વર્તુળ |z| = 2 માં અંતર્ગત સમબાજુ ત્રિકોણનાં શિરોબિંદુઓ z1, z2, z3 છે. z1 = 1 +  જો તો z2, z3 અનુક્રમે ........ થાય.  from Mathematics સંકર સંખ્યાઓ

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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

41. જો bold e to the power of bold log bold space bold 2 to the power of begin inline style fraction numerator bold vertical line bold z bold vertical line to the power of bold 2 bold minus bold vertical line bold z bold vertical line bold plus bold 4 over denominator bold vertical line bold z bold vertical line to the power of bold 2 bold plus bold 1 end fraction end style end exponent end exponent bold space bold equals bold space bold log subscript square root of bold 2 end subscript bold left parenthesis bold 16 bold right parenthesis તો |z| = ....... 
  • bold 1 over bold 4
  • bold 1 over bold 2
  • bold 1 over bold 3
  • 1


42.
જો z1, z2 અને z3 ઘડિયાળની વિરુદ્વ દિશામાં દર્શાવેલ bold increment bold ABC નાં શિરોબિંદુઓ છે. જો z0 એ bold increment bold ABC નું પરિકેન્દ્ર હોય, તો open parentheses fraction numerator bold z subscript bold 0 bold minus bold z subscript bold 1 over denominator bold z subscript bold 0 bold minus bold z subscript bold 2 end fraction close parentheses fraction numerator bold sin bold 2 bold A over denominator bold sin bold 2 bold B end fraction bold plus bold space open parentheses fraction numerator bold z subscript bold 0 bold minus bold z subscript bold 3 over denominator bold z subscript bold 0 bold minus bold z subscript bold 2 end fraction close parentheses bold space fraction numerator bold sin bold 2 bold space bold C over denominator bold sin bold 2 bold space bold B end fraction
  • 2

  • 1

  • -1

  • 0


43.
જો α, β એ u2 - 2u + 2 = 0 નાં બીજ હોય તથા cotθ = x + 1  હોય, તો fraction numerator bold left parenthesis bold x bold plus bold alpha bold right parenthesis to the power of bold n bold minus bold left parenthesis bold x bold plus bold beta bold right parenthesis to the power of bold n over denominator bold alpha bold minus bold beta end fraction= ...... 
  • fraction numerator bold sin bold space bold n bold space bold theta over denominator bold n bold space bold sin to the power of bold n bold space bold theta end fraction
  • fraction numerator bold sin bold space bold n bold space bold theta over denominator bold cos to the power of bold n bold space bold theta end fraction
  • fraction numerator bold sin bold space bold n bold space bold theta over denominator bold sin to the power of bold n bold space bold theta end fraction
  • fraction numerator cos space straight n space straight theta over denominator cos to the power of straight n space straight theta end fraction

44. Re (z)2 = 0 અને |z| = a square root of bold 2 bold space bold left parenthesis bold a bold space bold greater than bold space bold 0 bold right parenthesis સમીકરણને કેટલા ઉકેલ મળે ?
  • 3

  • 4

  • 2

  • 1


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45. જો |z| = 1 અને bold w bold space bold equals bold space fraction numerator bold z bold minus bold 1 over denominator bold z bold plus bold 1 end fraction bold space bold left parenthesis bold z bold not equal to bold minus bold 1 bold right parenthesis તો Re(w)
  • open vertical bar fraction numerator bold 1 over denominator bold z bold plus bold 1 end fraction close vertical bar fraction numerator bold 1 over denominator bold vertical line bold z bold plus bold 1 bold vertical line to the power of bold 2 end fraction
  • 0

  • fraction numerator square root of bold 2 over denominator bold vertical line bold z bold plus bold 1 bold vertical line to the power of bold 2 end fraction
  • fraction numerator bold 1 over denominator bold vertical line bold z bold plus bold 1 bold vertical line to the power of bold 2 end fraction

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46.
વર્તુળ |z| = 2 માં અંતર્ગત સમબાજુ ત્રિકોણનાં શિરોબિંદુઓ z1, z2, z3 છે. z1 = 1 + square root of bold 3 bold space bold i જો તો z2, z3 અનુક્રમે ........ થાય. 
  • bold minus bold 2 bold comma bold space bold 2 bold space bold plus bold space square root of bold 3 bold space bold i
  • bold minus bold 2 bold comma bold space bold minus bold 1 bold space bold plus bold space square root of bold 3 bold space bold i
  • bold minus bold 2 bold comma bold space bold 1 bold space bold minus bold space square root of bold 3 bold space bold i
  • bold minus bold 1 bold space bold plus bold space square root of bold 3 bold space bold i bold comma bold space bold minus bold 2

C.

bold minus bold 2 bold comma bold space bold 1 bold space bold minus bold space square root of bold 3 bold space bold i

