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 Multiple Choice QuestionsMultiple Choice Questions

1.

In ABC if θ is any angle, then bcosC + θ + ccosB - θ = ?

  • acotθ

  • acosθ

  • atanθ

  • asinθ


2.

The number of solutions of the trigonometric equation 1 + cosx . cos5x = sin2x in [0, 2π] is

  • 8

  • 12

  • 10

  • 6


3.

A value of θ for which fraction numerator 2 space plus space 3 straight i space sin space straight theta over denominator 1 minus 2 straight i space sin space straight theta end fraction is purely imaginary is

  • π/3

  • π/6

  • sin to the power of negative 1 end exponent open parentheses fraction numerator square root of 3 over denominator 4 end fraction close parentheses
  • sin to the power of negative 1 end exponent open parentheses fraction numerator square root of 3 over denominator 4 end fraction close parentheses
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4.

Consider f(x) = tan-1 open parentheses square root of fraction numerator 1 space plus sin space straight x over denominator 1 minus sin space straight x end fraction end root close parentheses comma space straight x space element of space open parentheses 0 comma space straight pi over 2 close parentheses. A normal to y = f (x) at x = π/6 also passes through the point

  • (0,0)

  • (0, 2π/3)

  • (π/6 ,0)

  • (π/6 ,0)


B.

(0, 2π/3)



We space have comma space straight f left parenthesis straight x right parenthesis space equals space tan to the power of negative 1 end exponent square root of fraction numerator 1 plus space sin space straight x over denominator 1 minus space sin space straight x end fraction end root comma space straight x element of space open parentheses 0 comma space straight pi over 2 close parentheses
rightwards double arrow space straight f left parenthesis straight x right parenthesis space equals space tan to the power of negative 1 end exponent space square root of open parentheses cos begin display style straight x over 2 end style plus space sin begin display style straight x over 2 end style close parentheses squared over open parentheses cos straight x over 2 minus space sin straight x over 2 close parentheses squared end root
space equals space tan to the power of negative 1 end exponent open parentheses fraction numerator cos space begin display style straight x over 2 end style space plus space sin space begin display style straight x over 2 end style over denominator cos space begin display style straight x over 2 end style space minus space sin space begin display style straight x over 2 end style end fraction close parentheses
open square brackets therefore space cos space straight x over 2 greater than space sin space straight x over 2 space for space 0 space less than thin space straight x over 2 less than straight x over 4 close square brackets
equals space tan to the power of negative 1 end exponent space open parentheses fraction numerator 1 plus tan space begin display style straight x over 2 end style over denominator 1 minus space tan space begin display style straight x over 2 end style end fraction close parentheses
equals space tan to the power of negative 1 end exponent space open square brackets tan space open parentheses straight pi over 4 plus straight pi over 2 close parentheses close square brackets

space equals space straight pi over 4 plus straight pi over 2
rightwards double arrow space straight f apostrophe left parenthesis straight x right parenthesis space equals space 1 half
rightwards double arrow space straight f apostrophe open parentheses straight pi over 6 close parentheses equals 1 half
Now comma space equation space of space normal space at space straight x space equals space straight pi over 6 space is space given space by
open square brackets straight y minus straight f open parentheses straight pi over 6 close parentheses close square brackets space equals space minus 2 open parentheses straight x minus straight pi over 6 close parentheses
rightwards double arrow open parentheses straight y minus straight pi over 3 close parentheses space equals space minus space 2 open parentheses straight x minus straight pi over 6 close parentheses
open square brackets therefore comma space straight f open parentheses straight pi over 6 close parentheses space equals space straight pi over 4 plus straight pi over 12 equals fraction numerator 4 straight pi over denominator 12 end fraction space equals straight pi over 3 close square brackets
Whcih space passes space through space the space point space open parentheses 0 comma space fraction numerator 2 straight pi over denominator 3 end fraction close parentheses
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5.

If 0≤x<2π, then the number of real values of x, which satisfy the equation cosx+cos2x+cos3x+cos4x=0, is :

  • 3

  • 5

  • 7

  • 7

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6. straight l im with straight x rightwards arrow 0 below space fraction numerator left parenthesis 1 minus cos 2 straight x right parenthesis left parenthesis 3 space plus cos space straight x right parenthesis over denominator straight x space tan space 4 straight x end fraction space is equal to 
  • 4

  • 3

  • 2

  • 2

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7.

Let tan to the power of negative 1 end exponent space straight y space equals tan to the power of negative 1 end exponent space straight x space plus space tan to the power of negative 1 end exponent space open parentheses fraction numerator 2 straight x over denominator 1 minus straight x squared end fraction close parentheses comma where vertical line straight x vertical line space less than space fraction numerator 1 over denominator square root of 3 space end fraction space.Then, a value of y is

  • fraction numerator 3 straight x squared minus straight x squared over denominator 1 minus 3 straight x squared end fraction
  • fraction numerator 3 straight x space plus space straight x cubed over denominator 1 minus 3 straight x squared end fraction
  • fraction numerator 3 straight x minus straight x cubed over denominator 1 plus 3 straight x squared end fraction
  • fraction numerator 3 straight x minus straight x cubed over denominator 1 plus 3 straight x squared end fraction
176 Views

8.

If y  = sec (tan-1 x), then dy/dx at x = 1 is equal to 

  • fraction numerator 1 over denominator square root of 2 end fraction
  • 1 half
  • 1

  • 1

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9.

If A = sin2 x + cos4 x, then for all real x

  • 3/4 ≤ A ≤ 1

  • 31/16 ≤ A ≤ 1

  • 1≤ A ≤2

  • 1≤ A ≤2

154 Views

10.

If cos-1 x - cos-1 y/2 = α, then 4x2 − 4xy cos α + y2 is equal to

  • 2 sin 2α

  • 4

  • 4 sin2 α

  • 4 sin2 α

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