Differential Equations PYQs Set 11 (PDF)

Set 11 of 11 · Questions 301-326 of 326

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Differential Equations · Set 11 (Q301-326)
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301. $Let y = y (x) be the solution of the differential equation cos x (lo g_{e} (cos x))^{2} d y + (sin x - 3 y sin x lo g_{e} (cos x)) d x = 0, x \in (0, \frac{π}{2}) . If y (\frac{π}{4}) = \frac{- 1}{l o g _{e} 2}, then y (\frac{π}{6}) is equal to:$ (1) $\frac{1}{l o g _{e} ( 3 ) - l o g _{e} ( 4 )}$ (2) $\frac{2}{l o g _{e} ( 3 ) - l o g _{e} ( 4 )}$ (3) $\frac{1}{l o g _{e} ( 4 ) - l o g _{e} ( 3 )}$ (4) $- \frac{1}{l o g _{e} ( 4 )}$ 302. $Let y = y (x) be the solution of the differential equation, x \frac{dy}{dx} + y = x log_{e} x, (x < 1) . If 2 y (2) = log_{e} 4 - 1, then y (e) is equal to :$ (1) $- \frac{e}{2}$ (2) $- \frac{e ^{2}}{2}$ (3) $\frac{e}{4}$ (4) $\frac{e ^{2}}{4}$ 303. $Solution of e^{\frac{d y}{d x}} = x when x = 1 and y = 0 is$ (1) $y = x (log x - 1) + 1$ (2) $y = x (log x - 1) + 4$ (3) $y = x (log x - 1) + 3$ (4) $y = x (log x + 1) + 1$ 304. $Statement I : The differential equation corresponding to the family of circles having their centres on Y-axis and fixed radius k is (x^{2} - k^{2}) (\frac{dy}{dx})^{2} + x^{2} = 0 Statement II : The differential equation of corresponding to the family of circles passing through the origin and having their centres on X -axis is x^{2} - y^{2} + 2 xy \frac{dy}{dx} = 0 Which of the above statements is (are) true?$ (1) $Statement I is true, but Statement II is false$ (2) $Statement II is true, but Statement I is false$ (3) $Both Statement I and Statement II are true$ (4) $Both Statement I and Statement II are false$ 305. $y = ax + b is$ (1) $General solution of \frac{d ^{3} y}{d x ^{3}} = 0$ (2) $General solution for \frac{dy}{dx} = a + b$ (3) $General solution for both \frac{d ^{2} y}{d x ^{2}} = 0 and \frac{d ^{3} y}{d x ^{3}} = 0$ (4) $General solution for \frac{d ^{2} y}{d x ^{2}} = 0$ 306. $Let y = y_{1} (x) and y = y_{2} (x) be two distinct solutions of the differential equation \frac{d y}{d x} = x + y, with y_{1} (0) = 0 and y_{2} (0) = 1 respectively. Then, the number of points of intersection of y = y_{1} (x) and y = y_{2} (x) is$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 307. $The general solution of the differential equation (x - y^{2})dx + y(5x + y^{2})dy = 0 is :$ (1) $(y^{2} + x)^{4} = C ∣ (y^{2} + 2 x)^{3} ∣$ (2) $(y^{2} + 2 x)^{4} = C\| (y^{2} + x)^{3} \|$ (3) $\| (y^{2} + x)^{3} \| = C (2 y^{2} + x)^{4}$ (4) $\| (y^{2} + 2 x)^{3} \| = C (2 