If limx→0x1 + acosx - bsinxx3 

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71.

If limx0x1 + acosx - bsinxx3 = 1, then

  • a = - 52, b = - 12

  • a = - 32, b = - 12

  • a = - 32, b = - 52

  • a = - 52, b = - 32


D.

a = - 52, b = - 32

limx0x1 + acosx - bsinxx3 = 1Using series of sin(x) and cos(x),limx0x1 + a1 - x22! + ... - bx - x33! + ...x3 = 1 limx0x1 + a - x32! + ... - bx + bx33! + ...x3 = 1 limx0x1 + a - b + x3b6 - a2 + ...x3 = 1For existence of finite limit,1 + a - b = 0           ...(i)Then, limx0x3b6 - a2 + ...x3 = 1 b6 - a2 = 1 b - 3a = 6On solvmg Eqs. (i) and (ii), we geta = - 52, b = - 32


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72.

If α and β are the rooots of ax2 + bx + c = 0, then limxα1 - cosax2 + bx + cx - α2 is equal to

  • a22α - β2

  • - a22α - β2

  • 0

  • 1


73.

If sin(x + y) = log(x + y), then dydx is equal to

  • - 1

  • 1

  • 2

  • - 2


74.

If x = acos3t and y = asin3t, then dydxt = π4 is equal to

  • 1

  • - 1

  • 0


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75.

The anti-derivative F of f defined by f(x) = 4x3 - 6x2 + 2x + 5, F(0) = 5, is

  • x4 - 2x3 + x2 + 5x

  • 12x2 - 12x + 2

  • 16x4 - 18x3 + 4x2 + 5x

  • x4 - 2x3 + x2 + 5x + 5


76.

limx0x- 11x

  • 0

  • 1

  • - 1

  • Does not exist


77.

If f(x) = sinxx for x  00         for x = 0

where [x] denotes the greatest integer less than or eual to x, then limx0fx is equal to

  • 1

  • - 1

  • 0

  • Does not exist


78.

If limnlogn + r - lognn = 2log2 - 12, then limn1nλn + 1λn + 2λ ... n + nλ1n is equal to

  • 4λe

  • 4eλ

  • 4e1λ

  • e4λ


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79.

limx0sinsinx - sinxax3 + bx5 + c = - 112, then

  • a = 2, b  R, c = 0

  • a = - 2, b  R, c = 0

  • a = 1, b  R, c = 0

  • a = - 1, b  R, c = 0


80.

If f(x) = sinxx, x  00,         x = 0, where [x] denotes the gretest function less than or equal to x, then limx0fx is equal to

  • 1

  • 0

  • - 1

  • Does not exist


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