#### 12th Standard Maths English Medium Ordinary Differential Equations Reduced Syllabus Important Questions 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 order and degree of the differential equation $\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0$are respectively

(a)

2, 3

(b)

3, 3

(c)

2, 6

(d)

2, 4

2. The solution of $\frac{dy}{dx}+$p(x)y=0 is

(a)

$y={ ce }^{ \int { pdx } }$

(b)

$y={ ce }^{ -\int { pdx } }$

(c)

$x={ ce }^{ -\int { pdy } }$

(d)

$x={ce }^{ \int { pdy } }$

3. The integrating factor of the differential equation $\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda }$ is

(a)

$\frac { x }{ { e }^{ \lambda } }$

(b)

$\frac { { e }^{ \lambda } }{ x }$

(c)

${ \lambda e }^{ x }$

(d)

ex

4. 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)

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

5. 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

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. 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

8. The number of arbitrary constants in the particular solution of a differential equation of third order is

(a)

3

(b)

2

(c)

1

(d)

0

9. 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

10. The solution of (x2-ay)dx=(ax-y2)dy is

(a)

y=x2+y2-a(x+y)

(b)

y=x2+y2-a(x+y)

(c)

x3+y2=3ayx+c

(d)

(x2-ay)(ax-y2)=0

11. The transformation y=vx reduces $\\ \\ \\ \frac { dy }{ dx } =\frac { x+y }{ 3x }$

(a)

$\frac { 3av }{ 4v+1 } =\frac { dx }{ x }$

(b)

$\frac { 3dv }{ v+1 } =\frac { dx }{ x }$

(c)

$2x\frac { dv }{ dx } =v$

(d)

$\frac { 3dv }{ 1-2v } ==\frac { dx }{ x }$

12. The I.F. of cosec x$\frac{dy}{dx}+y$sec2x=0 is

(a)

esec x

(b)

etan x

(c)

esec x tan x

(d)

esec2 x

13. The order and degree of y'+(y")2=(x+t")2 are _________.

(a)

1,1

(b)

1,2

(c)

2,1

(d)

2,2

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)
${ \left( \frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } \right) }^{ \frac { 2 }{ 3 } }-3\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +4=0$

17. A differential equation, determine its order, degree (if exists)
${ x }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }=0$

18. Express the physical statement in the form of the differential equation.
Radium decays at a rate proportional to the amount Q present.

19. Find value of m so that the function y = emx is a solution of the given differential equation.
y '+ 2y = 0

20. Find value of m so that the function y = emx is a solution of the given differential equation.
y''− 5y' + 6y = 0

21. Determine the order and degree (if exists) of the following differential equations:
$3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }$

22. Solve the Linear differential equation:
$\frac { dy }{ dx } =\frac { { sin }^{ 2 }x }{ 1+{ x }^{ 3 } } -\frac { { 3x }^{ 2 } }{ 1+{ x }^{ 3 } } y$

23. Form the differential equation satisfied by are the straight lines in my-plane.

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

25. Solve :$\cfrac { dy }{ dx } =\cfrac { 2x }{ { x }^{ 2 }+1 }$

1. 3 Marks

10 x 3 = 30
26. Find the differential equation corresponding to the family of curves represented by the equation y = Ae8x + Be-8x, where A and B are arbitrary constants.

27. Form the differential equation by eliminating the arbitrary constants A and B from y = Acos x + B sin x.

28. 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.

29. Solve the differential equation:
$sin\frac { dy }{ dx } =a,y(0)=1$

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

31. A radioactive isotope has an initial mass 200mg, which two years later is 50mg . Find the expression for the amount of the isotope remaining at any time. What is its half-life? (half-life means the time taken for the radioactivity of a specified isotope to fall to half its original value).

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

33. Form the D.E of family of curves represented by y=c(x-c)2.where c is the parameter.

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

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

1. 5 Marks

7 x 5 = 35
36. Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis.

37. Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin.

38. The velocity v , of a parachute falling vertically satisfies the equation $\\ \\ \\ \\ \\ \\ \\ v\frac { dv }{ dx } =g\left( 1-\frac { { v }^{ 2 } }{ { k }^{ 2 } } \right) \\ \\$, where g and k are constants. If v and x are both initially zero, find v in terms of x.

39. Solve the differential equation:
(ydx-xdy)cot$\left( \frac { x }{ y } \right)$=ny2 dx

40. A tank contains 1000 litres of water in which 100 grams of salt is dissolved. Brine (Brine is a high-concentration solution of salt (usually sodium chloride) in water) runs in a rate of 10 litres per minute, and each litre contains 5grams of dissolved salt. The mixture of the tank is kept uniform by stirring. Brine runs out at 10 litres per minute. Find the amount of salt at any time t.

41. Assume that the rate at which radioactive nuclei decay is proportional to the number of such nuclei that are present in a given sample. In a certain sample 10% of the original number of radioactive nuclei have undergone disintegration in a period of 100 years. What percentage of the original radioactive nuclei will remain after 1000 years?

42. A tank initially contains 50 litres of pure water. Starting at time t = 0 a brine containing with 2 grams of dissolved salt per litre flows into the tank at the rate of 3 litres per minute. The mixture is kept uniform by stirring and the well-stirred mixture simultaneously flows out of the tank at the same rate. Find the amount of salt present in the tank at any time t > 0.