Differential Equations PYQs Set 5 (PDF)

Set 5 of 11 · Questions 121-150 of 326

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Differential Equations · Set 5 (Q121-150)
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121. $The solution of the differential equation x \frac{dy}{dx} + 2 y = x^{2} (x \neq = 0) with y (1) = 1, is :$ (1) $y = \frac{4}{5} x^{3} + \frac{1}{5 x ^{2}}$ (2) $y = \frac{x ^{3}}{5} + \frac{1}{5 x ^{2}}$ (3) $y = \frac{x ^{2}}{4} + \frac{3}{4 x ^{2}}$ (4) $y = \frac{3}{4} x^{2} + \frac{1}{4 x ^{2}}$ 122. $If y = y (x) is the solution of the differential equation, \frac{d y}{d x} + 2 y tan x = sin x, y (\frac{π}{3}) = 0 then the maximum value of the function y (x) over R is equal to:$ (1) $8$ (2) $\frac{1}{2}$ (3) $- \frac{15}{4}$ (4) $\frac{1}{8}$ 123. $The differential equation of the family of curves, x^{2} = 4 b (y + b), b \in R, is$ (1) $x (\frac{dy}{dx})^{2} = 2 y \frac{dy}{dx} + x^{2}$ (2) $x (\frac{dy}{dx})^{2} = 2 y \frac{dy}{dx} + x$ (3) $x (\frac{dy}{dx})^{2} = y \frac{dy}{dx} + x^{2}$ (4) $x (\frac{dy}{dx})^{2} = y \frac{dy}{dx} + 2 x^{2}$ 124. $If m and n are respectively the order and the degree of the differential equation representing the family of curves y^{2} - 5 ax - 5 a^{\frac{3}{2}} = 0 (a > 0 is a parameter), then the value of m - n is$ (1) $1$ (2) $-1$ (3) $2$ (4) $-2$ 125. $The solution of the differential equation ydx - xdy = xydx is$ (1) $x^{2} = c e^{x} y^{2}$ (2) $x = cy e^{x}$ (3) $xy = c e^{x}$ (4) $x^{2} y^{2} = c log x$ 126. $The general solution of the differential equation (y^{3} - x^{3}) dx - xydy = 0 (x > 0) is :$ (1) $y^{2} - 2 x^{2} + c x^{3} = 0$ (2) $y^{2} + 2 x^{3} + c x^{2} = 0$ (3) $y^{2} + 2 x^{2} + c x^{3} = 0$ (4) $y^{2} - 2 x^{3} + c x^{2} = 0$ 127. $If c is a parameter, then the differential equation of the family of curves x^{2} = c (y + c)^{2} is$ (1) $x (\frac{dy}{dx})^{3} + y (\frac{dy}{dx})^{2} - 1 = 0$ (2) $x (\frac{dy}{dx})^{3} - y (\frac{dy}{dx})^{2} + 1 = 0$ (3) $x (\frac{dy}{dx})^{3} + y (\frac{dy}{dx})^{2} + 1 = 0$ (4) $x (\frac{dy}{dx})^{3} - y (\frac{dy}{dx})^{2} - 1 = 0$ 128. $If \frac{dy}{dx} = tan^{2} (x + y), then sin 2 (x + y) =$ (1) $x - y + c$ (2) $2 (x - y) + c$ (3) $x + y + c$ (4) $2 (x + y) + c$ 129. $The differential equation y^{2} dx + (2 xy - 1) dy = 0 is$ (1) $linear in y$ (2) $not a linear equation$ (3) $a homogenous equation$ (4) $linear in x$ 130. $Let y = y (x) be the solution of the differential equation, (x^{2} + 1)^{2} \frac{dy}{dx} + 2 x (x^{2} + 1) y = 1 such that y (0) = 0 . If a y (1) = \frac{π}{32}, then the value of ’ a ’ is :$ (1) $\frac{1}{4}$ (2) $\frac{1}{2}$ (3) $1$ (4) $\frac{1}{16}$ 131. $A population grows at the rate of 10 % of the population per year. How long does it take for the population to double?