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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

61. સંકર સંખ્યા z માટે 4 arg(z-5i) = 2arg(z+3)=cos^-1(-1) તો

$bold z with bold bar on top bold space bold equals bold space bold 2 bold italic i bold space bold plus bold space bold 3$
z એ x+y+1 = 0 પર છે.
|z| એ x² + y²+ 2x - 4y - 8 = 0 ની ત્રિજ્યા છે.
$bold z bold space bold plus bold space bold z with bold bar on top bold space bold less than bold space bold 0$

62.

P-ii, Q-iii, R-i, S-iv
P-i, Q-ii, R-iii, S-iv
P-iv, Q-iii, R-ii, S-i
P-ii, Q-i, R-iv, S-iiii

63. $bold sum from bold p bold equals bold 1 to bold 32 of bold left parenthesis bold 3 bold p bold plus bold 2 bold right parenthesis bold space open parentheses bold sum from bold q bold equals bold 1 to bold 10 of open parentheses bold sin fraction numerator bold 2 bold qπ over denominator bold 11 end fraction bold minus bold space bold i bold space bold cos bold space fraction numerator bold 2 bold qπ over denominator bold 11 end fraction close parentheses close parentheses to the power of bold p bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space bold.$

48 (1+i)
24(1-i)
48(1-i)
24(1+i)

64.

P-i, Q-ii, R-iii, S-iv
P-ii, Q-ii, R-iv, S-iii
P-iv, Q-iii, R-ii, S-i
P-iv, Q-i, R-ii, S-iii

65.

P-iii, Q-ii, R-iv, S-i
P-i, Q-ii, R-iii, S-iv
P-ii, Q-iii, R-iv, S-i
P-iv, Q-iii, R-ii, S-i

P-iii, Q-ii, R-iv, S-i

Tips: -

$bold left parenthesis bold P bold right parenthesis bold space fraction numerator bold 1 bold minus bold i over denominator bold 1 bold plus bold i end fraction bold equals bold i$

   $bold space bold therefore bold space open parentheses fraction numerator bold 1 bold minus bold i over denominator bold 1 bold plus bold i end fraction close parentheses to the power of bold n bold space bold equals bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold space bold space bold space bold space bold space bold space bold space bold n bold space bold equals bold space bold 1 bold comma bold space bold 3 bold comma bold space bold 5 bold comma bold space bold. bold. bold.$ શુદ્વ કાલ્પનિક સંખ્યા મળે.

n = 3 માટે $open parentheses fraction numerator bold 1 bold minus bold i over denominator bold 1 bold plus bold i end fraction close parentheses to the power of bold n$ નો શુદ્વ કાલ્પનિક ભાગ ધન સંખ્યા મળે. આથી P-iii
(અહીં P-iii વાળો એક જ વિક્લ્પ હોવાથી જવાબ નક્કી થઈ જાય.)

$bold Q bold space open parentheses bold sum from bold m bold equals bold 1 to bold 2 bold n bold plus bold 1 of bold i to the power of bold 2 bold m end exponent close parentheses bold equals bold space bold left curly bracket bold left parenthesis bold minus bold 1 bold right parenthesis bold space bold plus bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold 2 bold space bold plus bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold 3 bold space bold plus bold space bold. bold. bold. bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold 2 bold n bold plus bold 1 end exponent bold right curly bracket bold space bold equals bold space bold left parenthesis bold minus bold 1 bold right parenthesis bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold Q bold minus bold ii bold left parenthesis bold R bold right parenthesis bold space bold 1 bold plus bold space bold e to the power of bold iA bold space bold e to the power of bold iB bold space bold e to the power of bold iC bold equals bold 1 bold space bold plus bold space bold cos bold space bold left parenthesis bold A bold space bold plus bold space bold B bold space bold plus bold space bold C bold right parenthesis bold space bold plus bold space bold space bold i bold space bold sin bold space bold left parenthesis bold A bold space bold plus bold space bold B bold space bold plus bold space bold C bold right parenthesis bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold R bold minus bold iv bold equals bold space bold 1 bold space bold space bold plus bold space bold cos bold space bold pi bold space bold plus bold space bold i bold space bold sin bold space bold pi bold space bold equals bold space bold 0 bold space$

(S) ધારો કે $bold z bold space bold equals bold space bold r bold space bold left parenthesis bold cos bold space bold theta bold space bold plus bold space bold i bold space bold sin bold space bold theta bold right parenthesis$

