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

12th Standard

    Reg.No. :
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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℃.

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