General solution of ydydx + by2 = acosx,

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 Multiple Choice QuestionsMultiple Choice Questions

11.

If f''(0) = k, k  0, then the value of limx0 2f(x) - 3f(2x) +f(4x)x2 is

  • k

  • 2k

  • 3k

  • 4k


12.

If  y = emsin- 1x then 1 - x2d2ydx2 - xdydx - ky = 0, where k is equal to

  • m2

  • 2

  • - 1

  • - m2


13.

Solution of x + y2dydx = a2 ( 'a' being a constant) is

  • x + ya = tany + Ca, C is an arbitrary

  • xy = atanCx, C is an arbitrary

  • xa = tanyC, C is an arbitrary

  • xy = tan(x + C), C is an arbitrary


14.

The integrating factor of the first order differential equation

x2x2 - 1dydx + xx2 + 1y = x2 - 1 is

  • ex

  • x - 1x

  • x + 1x

  • 1x2


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15.

Let a and B be the roots of x2 + x + 1 = 0. If n be a positive integer, then αn + βn is

  • 2cos23

  • 2sin23

  • 2cos3

  • 2sin3


16.

For real x, the greatest value of x2 + 2x + 42x2 + 4x + 9 is

  • 1

  • - 1

  • 12

  • 14


17.

If the solution of the differential equation xdydx + y = xex be xy = exϕ(x) + C, then ϕx is equal to

  • x + 1

  • x - 1

  • 1 - x

  • x


18.

The order of the differential equation of all parabolas whose axis of symmetry along X-axis is

  • 2

  • 3

  • 1

  • None of these


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19.

General solution of ydydx + by2 = acosx, 0 < x < 1 is

  • y2 = 2a2bsinx + cosx + ce- 2bx

  • 4b2 + 1y2 = 2asinx + 2bcosx + ce- 2bx

  • 4b2 + 1y2 = 2asinx + 2bcosx + ce2bx

  • y2 = 2a2bsinx + cosx + ce- 2bx


B.

4b2 + 1y2 = 2asinx + 2bcosx + ce- 2bx

Given, ydydx + by2 = acosx, 0 < x < 1     ...(i)

Let y2 = z

 2ydydx = dzdx ydydx = dzdx             ...(ii) 12dzdx + by2 = acosx          using Eq. (ii) dzdx + 2by2 = 2acosx

Now, IF = e2bdx             = e2bx z.e2bx = 2a4b2 + 1sinx + 2bcosxe2bx + C 4b2 + 1y2 = 2asinx + 2bcosx + Ce- 2bx


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20.

The integrating factor of the differential equation

1 + x2dydx + y = etan-1x is

  • tan-1x

  • 1 + x2

  • etan-1x

  • loge1 + x2


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