from Mathematics Limits and Derivatives

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 Multiple Choice QuestionsMultiple Choice Questions

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1. limit as straight x space rightwards arrow 0 of space fraction numerator sin begin display style left parenthesis end style begin display style straight pi end style begin display style space end style begin display style begin display style cos end style squared end style begin display style straight x end style begin display style right parenthesis end style over denominator straight x squared end fraction space is space equal space to
  • π/2

  • 1


D.

limit as straight x space rightwards arrow 0 of space fraction numerator sin begin display style space end style begin display style left parenthesis end style begin display style straight pi end style begin display style space end style begin display style begin display style cos end style squared end style begin display style straight x end style begin display style right parenthesis end style over denominator straight x squared end fraction
space equals space limit as straight x space rightwards arrow 0 of space fraction numerator sin begin display style space end style begin display style straight pi end style begin display style left parenthesis end style begin display style 1 end style begin display style minus end style begin display style begin display style sin end style squared end style begin display style straight x end style begin display style right parenthesis end style over denominator straight x squared end fraction space
equals space limit as straight x space rightwards arrow 0 of space fraction numerator sin begin display style left parenthesis end style begin display style straight pi end style begin display style minus end style begin display style begin display style πsin end style squared end style begin display style straight x end style begin display style right parenthesis end style over denominator straight x squared end fraction
equals limit as straight x space rightwards arrow 0 of space fraction numerator sin begin display style left parenthesis end style begin display style straight pi end style begin display style space end style begin display style begin display style sin end style squared end style begin display style space end style begin display style straight pi end style begin display style right parenthesis end style over denominator straight x squared end fraction space space space space space left square bracket because space sin space left parenthesis straight pi space minus space straight theta space equals space sin space straight theta right square bracket
equals space limit as straight x space rightwards arrow 0 of space fraction numerator sin begin display style space end style begin display style begin display style πsin end style squared end style begin display style straight x end style over denominator straight pi space sin squared straight x end fraction space straight x space left parenthesis straight pi right parenthesis open parentheses fraction numerator begin display style sin squared end style begin display style straight x end style over denominator straight x squared end fraction close parentheses space equals space straight pi space open square brackets because space limit as straight theta rightwards arrow 0 of space fraction numerator sin begin display style space end style begin display style straight theta end style over denominator straight theta end fraction space equals 1 space close square brackets
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2. limit as straight x space rightwards arrow 0 of fraction numerator left parenthesis 1 minus cos space 2 straight x right parenthesis left parenthesis 3 plus cos space straight x right parenthesis over denominator straight x space tan space 4 straight x end fraction space is space equal space to space
  • -1/4

  • 1/2

  • 1

  • 1

188 Views

3.

The equation of the tangent to the curve y = x +4/x2, that is parallel to the x-axis, is

  • y= 0

  • y= 1

  • y= 2 

  • y= 2 

162 Views

4.

Let cos (α + β) = 4/5 and let sin (α - β) = 5/13, where 0 ≤α,β ≤ π/4. Then tan 2α is equal to

  • 25/16

  • 56/33

  • 19/12

  • 19/12

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5. stack lim space with straight x space rightwards arrow straight pi over 2 below space fraction numerator cot space straight x space minus cos space straight x over denominator left parenthesis straight pi minus 2 straight x right parenthesis cubed end fraction space equals
  • 1/4

  • 1/24

  • 1/16

  • 1/16

396 Views

6.

If 5(tan2x – cos2x) = 2cos 2x + 9, then the value of cos4x is

  • -7/9

  • -3/5

  • 1/3

  • 1/3

542 Views

7.

The differential equation which represents the family of curves y=c1ec2xe, where c1 and c2 are arbitrary constants, is

  • y' =y2

  •  y″ = y′ y

  • yy″ = y′

  • yy″ = y′

120 Views

8.

The solution of the differential equation dy over dx space equals space fraction numerator straight x plus straight y over denominator straight x end fraction   satisfying the condition y (1) = 1 is  

  • y = ln x + x

  • y = x ln x + x2

  •  y = xe(x−1)

  •  y = xe(x−1)

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9.

If limit as straight x space rightwards arrow infinity of space open parentheses 1 plus straight a over straight x plus straight b over straight x squared close parentheses to the power of 2 straight x end exponent space equals space straight e squared then the values of a and b, are

  • straight a element of space straight R with equals below space straight b element of space straight R with equals below
  • straight a space equals 1 comma space straight b element of space straight R with equals below
  • straight a element of space straight R with equals below space comma space straight b space equals 2
  • straight a element of space straight R with equals below space comma space straight b space equals 2
117 Views

10.

For each t ∈R, let [t] be the greatest integer less than or equal to t. Then

limx0+ x1x+2x+......+15x

  • does not exist (in R)

  • is equal to 0

  • is equal to 15

  • is equal to 120


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