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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

51. જો $bold alpha bold space bold equals bold space bold cos bold space open parentheses fraction numerator bold 8 bold pi over denominator bold 11 end fraction close parentheses bold space bold plus bold space bold i bold space bold sin bold space open parentheses fraction numerator bold 8 bold pi over denominator bold 11 end fraction close parentheses$ તો, $bold Re bold space bold left parenthesis bold space bold alpha bold space bold plus bold space bold alpha to the power of bold 2 bold space bold plus bold space bold alpha to the power of bold 3 bold space bold plus bold space bold alpha to the power of bold 4 bold space bold plus bold space bold alpha to the power of bold 5 bold right parenthesis bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space bold.$

1
0
$bold minus bold 1 over bold 2$
$bold 1 over bold 2$

52. જો |z| = 1 અને arf (z) = તો arg $open parentheses fraction numerator bold 1 bold plus bold z over denominator bold 1 bold plus bold z end fraction close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space bold.$

θ
$bold pi bold minus bold theta$
-θ
$bold pi over bold 2 bold minus bold theta$

53. જો f(z) = z⁴ - 8z³ + 4z² + 4z + 39 અને f (3 + 2i) = a + ib તો a:b = .........

-1 : 8
1 : 8
1 : 4
-1 : 4

54. z અને w એ સંકર સંખ્યાઓ છે. જો $bold z with bold bar on top bold space bold plus bold space bold i bold space bold w with bold bar on top bold space bold equals bold space bold 0$ અને arg (zw) = $bold pi$ તો argz = ............ .

$fraction numerator bold 5 bold pi over denominator bold 4 end fraction$
$fraction numerator bold minus bold pi over denominator bold 4 end fraction$
$bold pi over bold 4$
$bold pi over 2$

55. $bold sum from bold k bold equals bold 1 to bold 6 of bold space open parentheses bold sin open parentheses fraction numerator bold 2 bold kπ over denominator bold 7 end fraction close parentheses bold plus bold space bold i bold space bold cos bold space open parentheses fraction numerator bold 2 bold kπ over denominator bold 7 end fraction close parentheses close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space bold.$

-i
1
0
-1

-i

Tips: -

$bold z bold space bold equals bold space bold sum from bold k bold equals bold 1 to bold 6 of open parentheses bold sin open parentheses fraction numerator bold 2 bold kπ over denominator bold 7 end fraction close parentheses bold plus bold i bold space bold cos bold space open parentheses fraction numerator bold 2 bold kπ over denominator bold 7 end fraction close parentheses close parentheses bold space bold equals bold space bold i bold space bold sum from bold k bold equals bold 1 to bold 6 of bold space open parentheses bold cos open parentheses fraction numerator bold 2 bold kπ over denominator bold 7 end fraction close parentheses bold minus bold i bold space bold sin bold space open parentheses fraction numerator bold 2 bold kπ over denominator bold 7 end fraction close parentheses close parentheses$

ધારો કે $bold theta bold space bold equals bold space fraction numerator bold 2 bold pi over denominator bold 7 end fraction$

$bold therefore bold space bold z bold space bold equals bold space bold i bold space bold sum from bold k bold equals bold 1 to bold 6 of bold space bold e to the power of bold minus bold ikθ end exponent bold space bold space bold space bold space bold space bold space bold equals bold space bold i bold space bold left square bracket bold e to the power of bold minus bold 1 bold theta end exponent bold space bold plus bold space bold e to the power of bold minus bold 2 bold theta end exponent bold plus bold space bold. bold. bold. bold space bold plus bold space bold plus bold space bold e to the power of bold minus bold i bold 6 bold theta end exponent bold right square bracket bold space bold space bold space bold space bold space bold space bold equals bold space bold ie to the power of bold iθ bold space open parentheses fraction numerator bold e to the power of bold minus bold i bold 6 bold theta end exponent bold minus bold 1 over denominator bold e to the power of bold minus bold theta end exponent bold minus bold 1 end fraction close parentheses bold space bold space bold space bold space bold space bold space bold equals bold space bold ie to the power of bold minus bold iθ end exponent bold space open parentheses fraction numerator bold cos bold space bold 6 bold theta bold space bold minus bold space bold i bold space bold sin bold space bold 6 bold theta bold space bold minus bold space bold 1 over denominator bold cos bold space bold theta bold space bold minus bold space bold i bold space bold sin bold space bold theta bold space bold minus bold space bold 1 end fraction close parentheses bold space bold space bold space bold space bold space bold space bold equals bold space bold ie to the power of bold minus bold iθ end exponent bold space open parentheses fraction numerator bold 1 bold minus bold cos bold space bold 6 bold space bold theta bold space bold plus bold space bold i bold space bold sin bold space bold 6 bold space bold theta over denominator bold 1 bold space bold minus bold space bold cos bold space bold theta bold space bold plus bold space bold i bold space bold sin bold space bold theta end fraction close parentheses$