$fraction numerator bold z over denominator bold vertical line bold z bold vertical line end fraction bold space bold equals bold space bold left parenthesis bold cos bold space bold theta bold space bold plus bold space bold i bold space bold sin bold space bold theta bold right parenthesis bold therefore bold space open vertical bar fraction numerator bold z over denominator bold vertical line bold z bold vertical line end fraction bold minus bold 1 close vertical bar bold space bold equals bold space bold vertical line bold left parenthesis bold cos bold space bold theta bold space bold minus bold space bold 1 bold space bold plus bold space bold i bold space bold sin bold space bold theta bold space bold vertical line bold space$

   $bold vertical line bold equals bold space bold vertical line bold minus bold 2 bold sin to the power of bold 2 bold space bold theta over bold 2 bold space bold plus bold space bold 2 bold space bold i bold space bold sin bold space bold theta over bold 2 bold space bold cos bold space bold theta bold 2 bold space bold vertical line bold space bold equals bold space bold 2 bold space bold vertical line bold sin bold theta over bold 2 bold vertical line$

   $bold therefore bold space open vertical bar fraction numerator bold z over denominator bold vertical line bold z bold vertical line end fraction bold minus bold 1 close vertical bar$ નું મહત્તમ મૂલ્ય થાય. S-i

66. જો સંકર સંખ્યા z માટે $bold arg bold space open parentheses fraction numerator bold z bold minus bold 2 over denominator bold z bold plus bold 2 end fraction close parentheses bold equals bold space bold pi over bold 4$ તો |z-2i| ની કિંમત = ......

$bold 3 square root of bold 2$
$2 square root of bold 2$
$4 square root of bold 2$
$5 square root of bold 2$

67. જો z₁, z₂, .... z_n એ વર્તુળ |z| = r પર આવેલ હોય તથા હોય તો, $bold Re bold space open parentheses bold sum from bold j bold equals bold 1 to bold n of bold space bold sum from bold k bold equals bold 1 to bold n of bold space bold sum from bold z subscript bold k to bold z bold space bold j of close parentheses bold space bold equals bold space bold 0$ હોય તો,

$bold sum from bold j bold equals bold 1 to bold n of bold space bold z subscript bold j bold space bold equals bold space bold 0$
$open vertical bar bold sum from bold j bold equals bold 1 to bold n of bold space bold z subscript bold j close vertical bar bold space bold equals bold 0$
$bold sum from bold j bold equals bold 1 to bold n of bold space bold z with bold bar on top bold space bold j bold space bold equals bold space bold 0$
આપેલ પૈકી એક પણ નહી

68. P(x) અને Q (x) એ બહુપદી છે. ધારો કે x² + x + 1 એ f(x) = P(x³) + xQ(x³) નો અવયવ છે, તો

(x-1)એ P(x) તથા Q(x) બંનેનો અવયવ હોય.
(x-1)એ f(x)નો અવયવ હોય.
(x-1) એ P(x) નો અવયવ હોય પરંતુ Q(x) નો ન હોય.
(x-1)એ Q(x) નો અવયવ હોય પરંતુ P(x)નો ન હોય.

69. જો z₁, z₂ બે શુન્યેતર સંકર સંખ્યાઓ છે, જ્યાં |2z₁- 3z₂|² = |2z₁|² + |3z₂|² તો

$bold arg bold space open parentheses bold z subscript bold 2 over bold z subscript bold 1 close parentheses bold space bold equals bold space bold plus-or-minus bold space bold pi over bold 2$
$bold z subscript bold 1 bold space stack bold z subscript bold 2 with bold bar on top bold space bold plus bold space bold z subscript bold 2 bold space stack bold z subscript bold 1 with bold bar on top bold space bold equals bold space bold 0$
$bold z subscript bold 2 over bold z subscript bold 1$ એ વાસ્તવિક સંખ્યા હોય.
$bold z subscript bold 2 over bold z subscript bold 1$ એ શુદ્વ કાલ્પનિક સંખ્યા છે.

70. ધારો કે $bold z subscript bold k bold space bold equals bold space bold cos bold space open parentheses fraction numerator bold 2 bold k bold space bold pi over denominator bold 10 end fraction close parentheses bold space bold plus bold space bold i bold space bold sin bold space open parentheses fraction numerator bold 2 bold k bold space bold pi over denominator bold 10 end fraction close parentheses bold comma bold space bold k bold space bold equals bold space bold 1 bold comma bold space bold 2 bold comma bold space bold. bold. bold. bold space bold 9$

P-1, Q-2, R-4, S-3
P-2, Q-1, R-3, S-4
P-2, Q-1, R-4, S-3
P-1, Q-2, R-3, S-4

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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

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