Tips: -

bold z subscript bold 1 bold space bold equals bold space bold 1 bold space bold plus bold space square root of bold 3 bold space bold i bold space bold equals bold space bold 2 bold e begin inline style bold iπ over bold 3 end style

bold m bold angle bold z subscript bold 1 bold o bold space bold z subscript bold 2 bold space bold equals bold space fraction numerator bold 2 bold pi over denominator bold 3 end fraction અને bold m bold angle bold z subscript bold 2 bold o bold space bold z subscript bold 3 bold space bold equals bold space fraction numerator bold 2 bold pi over denominator bold 3 end fraction

bold therefore bold space bold z subscript bold 2 bold space bold equals bold space bold z subscript bold 1 bold e to the power of begin inline style fraction numerator bold 2 bold πi over denominator bold 3 end fraction end style end exponent bold space bold space bold space bold z subscript bold 3 bold space bold equals bold space bold z subscript bold 2 bold space bold e end subscript to the power of begin inline style fraction numerator bold 2 bold πi over denominator bold 3 end fraction end style end exponent
આથી bold z subscript bold 2 bold space bold equals bold space bold 2 bold e to the power of bold iπ bold space bold equals bold space bold minus bold 2 bold spaceઅને bold z subscript bold 3 bold space bold equals bold space bold minus bold 2 bold e to the power of begin inline style fraction numerator bold 2 bold πi over denominator bold 3 end fraction end style end exponent
bold therefore bold space bold z subscript bold 3 bold space bold equals bold space bold minus bold 2 bold space bold left square bracket bold cos bold space open parentheses fraction numerator bold 2 bold pi over denominator bold 3 end fraction close parentheses bold space bold plus bold space bold i bold space bold sin bold space open parentheses fraction numerator bold 2 bold pi over denominator bold 3 end fraction close parentheses bold space bold right square bracket

           bold equals bold space bold minus bold space bold 2 bold space open parentheses fraction numerator bold minus bold 1 over denominator bold 2 end fraction bold plus bold i fraction numerator square root of bold 3 over denominator bold 2 end fraction close parentheses bold space bold equals bold space bold 1 bold space bold minus square root of bold 3 bold space bold i

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47. જો (cos θ + i sin θ) (cos2θ + i sin2θ) ... (cos n θ + i sin n θ) = 1 તો θ = ....... 
  • fraction numerator bold 4 bold kπ over denominator bold n bold left parenthesis bold n bold plus bold 1 bold right parenthesis end fraction bold comma bold space bold n bold space bold element of bold space bold Z
  • fraction numerator bold 2 bold pi over denominator bold 2 bold left parenthesis bold n bold left parenthesis bold n bold plus bold 1 bold right parenthesis end fraction bold comma bold space bold k bold space bold element of bold space bold Z
  • fraction numerator bold kπ over denominator bold n bold left parenthesis bold n bold plus bold 1 bold right parenthesis end fraction bold comma bold space bold k bold space bold element of bold space bold Z
  • આપેલ પૈકી એક પણ નહી 


48.
a, b, c, a1, b1, c1 એ શૂન્યેતર સંકર સંખ્યાઓ છે, જ્યાં bold a over bold a subscript bold 1 bold space bold plus bold space bold b over bold b subscript bold 1 bold space bold plus bold space bold c over bold c subscript bold 1 bold space bold equals bold space bold 1 અને bold a subscript bold 1 over bold a bold space bold plus bold space bold b subscript bold 1 over bold b bold space bold plus bold space bold c subscript bold 1 over bold c bold space bold equals bold space bold 0 bold comma તો bold a to the power of bold 2 over bold a subscript bold 1 to the power of bold 2 bold plus bold b to the power of bold 2 over bold b subscript bold 1 to the power of bold 2 bold plus bold c to the power of bold 2 over bold c subscript bold 1 to the power of bold 2 bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold space bold.
  • 2 + 2i

  • 2i

  • 2

  • 1 + 2i


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49.
ધારો કે z = x + iy જ્યાં, x, y ∈ Z, સમીકરણ bold z bold space bold z with bold bar on top to the power of bold 3 bold space bold plus bold space bold z with bold bar on top bold space bold z to the power of bold 3 bold space bold equals bold space bold 350 ના બીજથી રચાતા લંબચોરસનું ક્ષેત્રફળ ......
  • 68

  • 175

  • 350

  • 48


50. જો શુન્યેતર સંકર સંખ્યાઓ z1, z2 માટે bold z subscript bold 1 over bold z subscript bold 2 bold plus bold z subscript bold 2 over bold z subscript bold 1 bold space bold equals bold space bold 1 તો O(0), P(z1) તથા Q(z2) એ 
  • સમબાજુ ત્રિકોણ બનાવે. 

  • રેખા પરનાં બિંદુઓ છે. 

  • કાટકોણ ત્રિકોણનાં શિરોબિંદુઓ છે. 

  • એકમ વર્તુળ પર આવેલ છે.


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