bold equals bold space bold ie to the power of bold minus bold iθ end exponent bold space fraction numerator bold sin bold space bold 3 bold theta over denominator bold sin begin display style bold theta over bold 2 end style end fraction bold space open parentheses fraction numerator bold sin bold space bold 3 bold space bold theta bold plus bold space bold i bold space bold cos bold space bold 3 bold space bold theta over denominator bold sin begin display style bold theta over bold 2 end style bold plus bold space bold i bold space bold cos bold space begin display style bold theta over bold 2 end style end fraction close parentheses bold equals bold space bold ie to the power of bold minus bold iθ end exponent bold space fraction numerator bold sin bold space bold 3 bold theta over denominator bold sin begin display style bold theta over bold 3 end style end fraction bold space bold e to the power of bold minus bold 3 bold iθ end exponent bold space bold space bold e to the power of bold iθ over bold 2 end exponent bold space bold minus bold space bold ie to the power of fraction numerator bold minus bold 7 bold iθ over denominator bold 2 end fraction end exponent bold space fraction numerator bold sin bold space bold 3 bold theta over denominator bold sin bold space begin display style bold theta over bold 2 end style end fraction bold equals bold space bold ie to the power of bold minus bold pi end exponent bold space fraction numerator bold sin begin display style fraction numerator bold 6 bold pi over denominator bold 7 end fraction end style over denominator bold sin begin display style bold pi over bold 7 end style end fraction bold equals bold space bold i bold space bold left parenthesis bold cos bold space bold pi bold space bold minus bold space bold i bold space bold sin bold space bold pi bold right parenthesis bold space fraction numerator bold sin bold space open parentheses bold pi bold minus begin display style bold pi over bold 7 end style close parentheses over denominator bold sin begin display style bold pi over bold 7 end style end fraction bold space bold equals bold space bold minus bold i

56. z અને w એ શૂન્યેતર સંકર સંખ્યાઓ છે. જો |zw| = 1 અને argz - argw = $bold pi over bold 2$ તો $bold z with bold bar on top bold space bold w bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold space$

-1
-i
i
1

57.

bold sin bold space bold italic alpha bold space bold plus bold space bold italic s bold italic i bold italic n bold space bold italic beta bold space bold plus bold space bold italic s bold italic i bold italic n bold space bold italic gamma bold space bold equals bold space bold 0 bold space bold equals bold space bold italic c bold italic o bold italic s bold space bold italic alpha bold space bold plus bold space bold italic c bold italic o bold italic s bold space bold italic beta bold space bold plus bold space bold italic c bold italic o bold italic s bold space bold italic gamma bold space

જ્યાં

bold minus bold space bold pi bold space bold less than bold space bold alpha bold space bold comma bold space bold beta bold comma bold space bold gamma bold space bold less or equal than bold space bold pi

bold જ ો bold space bold sin to the power of bold 2 bold space bold alpha bold space bold plus bold space bold sin to the power of bold 2 bold space bold beta bold space bold plus bold space bold sin to the power of bold n bold space bold gamma bold space bold equals bold space bold 3 over bold 2 bold space bold તથ ા bold space bold sum bold cos bold space bold left parenthesis bold alpha bold space bold plus bold space bold beta bold right parenthesis bold space bold equals bold space bold lambda bold space bold ત ો bold comma bold space bold left parenthesis bold n bold comma bold space bold lambda bold right parenthesis bold space bold equals bold space bold. bold. bold. bold. bold. bold space

(2, 3)
(0, 3)
(2, 0)
(2, 3)

58.

જો સંકર સંખ્યા $bold alpha bold space bold left parenthesis bold space bold alpha bold space bold not equal to bold space bold 1 bold right parenthesis$ એ સમીકરણ $bold z to the power of bold p bold plus bold q end exponent bold space bold minus bold space bold z to the power of bold minus bold space bold minus bold space bold z to the power of bold q bold space bold plus bold space bold 1 bold space bold equals bold space bold 0$ જ્યાં p, q ભિન્ન અવિભાજ્ય સંખ્યા છે, નું બીજ હોય તો

1 + α + α² + ... + α^p = 0
1 + α + α² + ... + α^q = 0 અને 1 + α + a² + ... + a^q-1 = 0
1 + α + a² + ... α^p-1 = 0 અને 1 + α + a² + ... + a^q-1 = 0
પરંતુ બંને એક સાથે ન ઉદભવે

59. જો |z| = 1 અને z ≠ ± 1 તો કોઈ પણ સંકર સંખ્યા z માટે $fraction numerator bold z over denominator bold 1 bold space bold minus bold space bold z to the power of bold 2 end fraction$ એ = ....... .

Y-અક્ષ પર હોય.
ઉંગમબિંદુમાંથી પસાર ન થતી રેખા પર હોય.
X-અક્ષ પર હોય.
$bold vertical line bold z bold vertical line bold space bold equals bold space square root of bold 2$

60. જો સંકર સંખ્યા z માટે |z - 3 - 2i| ≤ 2, તો |2z - 6 + 5i| નું મહત્વ મૂલ્ય ....... છે.

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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

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