ધારો કે z = x + iy જ્યાં, x, y ∈ Z, સમીકરણ  ના બીજથી રચાતા લંબચોરસનું ક્ષેત્રફળ ...... from Mathematics સંકર સંખ્યાઓ

Book Store

Download books and chapters from book store.
Currently only available for.
CBSE

Subject

Mathematics
Advertisement
zigya logo

Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

41. 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 


42.
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

43. 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

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


Advertisement
45.
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


46.
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

47. 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


48. 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.


Advertisement
Advertisement
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


D.

48

Tips: -

આપેલ 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 equals bold space bold 350 એ ચાર ઘાતવાળું છે. આથી તેને ચાર બીજ મળે. 

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 equals bold space bold 350

bold therefore bold space bold vertical line bold z bold vertical line to the power of bold 2 bold space bold z with bold bar on top to the power of bold 2 bold space bold plus bold space bold vertical line bold z bold vertical line to the power of bold 2 bold space bold z to the power of bold 2 bold space bold equals bold space bold 350 bold space bold space bold space bold space bold space bold space bold space bold space bold left parenthesis bold z bold space bold z with bold bar on top bold space bold equals bold space bold vertical line bold z bold vertical line to the power of bold 2 bold right parenthesis

bold therefore bold space bold vertical line bold z bold vertical line to the power of bold 2 bold space bold left parenthesis top enclose bold z to the power of bold 2 bold plus bold z to the power of bold 2 bold right parenthesis bold space bold equals bold space bold 350

bold therefore bold space bold left parenthesis bold x to the power of bold 2 bold space bold plus bold space bold y to the power of bold 2 bold right parenthesis bold space bold right square bracket bold left parenthesis bold x bold minus bold iy bold right parenthesis to the power of bold 2 bold space bold plus bold space bold left parenthesis bold x bold plus bold iy bold right parenthesis to the power of bold 2 bold right square bracket bold space bold equals bold space bold 350

bold therefore bold space bold left parenthesis bold x to the power of bold 2 bold space bold plus bold space bold y to the power of bold 2 bold right parenthesis bold space bold left parenthesis bold 2 bold x to the power of bold 2 bold minus bold 2 bold y to the power of bold 2 bold right parenthesis bold space bold equals bold space bold 350

bold therefore bold space bold left parenthesis bold x to the power of bold 2 bold plus bold y to the power of bold 2 bold right parenthesis bold thin space bold left parenthesis bold x to the power of bold 2 bold minus bold y to the power of bold 2 bold right parenthesis bold space bold equals bold space bold 175

x અને y પૂર્ણાંક સંખ્યા હોવાથી x2 + y2 અને x2 - y2 પણ પૂર્ણાંક સંખ્યા થશે.
 
175 = 25 × 7 = (42 + 32) લખી શકાય. 
bold therefore bold space bold x to the power of bold 2 bold space bold equals bold space bold 16 bold comma bold space bold y to the power of bold 2 bold space bold equals bold space bold 9

bold therefore bold space bold x bold space bold equals bold space bold plus-or-minus bold space bold 4 bold comma bold space bold y bold space bold equals bold space bold plus-or-minus bold space bold 3

∴ લંબચોરસનાં ચાર શિરોબિંદુ (4, 3), (-4, 3), (4, -3) થશે. 

તેનું ક્ષેત્રફળ = 8 × 6 = 48 થાય.

Advertisement
50.
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


Advertisement

Switch