Differential Equations PYQs Set 1 (PDF)

Set 1 of 11 · Questions 1-30 of 326

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Differential Equations · Set 1 (Q1-30)
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1. $The general solution of the differential equation 1 - x^{2} y^{2} \cdot dx = y \cdot dx + x \cdot dy is$ (1) $sin (xy) = x + c$ (2) $sin^{- 1} (xy) + x = c$ (3) $sin (x + c) = xy$ (4) $sin (xy) + x = c$ 2. $The general solution of the differential equation 1 + x^{2} + y^{2} + x^{2} y^{2} + xy \frac{dy}{dx} = 0 is : (where C is a constant of integration).$ (1) $1 + y^{2} + 1 + x^{2} = \frac{1}{2} log_{e} (\frac{1 + x ^{2} + 1}{1 + x ^{2} - 1}) + C$ (2) $1 + y^{2} + 1 + x^{2} = \frac{1}{2} log_{e} (\frac{1 + x ^{2} - 1}{1 + x ^{2} + 1}) + C$ (3) $1 + y^{2} - 1 + x^{2} = \frac{1}{2} log_{e} (\frac{1 + x ^{2} - 1}{1 + x ^{2} + 1}) + C$ (4) $1 + y^{2} - 1 + x^{2} = \frac{1}{2} log_{e} (\frac{1 + x ^{2} + 1}{1 + x ^{2} - 1}) + c$ 3. $The general solution of the differential equation \frac{dy}{dx} + \frac{y}{x} = x^{2} is$ (1) $xy = \frac{x ^{2}}{2} + c$ (2) $xy = \frac{x ^{3}}{3} + c$ (3) $xy = \frac{x ^{4}}{4} + c$ (4) $xy = \frac{x ^{5}}{5} + c$ 4. $The general solution of x \frac{dy}{dx} = y - x tan (\frac{y}{x}) is$ (1) $x^{2} sin (\frac{x}{y}) = c$ (2) $x sin (\frac{x}{y}) = c$ (3) $x sin (\frac{y}{x}) = c$ (4) $x^{2} sin (\frac{y}{x}) = c$ 5. $a, b, c, d are arbitrary constants. Then the corresponding differential equation to y = a e^{x} + b e^{- x} + ccos x + dsin x is$ (1) $y^{(4)} = y$ (2) $y^{(4)} + y = 0$ (3) $y^{(4)} - y^{(2)} + 1 = 0$ (4) $y^{(4)} + 2 y^{(2)} + 1 = 0$ 6. $A gardener is digging a plot of land. As he gets tired, he works more slowly. After ' t' minutes he is digging at a rate of \frac{2}{t} square metres per minute. How long will it take him to dig an area of 40 square metres?$ (1) $100 minutes$ (2) $10 minutes$ (3) $30 minutes$ (4) $40 minutes$ 7. $The differential equation satisfied by the system of parabolas y^{2} = 4 a (x + a) is:$ (1) $y (\frac{dy}{dx})^{2} - 2 x (\frac{dy}{dx}) - y = 0$ (2) $y (\frac{dy}{dx})^{2} - 2 x (\frac{dy}{dx}) + y = 0$ (3) $y (\frac{dy}{dx})^{2} + 2 x (\frac{dy}{dx}) - y = 0$ (4) $y (\frac{dy}{dx}) + 2 x (\frac{dy}{dx}) - y = 0$ 8. $The order of the differential equation of all circles whose radius is 4 , is$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 9. $The general solution of the differential equation (1 + y^{2}) dx = (tan^{- 1} y - x) dy is$ (1) $2 x e^{t a n^{- 1} y} = e^{2 t a n^{- 1} y} + c$ (2) $xy + tan^{- 1} y = c$ (3) $2 tan^{- 1} y = y^{2} - x + c$ (4) $x e^{t a n^{- 1} y} = e^{t a n^{- 1} y} (tan^{- 1} y - 1) + c$ 10. $The sum of the degree and order of the differential equation (1 + y_{1}^{2})^{2/3} = y_{2} is$ (1) $4$ (2) $5$ (3) $6$ (4) $7$ 11. $Let y = y (x) be the solution of the differential equation x^{4} dy + (4 x^{3} y + 2 sinx) dx = 0, x > 0, y (\frac{π}{2}) = 0 . Then π^{4} y (\frac{π}{3}) is equal to:$ (1) $72$ (2) $92$ (3) $64$ (4) $81$ 12. $Find the sum of the order and degree of the differential equation y = x (\frac{dy}{dx})^{3} + \frac{d ^{2} y}{d x ^{2}}$ (1) $2$ (2) $3$ (3) $4$ (4) $5$ 13. $If the length of the sub tangent at any point p (x, y) on a curve f (x, y) = 0 is x + 7 y^{2}, then f (x, y) =$ (1) $xy + cy - 7 x$ (2) $\frac{x}{y} + 7 x - c$ (3) $7 y^{2} + cy - x$ (4) $7 xy + cy - x$ 14. $The family of curves