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Question Bank No: 1

1. The orthogonal trajectories of the family of curves $a^{(n - 1)}$ y = $x^{n}$ are given by

a)		$x^{n}$ + $n^{2}$ y = constant
b)		n $y^{2}$ + $x^{2}$ = constant
c)		$n^{2}$ x + $y^{n}$ = constant
d)		$n^{2}$ x $-$ $y^{2}$ = constant

2. The slope of a curve at any point is the reciprocal of twice the ordinate at the point and and it passes through the point (2, 3). The equation of curve is

a)		$x^{2}$ = x + 7
b)		$x^{2}$ = y $-$ 7
c)		$y^{2}$ = x $-$ 7
d)		$y^{2}$ = x + 7

3. Equation of the curve whose subnormal is constant is

a)		y = ax + b
b)		$y^{2}$ = 2ax + b
c)		c $y^{2}$ $-$ $x^{2}$ = a
d)		None of these

4. The curves for which the length of the normal is equal to the length of radius vector, are

a)		only circles
b)		either circles or rectangular hyperbolas
c)		only rectangular hyperbolas
d)		only parabolas

5. The differential equation y $\frac{dy}{dx}$ + x = a (where ‘a’ is a constant) represents

a)		a set of circles having centre on y-axis
b)		a set of circles with centre on x-axis
c)		a set of ellipse
d)		None of these

6. The curve for which slope of the tangent at any point equals the ratio of the abscissa to the ordinate of the point is a/an

a)		ellipse
b)		rectangular hyperbola
c)		circle
d)		parabola

7. The equation of the curve whose subnormal is twice the abscissa is a/an

a)		circle
b)		parabola
c)		hyperbola
d)		ellipse

8. The particular solution of log $\frac{dy}{dx}$ = 3x + 4y, y(0) = 0 is

a)		$e^{3 x}$ + 3 $e^{- 4 y}$ = 7
b)		4 $e^{3 x}$ $-$ $e^{- 4 y}$ = 4
c)		3 $e^{3 x}$ + 4 $e^{4 y x}$ = 3
d)		4 $e^{3 x}$ + 3 $e^{- 4 y}$ = 7

9. The equation of the curve passing through $(2, \frac{7}{2})$ and having slope 1 $- \frac{1}{x^{2}}$ at (x,y) is

a)		y = $x^{2}$ + x + 1
b)		xy = x + 1
c)		xy = $x^{2}$ + x + 1
d)		xy = y + 1

10. The equation of curve, whose slope at any point different from origin is y + $\frac{y}{x}$ is

a)		xy = $e^{x}$
b)		y = cx $e^{x}$ (c $\neq$ 0)
c)		y = x $e^{x}$
d)		y + x $e^{x}$ = c

11. Solution of $\frac{dy}{dx}$ + $\sqrt{\frac{1 - y^{2}}{1 - x^{2}}}$ = 0 is

a)		$\sin^{- 1}$ x $-$ $\sin^{- 1}$ y = c
b)		$\sin^{- 1}$ x + $\sin^{- 1}$ y = c
c)		$\sin^{- 1}$ x = c $\sin^{- 1}$ y
d)		( $\sin^{- 1}$ x) ( $\sin^{- 1}$ y) = c

12. The solution of $\frac{dy}{dx}$ = $\frac{x + 2 y}{x}$ is

a)		x + y = c
b)		$x^{2}$ + $y^{2}$ = c
c)		x + y = c $x^{2}$
d)		x + y = cx

13. The complete solution of the differential equation $\frac{dy}{dx}$ = 2x + 4 is

a)		y = $x^{2}$ + 4x
b)		y = $x^{2}$ + 4x + 1
c)		y = $x^{2}$ + 4x + 2
d)		y = $x^{2}$ + 5x + c

