The function f (x) = f(x) = asinx + bex&

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 Multiple Choice QuestionsMultiple Choice Questions

41.

The integrating factor of the differential equation

dydx + 3x2tan-1y - x31 + y2 = 0 is

  • ex2

  • ex3

  • e3x2

  • e3x3


42.

If y = e- xcos2x, then which of the following differential equation is satisfied?

  • d2ydx2 + 2dydx + 5y = 0

  • d2ydx2 + 5dydx + 2y = 0

  • d2ydx2 - 5dydx - 2y = 0

  • d2ydx2 + 2dydx - 5y = 0


43.

Let f be any continuously differentiable function on [a, b] and twice differentiable on (a, b) such that f(a) = f"(a) = 0 and f(b) = 0. Then,

  • f''(a) = 0

  • f'(x) = 0 for some x  a, b

  • f''(x)  0 for some x  a, b

  • f'''x = 0 for some x  a, b


44.

Let f : R  R  be such that f(2x - 1) = f(x) for all x  R. If f is continuous at x = 1 and f(1) = 1, then

  • f(2) = 1

  • f(2) = 2

  • f is continuous only at x = 1

  • f is continuous at all points


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45.

Let f(x) be a differentiable function in [2, 7]. If f(2) = 3 and f'(x)  5 for all x in (2, 7), then the maximum possible value of f(x) at x = 7 is

  • 7

  • 15

  • 28

  • 14


46.

The function f(x) = tanπx - π22 + x2, where x denotes the greatest integer  x, is

  • continuous for all values of x

  • discontinuous at x = π2

  • not differentiable for some values of x

  • discontinuous at x = - 2


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47.

The function f (x) = f(x) = asinx + bex is differentiable at x = 0 when

  • 3a + b = 0

  • 3a - b = 0

  • a + b = 0

  • a - b = 0


C.

a + b = 0

Given, f(x) = asinx + bexWe know that sinx and ex is not differentiable at x = 0.

Therefore, for f(x) to differentiable at x = 0, we must have a = b = 0.

Thus, a + b = 0


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48.

Let R be the set of all real numbers and f : [- 1, 1]  R be defined by

f(x) = xsin1x,     x  00,                 x = 0 Then,

  • f satisfies tile conditions of Rolle's theorem on [-1, 1]

  • f satisfies the conditions of Lagrange's mean value theorem on [-1, 1]

  • f satisfies the conditions of Rolle's theorem on [0, 1]

  • f satisfies the conditions of Lagrange's mean value theorem on [0, 1]


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49.

Suppose that f(x) is a differentiable function such that f'(x) is continuous, f'(0) = 1 and f''(0) does not exist. Let g(x) = xf'(x), Then,

  • g'(0) does not exist

  • g'(0) = 0

  • g'(0) = 1

  • g'(0) = 2


50.

Applying Lagrange's Mean Value Theorem for a suitable function f(x) in [0, h], we have f(h) = f(0) + hf'(θh), 0 < θ < 1. Then, for f(x) = cos(x), the value of limh0+θ is

  • 1

  • 0

  • 1/2

  • 1/3


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