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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

51. 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

52. 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 = ............ .

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

53. 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
$bold pi over bold 2 bold minus bold theta$

54.

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

55. 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.
$bold vertical line bold z bold vertical line bold space bold equals bold space square root of bold 2$

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)

(2, 0)

Tips: -

$bold sum bold cos bold space bold alpha bold space bold equals bold space bold 0 bold space bold sum bold sin bold space bold alpha bold comma bold space bold જ ્ ય ાં bold space bold minus bold pi bold space bold less than bold space bold alpha bold comma bold space bold beta bold comma bold space bold gamma bold space bold less or equal than bold space bold pi$

ધારો કે તો, $bold A bold space bold equals bold space bold e to the power of bold iα bold comma bold space bold B bold space bold equals bold space bold e to the power of bold iβ bold comma bold space bold C bold space bold equals bold space bold e to the power of bold iγ$ તો
A + B + C = 0 અને $bold 1 over bold A bold space bold plus bold space bold 1 over bold B bold space bold plus bold space bold 1 over bold C bold space bold equals bold space bold 0$

$bold therefore bold space bold A bold space bold plus bold space bold B bold space bold plus bold space bold C bold space bold equals bold space bold 0 bold space bold અન ે bold space bold AB bold space bold plus bold space bold BC bold space bold plus bold space bold CA bold space bold equals bold space bold 0 bold space bold therefore bold space bold A to the power of bold 2 bold space bold plus bold space bold B to the power of bold 2 bold space bold plus bold space bold C to the power of bold 2 bold space bold equals bold space bold 0 bold space bold અન ે bold space bold AB bold space bold plus bold space bold BC bold space bold plus bold space bold Ca bold space bold equals bold space bold 0 bold therefore bold space bold e to the power of bold i bold 2 bold alpha end exponent bold space bold plus bold space bold e to the power of bold i bold 2 bold beta end exponent bold space bold plus bold space bold e to the power of bold i bold 2 bold gamma end exponent bold space bold equals bold space bold 0 bold space bold અન ે bold space bold e to the power of bold i bold left parenthesis bold alpha bold plus bold beta bold right parenthesis end exponent bold space bold plus bold space bold e to the power of bold i bold left parenthesis bold gamma bold plus bold alpha bold right parenthesis end exponent bold space bold plus bold space bold e bold space bold equals bold space bold 0 bold therefore bold space bold cos bold space bold 2 bold alpha bold space bold plus bold space bold cos bold space bold 2 bold beta bold space bold plus bold space bold cos bold space bold 2 bold gamma bold space bold equals bold space bold 0 bold space bold અન ે bold space bold space bold cos bold space bold left parenthesis bold alpha bold space bold plus bold space bold beta bold right parenthesis bold space bold plus bold space bold cos bold space bold left parenthesis bold beta bold space bold plus bold space bold gamma bold right parenthesis bold space bold plus bold space bold cos bold space bold left parenthesis bold gamma bold space bold plus bold space bold alpha bold right parenthesis bold space bold equals bold space bold 0$ (વાસ્તવિક ભાગ સરખાવતાં)

$bold therefore bold space bold 1 bold space bold minus bold space bold 2 bold space bold sin to the power of bold 2 bold space bold alpha bold space bold plus bold space bold 1 bold space bold minus bold space bold 2 bold space bold sin to the power of bold 2 bold space bold beta bold space bold plus bold space bold 1 bold space bold minus bold space bold 2 bold space bold sin to the power of bold 2 bold space bold gamma bold space bold equals bold space bold 0 bold space bold therefore bold space bold sin to the power of bold 2 bold space bold alpha bold space bold plus bold space bold sin to the power of bold 2 bold beta bold space bold plus bold space bold sin to the power of bold 2 bold gamma bold space bold space bold equals bold space bold 3 over bold 2 bold તથ ા bold space bold sum bold cos bold left parenthesis bold alpha bold plus bold beta bold right parenthesis bold space bold equals bold space bold 0$

$bold therefore bold space bold n bold space bold equals bold space bold 2 bold comma bold space bold lambda bold space bold equals bold space bold 0 bold.$ આથી $bold left parenthesis bold n bold comma bold space bold lambda bold right parenthesis bold space bold equals bold space bold left parenthesis bold 2 bold comma bold space bold 0 bold right parenthesis$

58. 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
$bold minus bold 1 over bold 2$
$bold 1 over bold 2$

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

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

-i
1
0
-1

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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

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