y^{2} + x)^{4}$ 308. $If \frac{d y}{d x} = y + 3 > 0 and y(0) = 2, then y (ln 2) is equal to :$ (1) $7$ (2) $5$ (3) $13$ (4) $-2$ 309. $If \frac{d y}{d x} = \frac{2 ^{x + y} - 2 ^{x}}{2 ^{y}}, y (0) = 1, then y(1) is equal to :$ (1) $l o g_{2} (2 + e)$ (2) $l o g_{2} (1 + e)$ (3) $log2 (2e)$ (4) $l o g_{2} (1 + e2)$ 310. $If x = x(y) is the solution of the differential equation y \frac{d x}{d y} = 2x + y^{3} (y + 1) e^{y}, x(1) = 0; then x(e) is equal to :$ (1) $e^{3} (e^{e} - 1)$ (2) $e^{e} (e^{3} -1)$ (3) $e^{2} (e^{e} + 1)$ (4) $e^{e} (e^{2} -1)$ 311. $Let the solution curve of the differential equation x \frac{d y}{d x} - y = y^{2} + 16 x^{2}, y (1) = 3 be y = y_{(} x_{)} . Then y(2) is equal to :$ (1) $15$ (2) $11$ (3) $13$ (4) $17$ 312. $The general solution of (1 + e^{\frac{x}{y}}) dx + e^{\frac{x}{y}} (1 - \frac{x}{y}) dy = 0$ (1) $y e^{\frac{y}{x}} + x = c$ (2) $y e^{\frac{x}{y}} - x = c$ (3) $y e^{\frac{x}{y}} + y = c$ (4) $y e^{\frac{x}{y}} + x = c$ 313. $The general solution of the differential equation \frac{dy}{dx} = \frac{2 x - 3 y + 5}{6 x - 9 y + 7} is$ (1) $x - 3 y + \frac{22}{3} lo g ∣3 x - 7∣ + c = 0$ (2) $x - 3 y + \frac{8}{3} lo g ∣6 x - 9 y - 1∣ + c = 0$ (3) $3 x - 3 y + \frac{8}{3} lo g ∣3 x - 9 y + 1∣ + c = 0$ (4) $3 x - 2 y + \frac{22}{3} lo g ∣2 x - 3 y - 7∣ + c = 0$ 314. $If the general solution of \frac{dy}{dx} = \frac{y ^{2}}{xy - y ^{2} - x ^{2}} is tan^{- 1} (\frac{y}{x}) = f (y) + C, then f (e^{3}) =$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 315. $The solution of the differential equation \frac{dy}{dx} = (4 x + y + 1)^{2}, when y (0) = 1 is$ (1) $y = 2 x^{2} - 1 - \frac{π}{8}$ (2) $y = 4 x - (1 + \frac{π}{8})$ (3) $y = 2 tan (2 x + \frac{π}{4}) - 4 x - 1$ (4) $y = 2 tan (x + \frac{π}{8}) + 4 x - 1$ 316. $The solution of the differential equation x \frac{dy}{dx} - y = 3 represents a family of$ (1) $straight lines$ (2) $circles$ (3) $parabolas$ (4) $ellipses$ 317. $If the family of curves y = a e^{4 x} + b e^{- x}, where a, b are arbitrary constants represents the general solution of the differential equation f (x, y \frac{dy}{dx}, \frac{d ^{2} y}{d x ^{2}}) = 0, then \frac{df}{dx} =$ (1) $\frac{d ^{2} y}{d x ^{2}} - 3 \frac{dy}{dx} - 4 y$ (2) $\frac{d ^{3} y}{d x ^{3}} - 3 \frac{d ^{2} y}{d x ^{2}} - 4 \frac{dy}{dx}$ (3) $\frac{d ^{3} y}{d x ^{3}} - \frac{d ^{2} y}{d x ^{2}} - 3 \frac{dy}{dx} + 2$ (4) $\frac{d ^{3} y}{d x ^{3}} - \frac{d ^{2} y}{d x ^{2}} + 3$ 318. $Find the solution of the following differential equation (xcos (\frac{y}{x}) + ysin (\frac{y}{x})) ydx = (ysin (\frac{y}{x}) - xcos (\frac{y}{x})) xdy$ (1) $ycos (\frac{x}{y}) = \pm e^{- c}$ (2) $xcos (\frac{y}{x}) = \pm e^{- c}$ (3) $xycos (\frac{y}{x}) = \pm e^{- c}$ (4) $xysin (\frac{y}{x}) = \pm e^{- c}$ 319. $The solution of \frac{dy}{dx} = \frac{x + y}{x - y} is$ (1) $tan^{- 1} (\frac{y}{x}) = log x^{2} + y^{2} + C$ (2) $tan^{- 1} (\frac{y}{x}) = log x^{2} - y^{2} + C$ (3) $sin^{- 1} (\frac{y}{x}) = log x^{2} + y^{2} + C$ (4) $cos^{- 1} (\frac{y}{x}) = log x^{2} - y^{2} + C$ 320. $The differential equaiton of the simple harmonic motion given by x = Acos (nt + α) is$ (1) $\frac{d ^{2} x}{d t ^{2}} - n^{2} x = 0$ (2) $\frac{d ^{2} x}{d t ^{2}} + n^{2} x = 0$ (3) $\frac{dx}{dt} - \frac{d ^{2} x}{d t ^{2}} = 0$ (4) $\frac{d ^{2} x}{d t ^{2}} - \frac{dx}{dt} + nx = 0$ 321. $The general solution of the differential equation (x - 2 y + 1) d y - (3 x - 6 y + 2) d x = 0 is$ (1) $(x + 2 y + \frac{3}{5})^{2/25} \cdot e^{115 (x + 2 y)} = C$ (2) $(x - 2 y + \frac{3}{5})^{2/25} . e^{1/5 (x - 2 y)} = C$ (3) $(x - 2 y + \frac{3}{5})^{2/25} \cdot e^{1/5 (6 x - 2 y)} = C$ (4) $(x - 2 y + \frac{1}{5})^{2/25} \cdot e^{1/5 (x - 2 y)} = C$ 322. $Match the differential equations in List-I to their integrating factors in List-II. List-I (Differential equation) (P) (x^{3} + 1) \frac{dy}{dx} + x^{2} y = 3 x^{2} (Q) x^{2} \frac{dy}{dx} + 3 xy = x^{6} (R) (x^{3} + 1)^{2} \frac{dy}{dx} + 6 x^{2} (x^{3} + 1) y = x^{2} (S) (x^{2} + 1) \frac{dy}{dx} + 4 xy = ln x List-II (Integrating factor) (1) x^{3} (2) (x^{3} + 1)^{2} (3) (x^{2} + 1)^{2} (4) x^{2} + 1 (5) (x^{3} + 1)^{1/3} (6) (x^{3} + 1)^{1/2} The correct answer is P Q R S$ (1) $4 1 2 3$ (2) $5 1 2 3$ (3) $5 2 3 6$ (4) $5 1 3 4$ 323. $Let f be a twice differentiable non-negative function such that (f (x))^{2} = 25 + \sum ((f (t))^{2} + (f^{’} (t))^{2}) dt . Then the mean of f (log_{e} (1)), f (log_{e} (2)), \dots \dots, f (log_{e} (625)) is equal to____.$ 324. $Let y = y (x) be the solution of the differential equation (x + y + 2)^{2} dx = dy, y (0) = - 2 . Let the maximum and minimum values of the function y = y (x) in (0, \frac{π}{3}) be α and β, respectively. If (3 α + π)^{2} + β^{2} = γ + δ 3, γ, δ \in Z, then γ + δ equals________$ 325. $Let f be a real-valued differentiable function on R (the set of all real numbers) such that f (1) = 1 . If the y-intercept of the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P, then the value of f (- 3) is equal to$ 326. $Let y = y(x) be the solution of the differential equation \frac{d y}{d x} = \frac{4 y ^{3} + 2 y x ^{2}}{3 x y ^{2} + x ^{2}},y(l) = 1. If for some n \in N, y_{(} 2_{)} \in [n - 1, n), then n is equal to ________$