$ (1) $5 log 2 years$ (2) $2 log 10 years$ (3) $20 log 2 years$ (4) $10 log 2 years$ 132. $The differential equation corresponding to the family of curves given by a x^{2} + b y^{2} = 1 where a and b are arbitrary constants is.$ (1) $x \frac{d ^{2} y}{d x ^{2}} = \frac{dy}{dx}$ (2) $xy \frac{d ^{2} y}{d x ^{2}} + x (\frac{dy}{dx})^{2} - y \frac{dy}{dx} = 0$ (3) $xy \frac{d ^{2} y}{d x ^{2}} + y (\frac{dy}{dx})^{2} - x \frac{dy}{dx} = 0$ (4) $xy \frac{d ^{2} y}{d x ^{2}} - x (\frac{dy}{dx})^{2} + y \frac{dy}{dx} = 0$ 133. $Let y = y (x) be the solution of the differential equation sec xdy + 2 (1-x) tan x+x (2-x) dx = 0 such that y (0) = 2 . Then y (2) is equal to :$ (1) $2$ (2) $2 {1 - sin (2)}$ (3) $2 {sin (2) + 1}$ (4) $1$ 134. $If ' m' and ' n' are the order and degree of the differential equation (y^{’ ’})^{5} + 4 \cdot \frac{( y ^{’ ’} ) ^{3}}{y ^{’ ’ ’}} + y^{’ ’ ’} = sin x, then$ (1) $m = 3, n = 5$ (2) $m = 3, n = 3$ (3) $m = 3, n = 1$ (4) $m = 3, n = 2$ 135. $Let y = y (x) be the solution of the differential equation secx \frac{dy}{dx} - 2 y = 2 + 3 sinx, x \in (- \frac{π}{2}, \frac{π}{2}), y (0) = - \frac{7}{4} . Then y (\frac{π}{6}) is equal to:$ (1) $- \frac{5}{4}$ (2) $- \frac{5}{2}$ (3) $- 33 - 7$ (4) $- 32 - 7$ 136. $If the differential equation obtained by eliminating A, B from y = (sin^{- 1} x)^{2} + A cos^{- 1} x + B is (a - x^{2}) y^{’ ’} - x y^{’} = b, then \frac{b + a}{b - a} =$ (1) $2$ (2) $-2$ (3) $3$ (4) $-3$ 137. $The solution of the differential equation 3 x y^{’} - 3 y + (x^{2} - y^{2})^{1/2} = 0, satisfying the condition y (1) = 1 is$ (1) $3 cos^{- 1} (\frac{y}{x}) = ln \| x \|$ (2) $3 cos (\frac{y}{x}) = ln \| x \|$ (3) $3 cos^{- 1} (\frac{y}{x}) = 2 ln \| x \|$ (4) $3 sin^{- 1} (\frac{y}{x}) = ln \| x \|$ 138. $The solution of \frac{dy}{dx} + \frac{1}{x} = \frac{e ^{y}}{x ^{2}} is$ (1) $2 x = (1 + C x^{2}) e^{y}$ (2) $x = (1 + C x^{2}) e^{y}$ (3) $2 x^{2} = (1 + C x^{2}) e^{- y}$ (4) $x^{2} = (1 + C x^{2}) e^{- y}$ 139. $The solution of the differential equation (1 + y^{2}) + (x - e^{tan^{- 1} y}) \frac{dy}{dx} = 0, is (C is an arbitrary constant)$ (1) $x e^{t a n^{- 1} y} = tan^{- 1} y + C$ (2) $x e^{2 t a n^{- 1} y} = e^{- t a n^{- 1} y} + C$ (3) $2 x e^{t a n^{- 1} y} = e^{2 t a n^{- 1} y} + C$ (4) $x^{2} e^{t a n^{- 1} y} = 