#### 12th Standard Maths English Medium Ordinary Differential Equations Reduced Syllabus Important Questions With Answer Key 2021

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

Multiple Choice Questions

15 x 1 = 15
1. The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters, is

(a)

$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -y=0$

(b)

$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }+y=0$

(c)

$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }=0$

(d)

$\frac { { d }^{ 2 }x }{ { dy }^{ 2 } }=0$

2. The integrating factor of the differential equation $\frac{dy}{dx}$+P(x)y=Q(x)is x, then P(x)

(a)

x

(b)

$\frac { { x }^{ 2 } }{ 2 }$

(c)

$\frac{1}{x}$

(d)

$\frac{1}{x^2}$

3. The degree of the differential equation y $y(x)=1+\frac { dy }{ dx } +\frac { 1 }{ 1.2 } { \left( \frac { dy }{ dx } \right) }^{ 2 }+\frac { 1 }{ 1.2.3 } { \left( \frac { dy }{ dx } \right) }^{ 3 }+....$ is

(a)

2

(b)

3

(c)

1

(d)

4

4. If p and q are the order and degree of the differential equation $y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,$When

(a)

p < q

(b)

p = q

(c)

p>q

(d)

p exists and q does not exist

5. The solution of $\frac { dy }{ dx } ={ 2 }^{ y-x }$is

(a)

2x+2y=C

(b)

2x-2y=C

(c)

$\frac { 1 }{ { 2 }^{ x } } -\frac { 1 }{ { 2 }^{ y } } =C$

(d)

x+y=C

6. The solution of the differential equation $\frac { dy }{ dx } =\frac { y }{ x } +\frac { \phi \left( \frac { y }{ x } \right) }{ \phi '\left( \frac { y }{ x } \right) }$is

(a)

$x\phi \left( \frac { y }{ x } \right) =k$

(b)

$\phi \left( \frac { y }{ x } \right) =kx$

(c)

$y\phi \left( \frac { y }{ x } \right) =k$

(d)

$\phi \left( \frac { y }{ x } \right) =ky$

7. If sin x is the integrating factor of the linear differential equation $\frac { dy }{ dx } +Pt=Q,$Then P is

(a)

log sin x

(b)

cos x

(c)

tan x

(d)

cot x

8. The number of arbitrary constants in the general solutions of order n and n +1are respectively

(a)

n-1,n

(b)

n,n+1

(c)

n+1,n+2

(d)

n+1,n

9. If the solution of the differential equation $\frac{dy}{dx}=\frac{ax+3}{2y+f}$represents a circle, then the value of a is

(a)

2

(b)

-2

(c)

1

(d)

-1

10. The slope at any point of a curve y = f (x) is given by $\frac{dy}{dx}=3x^2$ and it passes through (-1,1). Then the equation of the curve is

(a)

y=x3+2

(b)

y=3x2+4

(c)

y=3x4+4

(d)

y=3x2+5

11. The order and degree of the differential equation ${ \left[ \left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +\left( \frac { dy }{ dx } \right) \right] }^{ \frac { 1 }{ 2 } }=\frac { { d }^{ 3 }y }{ { dx }^{ 3 } }$are

(a)

1,2

(b)

2,1

(c)

3,2

(d)

2,3

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

(a)

y sin x=$\frac{2}{3}$sin3x+c

(b)

y sec x=$\frac{x^2}{2}+c$

(c)

y sin x =c+x

(d)

2y sin x=sin x-$\frac{sin\ 3x}{3}+c$

13. The solution of log $\left( \frac { dy }{ dx } \right)$=ax+by is______.

(a)

$\frac { { e }^{ ax } }{ a } +\frac { { e }^{ -by } }{ b } +c=0$

(b)

aeax-be-by+c=0

(c)

aex+bey=k

(d)

beax+ae-by=k

14. The differential equation of x2y = k is _________.

(a)

${ x }^{ 2 }\frac { dy }{ dx } =0$

(b)

${ x }^{ 2 }\frac { dy }{ dx } +y=0$

(c)

${ x }\frac { dy }{ dx } +2y=0$

(d)

$y\frac { dy }{ dx } +2x=0$

15. The I.F. of (1+y2)dx=(tan-1-t-x)dy is ________.

(a)

etan-1 y

(b)

etan-1 x

(c)

tan-1 y

(d)

tan-1x

1. 2 Marks

10 x 2 = 20
16. A differential equation, determine its order, degree (if exists)
$\frac { dy }{ dx } +xy=cotx$

17. A differential equation, determine its order, degree (if exists)
${ \left( \frac { dy }{ dx } \right) }^{ 3 }=\sqrt { 1+\left( \frac { dy }{ dx } \right) }$

18. Assume that a spherical rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.

19. Find the differential equation of the family of all non-vertical lines in a plane.

20. Show that y = e−x + mx + n is a solution of the differential equation ex $\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right)$ -1 = 0

21. Determine the order and degree (if exists) of the following differential equations:
${ \left( \frac { { d }^{ 4 }y }{ { dx }^{ 4 } } \right) }^{ 3 }+4{ \left( \frac { dy }{ dx } \right) }^{ 7 }+6y=5cos3x$

22. Find the differential equation of the family of parabolas y2 ax = 4, where a is an arbitrary constant.

23. Solve: $\frac{dy}{dx}+y=e^{-x}$

24. Determine the order and degree of $\cfrac { \left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 3 }{ 2 } } }{ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } } =k$

25. Form the Differential Equation representing the family of curves y=Acos(x+B) where A and B are parameters.

1. 3 Marks

10 x 3 = 30
26. Find the differential equation of the family of circles passing through the origin and having their centres on the x -axis.

27. Find the differential equation of the family of parabolas with vertex at (0,−1) and having
axis along the y-axis.

28. Find the particular solution of (1+ x3 )dy − x2 ydx = 0 satisfying the condition y(1) = 2.

29. If F is the constant force generated by the motor of an automobile of mass M, its velocity is given by M $\frac{dV}{dt}$=F-kV, where k is a constant. Express V in terms of t given that V = 0 when t = 0.

30. Solve $\frac { dy }{ dx } +\frac { { y }^{ 2 } }{ { x }^{ 2 } } =\frac { y }{ x }$

31. Form the differential equation for y=e-2x [A coos 3x-B sin 3x]

32. Solve: x$\frac{dy}{dx}$+2y=x2

33. Verify that y=-x-1 is a solution of the D.E (y-x)dy-(y2-x2)dx=0

34. Solve:$\cfrac { dy }{ dx } =\cfrac { 1-cosx }{ 1+cosx }$

35. Solve : ydx+(x-y2)dy=0

1. 5 Marks

7 x 5 = 35
36. Find the equation of the curve whose slope is $\frac { y-1 }{ { x }^{ 2 }+x }$ and which passes through the point (1,0).

37. Solve (x2 -3y2) dx xydy= 0.

38. Solve the Linear differential equation:
$(1+x+{ xy }^{ 2 })\frac { dy }{ dx } +(y+{ y }^{ 3 })=0$

39. Solve : (1+y2)(1+logx)dx+xdy=0, given that x=1,y=1.

40. Solve : ${ 2x }^{ 2 }\left( \cfrac { dy }{ dx } \right) -2xy+{ y }^{ 2 }=0,y(e)=e$

41. Solve : $\cfrac { dy }{ dx } =-\cfrac { x+ycos }{ 1+sinx }$ .Also find the domain of the function.

42. Water at temperature 100℃ cools in 5 minutes to 80℃ in a room of temperature 30℃. Find (i) the temperature of water after 10 minutes. (ii) the time when the temperature is 40℃.