The solution of 25d2ydx2 - 10dydx + y&nb

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 Multiple Choice QuestionsMultiple Choice Questions

51.

Let f(x) = 0x1 - tdt,    x > 0x - 12,         x  1. Then

  • f(x) is continuous at x = 1

  • f(x) is not continuous at x = 1

  • f(x) is differentiable at x = 1

  • f(x) is not differentiable at x = 1


52.

If f(x) = x3 - 3x + 2,                      x < 2,x3 - 6x2 + 9x + 2,           x  2

then

  • limx2f(x) does not exist

  • f is not continuous at x = 2

  • f is continuous but not differentiable at x = 2

  • f is continuous and differentiable at x = 2


53.

Let y = 3x - 13x + 1sinx + loge1 + x, x > - 1. Then, at x = 0, dydx equals

  • 1

  • 0

  • - 1

  • - 2


54.

If f is a real-valued differentiable function such that f(x)f' (x) < 0 for all real x, then

  • f(x) must be an increasing function

  • f(x) must be a decreasing function

  • f(x) must be an increasing function

  • f(x) must be a decreasing function


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55.

Rolle's theorem is applicable in the interval [- 2, 2] for the function

  • f(x) = x3

  • f(x) = 4x4

  • f(x) = 2x3 + 3

  • f(x) = πx


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56.

The solution of 25d2ydx2 - 10dydx + y = 0, y(0) = 1, y(1) = 2e15 is

  • y = e5x + e- 5x

  • y = 1 + xe5x

  • y = 1 + xex5

  • y = 1 + xe- x5


C.

y = 1 + xex5

Let y = emx be the solution of given differential equation,

     dydx = memx  d2ydx2 = m2emx       25d2ydx2 - 10dydx + y = 025m2emx - 10 memx + emx = 0   emx25m2 - 10m + 1 = 0 Auxiliary equation 25m2 - 10m + 1 = 0                            emx  0 5m2 - 25m × 1 + 1 = 0                       5m - 12 = 0 m = 15, 15

Since, roots are real and euqal.

 General solution is y = c1 +c2xex5      ...(i)y0 = 1  c1 = 1y1 = 2e15  2ee5 = c1 + c2e15 c1 + c2 = 2  c1 = 1

Putting the value of c1 and c2 in Eq. (i), we get particular solution

y = 1 + xx5


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57.

If f(x) and g(x) are twice differentiable functions on (0, 3) satisfying f"(x) = g''(c), f'(1) = 4g'(D) = 6, f(2) = 3, g(2) = 9, then f(1) - g(1) is

  • 4

  • - 4

  • 0

  • - 2


58.

For function f(x) = ecosx, Rolle's theorem is

  • applicable, when π2  x  3π2

  • applicable, when 0  x  π2

  • applicable, when 0  x  π

  • applicable, when π4  x  π2


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59.

f(x) = 0,            x = 0x - 3,     x > 0 the function f(x) is

  • increasing when x  0

  • strictly increasing when x > 0

  • strictly increasing at x = 0

  • not continuous at x = 0 and so it is not increasing when x > 0


60.

The function f(ax) = ax + b is strictly increasing for all real x, if

  • a > 0

  • a < 0

  • a = 0

  •  0


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