4 e^{2 t a n^{- 1} y} + C$ 140. $Let y = y (x) be the solution of the differential equation x \frac{dy}{dx} - sin 2 y = x^{3} (2 - x^{3}) cos^{2} y, x \neq = 0 . If y (2) = 0, then tan (y (1)) is equal to$ (1) $- \frac{3}{4}$ (2) $\frac{3}{4}$ (3) $- \frac{7}{4}$ (4) $\frac{7}{4}$ 141. $The differential equation of y = e^{x} (a cos x + b sin x) is$ (1) $\frac{d ^{2} y}{d x ^{2}} - 2 \frac{dy}{dx} - y = 0$ (2) $\frac{d ^{2} y}{d x ^{2}} + 2 \frac{dy}{dx} + 2 y = 0$ (3) $\frac{d ^{2} y}{d x ^{2}} - 2 \frac{dy}{dx} + y = 0$ (4) $\frac{d ^{2} y}{d x ^{2}} - 2 \frac{dy}{dx} + 2 y = 0$ 142. $The solution of \frac{xdy}{dx} = y (ln y - ln x + 1) is$ (1) $x ln \frac{y}{x} = cy$ (2) $y ln \frac{x}{y} = cx$ (3) $ln \frac{x}{y} = cy$ (4) $ln \frac{y}{x} = cx$ 143. $If a body cools from 80^{\circ} C to 50^{\circ} C in the room temperature of 25^{\circ} C in 30 minutes, then the temperature of the body after 1 hour is$ (1) $31.36^{\circ} C$ (2) $32.25^{\circ} C$ (3) $36.36^{\circ} C$ (4) $33.25^{\circ} C$ 144. $The solution of the differential equation \frac{dy}{dx} - \frac{y + 3 x}{log _{e} ( y + 3 x )} + 3 = 0 is: (where C is a constant of integration).$ (1) $x - \frac{1}{2} (log_{e} (y + 3 x))^{2} = C$ (2) $y + 3 x - \frac{1}{2} (log_{e} x)^{2} = C$ (3) $x - 2 log_{e} (y + 3 x) = c$ (4) $x - log_{e} (y + 3 x) = c$ 145. $The curve that satisfies the differential equation xydy - (1 + y^{2}) dx = 0 passes through (1, 0) and intersects the curve x^{2} + 3 y^{2} = 3 at an angle θ . Then \frac{2 θ}{π} =$ (1) $2$ (2) $0$ (3) $4$ (4) $1$ 146. $If \frac{dy}{dx} + \frac{y}{x} = x^{2}, then 2 y (2) - y (1) =$ (1) $\frac{11}{4}$ (2) $\frac{9}{4}$ (3) $\frac{15}{4}$ (4) $\frac{13}{4}$ 147. $Let f (x) = x - 1 and g (x) = e^{x} for x \in R . If \frac{dy}{dx} = (e^{- 2 x} g (f (f (x))) - \frac{y}{x}), y (0) = 0, then y (1) is$ (1) $\frac{1 - e ^{3}}{e ^{4}}$ (2) $\frac{e - 1}{e ^{4}}$ (3) $\frac{1 - e ^{2}}{e ^{4}}$ (4) $\frac{2 e - 1}{e ^{3}}$ 148. $The general solution of sin y \cdot \frac{dy}{dx} = cos y (1 - xcos y) is$ (1) $sec y = x - 1 - c e^{x}$ (2) $sec y = x + 1 + c e^{x}$ (3) $sec y = x + e^{x} + c$ (4) $sec y = x - e^{x} + c$ 149. $Let y = y (x) be the solution of the differential equation (1 + y^{2}) e^{tan x} dx + cos^{2} x (1 + e^{2 tan x}) dy = 0, y (0) = 1 . Then y (\frac{π}{4}) is equal to$ (1) $\frac{2}{e}$ (2) $\frac{2}{e ^{2}}$ (3) $\frac{1}{e}$ (4) $\frac{1}{e ^{2}}$ 150. $If x = x (t) is the solution of the differential equation (t + 1) dx = (2 x + (t + 1)^{4}) dt, x (0) = 2, then, x (1) equals$