Let Xn = 1 - 1321 - 1621 

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 Multiple Choice QuestionsMultiple Choice Questions

11.

Let f(x) = x13 + x11 + x9 + x7 + x5 + x3 + x + 19. Then , f(x) = 0 has

  • 13 real roots

  • only one positive and only two negative real roots

  • not more than one real root

  • has two positive and one negative real root


12.

limx11 + x2 + x1 - x1 - x is equal to

  • 1

  • does not exist

  • 23

  • ln2


13.

The value of limnn + 1 + n + 2 + ... + 2n - 1n32 is

  • 2322 - 1

  • 232 - 1

  • 232 + 1

  • 2322 + 1


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14.

Let Xn = 1 - 1321 - 1621 - 1102 ... 1 - 1nn + 12, n  2

Then, the value of limnxn is

  • 1/3

  • 1/9

  • 1/81

  • 0


B.

1/9

We have,

Xn = 1 - 1321 - 1621 - 1102 ... 1 - 1nn + 12

 xn = n = 2nn2 + n - 2nn + 12         = n = 2nn + 2n - 1nn + 12        = n = 2nn + 2n + 1 . n = 2nn - 1n2        = n = 2nn + 2n + 12n = 2nn - 1n2

 xn = 43 . 54 . 65 ... n +2n + 1212 . 23 . 34 ... n - 1n2 xn = n + 2321n2 xn = 19n + 2n2 xn = 191 + 2n2 limnxn = 191 + 02 = 19


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15.

limn1 + 2 + ... + n - 1nn is equal to

  • 12

  • 13

  • 23

  • 0


16.

Let f: R  R be differentiable at x = 0. If f(0) = 0 and f'(0) = 2, then the value of

limx01xf(x) + f(2x) + f(3x) + ... + f(2015x) is

  • 2015

  • 0

  • 2015 × 2016

  • 2015 × 2014


17.

If limx02asinx - sin2xtan3x exists and is equal to 1, then the value of α is

  • 2

  • 1

  • 0

  • - 1


18.

Let f(x) be a differentiable function and f'(4) = 5. Then

limx2f4 - fx2x - 2 equals

  • 0

  • 5

  • 20

  • - 20


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19.

Let [x] denote the greatest integer less than or equal to x for any real number x. Then,

limnn2n is equal to

  • 0

  • 2

  • 2

  • 1


20.

The limit of 1x2 + 2013xex - 1 - 1ex - 1 as x  0

  • approaches + 

  • approaches - 

  • is equal to loge2013

  • does not exist


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