x \frac{d y}{d x} + y = x^{2} lo g_{x} x, (x, 1) . If y (2) = 2, then y (e) is equal to$ (1) $\frac{4 + e ^{2}}{4}$ (2) $\frac{1 + e ^{2}}{4}$ (3) $\frac{2 + e ^{2}}{2}$ (4) $\frac{1 + e ^{2}}{2}$ 287. $Statement (I) : The elimination of arbitrary constants α, β and γ from y = (α + β + γ) x results in a differential equation of order three. Statement (II): The elimination of arbitrary constants α, β and γ from y = α x + β sin x + γ e^{x} results in a differential equation of order three.$ (1) $I is true and II is false$ (2) $I is false and II is false$ (3) $I is true and II is true$ (4) $I is false and II is true$ 288. $Let f : [1, \infty) \to R be a differentiable function. If 6 \int_{1}^{x} f (t) d t = 3 x f (x) + x^{3} - 4 for all x \geq 1, then the value of f (2) - f (3) is$ (1) $-4$ (2) $3$ (3) $4$ (4) $-3$ 289. $If a curve y = f (x) passes through the point (1, 2) and satisfies x \frac{d y}{d x} + y = b x^{4}, then for what value of b, \int_{1}^{2} f (x) d x = \frac{62}{5} ?$ (1) $5$ (2) $\frac{62}{5}$ (3) $\frac{31}{5}$ (4) $10$ 290. $Let g be a differentiable function such that \int_{0}^{x} g (t) d t = x - \int_{0}^{x} t g (t) d t, x \geq 0 and let y = y (x) satisfy the differential equation \frac{d y}{d x} - y tan x = 2 (x + 1) sec x g (x), x \in (0, \frac{π}{2}) . If y (0) = 0, then y (\frac{π}{3}) is equal to$ (1) $\frac{2 π}{3 3}$ (2) $\frac{4 π}{3}$ (3) $\frac{2 π}{3}$ (4) $\frac{4 π}{3 3}$ 291. $If y = y (x) is the solution curve of the differential equation \frac{d y}{d x} + y tan x = x sec x, 0 \leq x \leq \frac{π}{3}$ (1) $\frac{π}{12} - \frac{3}{2} lo g_{e} (\frac{2}{e 3})$ (2) $\frac{π}{12} + \frac{3}{2} lo g_{e} (\frac{2 3}{e})$ (3) $\frac{π}{12} - \frac{3}{2} lo g_{e} (\frac{2 3}{e})$ (4) $\frac{π}{12} + \frac{3}{2} lo g_{e} (\frac{2}{e 3})$ 292. $Let y = y (x) be the solution of the differential equation cos x (3 sin x + cos x + 3) dy = (1 + y sin x (3 sin x + cos x + 3)) dx 0 \leq x \leq \frac{π}{2}, y (0) = 0 . Then, y (\frac{π}{3}) is equal to :$ (1) $2 log_{e} (\frac{2 3 + 9}{6})$ (2) $2 log_{e} (\frac{2 3 + 10}{11})$ (3) $2 log_{e} (\frac{3 + 7}{2})$ (4) $2 lo g_{e} (\frac{3 3 - 8}{4})$ 293. $Let a curve y = f(x) pass through the point (2, (lo g_{e} 2)^{2}), and have slope \frac{2 y}{x l o g _{e} x} for all positive real value of x. Then the value of f(e) is equal to ___$ 294. $Let y = y (x) be the solution of the differential equation \frac{d y}{d x} + \frac{2 x}{( 1 + x ^{2} ) ^{2}} y = x e^{\frac{1}{( 1 + x ^{2} )}}; y (0) = 0 . Then the area enclosed by the curve f (x) = y (x) e^{- \frac{1}{( 1 + x ^{2} )}} and the line y - x = 4 is______$ 295. $If the solution curve of the differential equation (y - 2 l o g_{e} x)dx + (x l o g_{e} x^{2})dy = 0, x > 1 passes through the points (e, \frac{4}{3}) and (e^{4}, α), then a is equal to$ 296. $Let y'(x) + y(x) g'(x) = g(x) g'(x), y(0) = 0, x \in R , where f'(x) denotes \frac{df ( x )}{d x} and g(x) is a given non-constant differentiable function on R with g(0) = g(b) = 0. Then the value of y(b) is$ 297. $Let f be a differentiable function such that 2 (x + 2)^{2} f (x) - 3 (x + 2)^{2} = 10 \int_{0}^{x} (t + 2) f (t) d t, x \geq 0 . Then f (2) is equal to .$ 298. $Let f : R \to R be a differentiable function with f(0) = 1 and satisfying the equation f(x + y) = f(x) f '(y) + f '(x)f(y) for all x, y \in R. Then, the value of lo g_{e} (f (4)) is ______$ 299. $If for x \geq 0, y = y (x) is the solution of the differential equation (1 + x) \frac{d y}{d x} = (1 + x)^{2} + (y - 3), if y (2) = 0 then y (3) = ?$ 300. $If the curve y = y (x) represented by the solution of the differential equation (2 x y^{2} - y) d x + x d y = 0, passes through the intersection of the lines, 2 x - 3 y = 1 and 3 x + 2 y = 8, then ∣ y (1) ∣ is equal to$