represented by the general solution of y^{’} = \frac{y}{2 x} contains$ (1) $circles$ (2) $ellipses$ (3) $hyperbolas$ (4) $parabolas$ 15. $The solution of \frac{d ^{2} y}{d x ^{2}} = 0 represents$ (1) $straight lines$ (2) $a circle$ (3) $a parabola$ (4) $a point$ 16. $The general solution of the differential equation of all circles having centre at A (- 1, 2) is$ (1) $x^{2} + y^{2} + x - 2 y + c = 0$ (2) $x^{2} + y^{2} - 2 x + 4 y + c = 0$ (3) $x^{2} + y^{2} - x + 2 y + c = 0$ (4) $x^{2} + y^{2} + 2 x - 4 y + c = 0$ 17. $The solution curve, of the differential equation 2 y \frac{d y}{d x} + 3 = 5 \frac{d y}{d x}, passing through the point (0, 1) is a conic, whose vertex lies on the line:$ (1) $2 x + 3 y = 9$ (2) $2 x + 3 y = - 9$ (3) $2 x + 3 y = - 6$ (4) $2 x + 3 y = 6$ 18. $Let y = y (x) be the solution of the differential equation (x^{2} - 3 y^{2}) dx + 3 xy dy = 0, y (1) = 1 . The 6 y^{2} (e) is equal to$ (1) $3 e^{2}$ (2) $e^{2}$ (3) $2 e^{2}$ (4) $\frac{3 e ^{2}}{2}$ 19. $The solution of the differential equation 2 x (\frac{dy}{dx}) - y = 4 represents a family of$ (1) $ellipse$ (2) $parabola$ (3) $straight line$ (4) $circle$ 20. $If a and b are respectively the order and degree of a differential equation y^{2} (y^{’ ’})^{2} + 3 x (y^{’})^{1/3} + x^{2} y^{2} = sin x, then$ (1) $b = a$ (2) $a = 3 b$ (3) $b = 3 a$ (4) $ab = 6$ 21. $The degree of the differential equation (1 + (\frac{dy}{dx})^{2})^{2} = \frac{d ^{2} y}{d x ^{2}} is$ (1) $1$ (2) $4$ (3) $2$ (4) $3$ 22. $The solution of (1 + y^{2}) dx - xydy = 0, y (1) = 0 represents a conic. Its eccentricity is$ (1) $2$ (2) $1/ e$ (3) $1$ (4) $2$ 23. $The curve passing through the point (1, 2) given that the slope of the tangent at any point (x, y) is 2 x / y represents$ (1) $Parabola$ (2) $Ellipse$ (3) $Hyperbola$ (4) $Circle$ 24. $The differential equation y \frac{dy}{dx} + x = c represents$ (1) $a family of parabolas$ (2) $a family of circles whose centres are on the x -axis$ (3) $a family of hyperbolas$ (4) $a family of circles whose centres are on the y -axis$ 25. $The substitution \frac{dy}{dx} = z, reduces the differential equation \frac{d ^{2} y}{d x ^{2}} - \frac{dy}{dx} = 0 to a differential equation whose solution is z =$ (1) $log x + C$ (2) $x + C$ (3) $Ae^{x}$ (4) $x^{2} + C$ 26. $If the slope of the tangent at a point (x, y) on a curve is \frac{y - 4}{x - 3} and the curve passes through (4, 3), then the point where it cuts the line y = x is$ (1) $(1, 1)$ (2) $(3, 3)$ (3) $(\frac{7}{2}, \frac{7}{2})$ (4) $(- \frac{5}{2}, - \frac{5}{2})$ 27. $The general solution of the differential equation x^{2} ydx - (x^{3} + y^{3}) dy = 0 is$ (1) $y^{3} = 3 x^{3} log (cx)$ (2) $c (x^{3} - y^{3}) = x^{2}$ (3) $log \| y \| - \frac{x ^{3}}{3 y ^{3}} = 0$ (4) $y^{2} - x^{2} = c^{2} (y^{2} - x^{2})$ 28. $Let y = y (x) be the solution of the differential equation 2 co s x \frac{d y}{d x} = si n 2 x - 4 y sin x, x (0, \frac{π}{2}) . If y (\frac{π}{3}) = 0, then y^{’} (\frac{π}{4}) + y (\frac{π}{4}) is equal to .$ 29. $If the solution y (x) of the given differential equation (e^{y} + 1) cos x d x + e^{y} sin x d y = 0 passes through the point (\frac{π}{2}, 0), then the value of e^{y (\frac{π}{6})} is equal to_______$ 30. $Let α \| x \| = \| y \| e^{xy - β}, α, β \in N be the solution of the differential equation x d y - y d x + xy (x d y + y d x) = 0, y (1) = 2 . Then α + β is equal to_______$