Define f on R into itself byfx = xsin1x, when 

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 Multiple Choice QuestionsMultiple Choice Questions

261.

The vectors AB = 3i^ + 5j^ + 4k^ and AC = 5i^ - 5j^ + 2k^ are the sides of a triangle ABC. The length ofthe median through A is :

  • 13 unit

  • 25 unit

  • 5 unit

  • 10 unit


262.

A function f on R into itself is continuous at a point ain R, iff for each  > 0, there exists, δ > 0 such that :

  • fx - fa <   x - a < δ

  • fx - fa >   x - a > δ

  • x - a > δ  fx - fa > 

  • x - a < δ  fx - fa < 


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263.

Define f on R into itself by

fx = xsin1x, when x  00,           when x = 0, then :

  • f is continuous at 0 but not differentiable at 0

  • f is both continuous and differentiable at 0

  • f is differentiable but not continuous at 0

  • None of these


A.

f is continuous at 0 but not differentiable at 0

fx = xsin1x, when x  00,           when x = 0LHL = limx0-fx = limh0- hsin1hRHL = limx0+fx = limh0 hsin1h It is continuous at x = 0limx0-fx = limx0+fx = f0 = 0LHD = limx0-fx = limh0f0 - h - f0- h       = limh0- hsin1h - 0- h = does not existRHD = limx0+fx = limh0f0 + h - f0h       = limh0hsin1h - 0h = does not exist f(x) is not differentiable at x =0. f(x) is continuous at x = 0 but not differentiable at x =0.


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264.

If x = Acos4t + Bsin4t, then d2xdt2 is equal to

  • - 16x

  • 16x

  • x

  • - x


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265.

If f(x) = 1 - cosxx, x  0k                , x = 0 is continuous at x = 0, then the value of k is

  • 0

  • 12

  • 14

  • 12


266.

The derivative of cos-11 - x21 + x2 w. r. t. cot-11 - 3x23x - x3 is

  • 32

  • 1

  • 12

  • 23


267.

If y = 1 - x + x22! - x33! + x44! ..., then d2ydx2 is equal to

  • - x

  • x

  • y

  • - y


268.

The function f(x) = x + xx is

  • discontinuous at origin because x is discontinuous there

  • continuous at origin

  • discontinuous at origin because both x and xx are discontinuous there

  • discontinuous at the origin because xx is discontinuous there


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269.

Let f(x) = sinπx5x, x 0k,           x = 0. If f(x) continuous at x = 0, the value of k is

  • 5π

  • π5

  • zero

  • 1


270.

If f(x) = xsin1x, if x  00,             if x = 0', then at x = 0 the function f is

  • continuous but not differentiable

  • differentiable but not continuous

  • continuous and differentiable

  • not continuous


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