14. The degree of the differential equation of all curves having normal of constant length k is

a)		1
b)		3
c)		4
d)		2

15. The degree and order of the differential equation corresponding to the family of curves y = a ${(x + a)}^{2}$ , where a is an arbitary constant is

a)		1, 1
b)		2, 1
c)		3, 1
d)		None of these

16. The differential equation of all parabolas whose axis of symmetry is parallel to x-axis is of order

a)		1
b)		2
c)		3
d)		None of these

17. The differential equation of all non-vertical lines in a plane is

a)		$\frac{dy}{dx}$ =0
b)		$\frac{dx}{dy}$ =0
c)		$\frac{d^{2} y}{d x^{2}}$ =0
d)		$\frac{d^{2} x}{d y^{2}}$ =0

18. The differential equation of all conics with centre at origin is of order

a)		2
b)		3
c)		4
d)		None of these

19. The degree of the differential equation satisfying $\sqrt{1 + x^{2}} + \sqrt{1 + y^{2}}$ =A(x $\sqrt{1 + y^{2}} - y \sqrt{1 - x^{2})}$ is

a)		2
b)		3
c)		1
d)		None of these

20. The order of the differential equation ${[1 + 5 {(\frac{dy}{dx})}^{2}]}^{3 / 2}$ =11 ${(\frac{d^{2} y}{d x^{2}})}^{5}$ is

a)		1
b)		2
c)		3
d)		4

21. The general solution of $ydx - xdy - 3 x^{2} y^{2} e^{x^{3}} dx = 0$ is

a)		$\frac{x}{y} = e^{x^{3}} + c$
b)		$\frac{y}{x} = e^{x^{3}} + c$
c)		$xy = e^{x^{3}} + c$
d)		$xy e^{x^{3}} = c$

22. The differntial equation obtained on eliminating A and B from $y = A \cos ω t + B \sin ω t$ is

a)		$y^{″} + y^{'} = 0$
b)		$y^{″} - ω^{2} y = 0$
c)		$y^{″} = - ω^{2} y$
d)		$y^{″} + y = 0$

23. The solution of differential equation $\frac{dy}{dx} = \frac{y - x}{y + x}$ is

a)		log $(x^{2} + y^{2}) + 2 \tan^{- 1} (\frac{y}{x})$ = C
b)		$\frac{y^{2}}{2} + xy = x_{3} - \frac{x^{2}}{2}$
c)		$(1 + \frac{x}{y}) y = (1 - \frac{x}{y}) x + C$
d)		y=x-2 $\log_{e} y + C$

24. Solution of the differential equation x dy - y dx = 0 represents

a)		a rectangular hyperbola
b)		straight line through orgin
c)		parabola where vertex is at origin
d)		circle whose centre at origin

25. Integrating factor of the differential equation cos x $\frac{dy}{dx} + y \sin x = 1$

a)		cos x
b)		tan x
c)		sec x
d)		sin x

26. Solution of the differential equation $\frac{dy}{dx} + \frac{y}{x} = x$ is

a)		x+y= $\frac{x^{2}}{2} + c$
b)		x-y= $\frac{1}{2} x^{3} + c$
c)		xy= $\frac{1}{4} x^{4} + c$
d)		y-x= $\frac{1}{4} x^{4} + c$

27. The general solution of the differential equation $\frac{dy}{dx} = \frac{y}{x}$ is

a)		y= $\frac{k}{x}$ , k is constant
b)		y= k log x
c)		y=kx
d)		log y = kx

28. The differential equation of all non vertical lines in a plane is

a)		$\frac{d^{2} y}{d x^{2}} = 0$
b)		$\frac{d^{2} x}{d y^{2}} = 0$
c)		$\frac{dy}{dx} = 0$
d)		$\frac{dy}{dx} = 0$

29. The order of the differentiable equation of all conic whose centre lie at the origin is given by

a)		2
b)		3
c)		4
d)		none of these

30. The degree of the differential equation ${[1 + {(\frac{dy}{dx})}^{2}]}^{3 / 2} = K \frac{d^{2} y}{d x^{2}}$ is

a)		1
b)		2
c)		3
d)		4

31. The order and degree of the differential equation $\frac{d^{2} y}{d x^{2}} - {(\frac{dy}{dx})}^{1 / 3} + x^{1 / 4} = 0$ are

a)		2,3
b)		3,3
c)		2,6
d)		2,4