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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

11. જો z એ વાસ્તવિક ન હોય તેવી સંકર સંખ્યા વર્તુળ |z| = 1 પર આવેલ છે, તો z = ...... .

$fraction numerator 1 plus itan left parenthesis arg space straight z right parenthesis over denominator 1 minus itan space left parenthesis arg space straight z right parenthesis end fraction$
$fraction numerator 1 space minus space itan space left parenthesis arg space straight z right parenthesis over denominator 1 space plus space itan space left parenthesis arg space straight z right parenthesis end fraction$
$fraction numerator bold 1 bold plus bold itan bold space open parentheses begin display style fraction numerator bold arg bold space bold z over denominator bold 2 end fraction end style close parentheses over denominator bold space bold 1 bold space bold minus bold space bold itan bold space open parentheses begin display style fraction numerator bold arg bold space bold z over denominator bold 2 end fraction end style close parentheses end fraction$
$fraction numerator bold 1 bold minus bold itan bold space open parentheses begin display style fraction numerator bold arg bold space bold z over denominator bold 2 end fraction end style close parentheses over denominator bold space bold 1 bold space bold plus bold space bold itan bold space open parentheses begin display style fraction numerator bold arg bold space bold z over denominator bold 2 end fraction end style close parentheses end fraction$

12.

$open vertical bar fraction numerator bold z bold minus bold 12 over denominator bold z bold minus bold 8 bold i end fraction close vertical bar bold space bold equals bold space bold 5 over bold 3$ અને $open vertical bar fraction numerator bold z bold minus bold 4 over denominator bold z bold minus bold 8 end fraction close vertical bar bold space bold equals bold space bold 1$ બંને શરતનું પાલન કરતી બધી સંકર સંખ્યાઓના કાલ્પનિક ભાગનો સરવાળો ....... થાય.

13. cos y sin y + cos² y sin 2y + cos³y sin3y + ...... n પદ ......

tan y (1 - cosⁿ y cosny)
cot y (1 - cosⁿ cosny)
cot y (1 - sinⁿ y sinny)
tan y (1 - sinⁿ y sin n y)

14.

bold z with bold minus on top bold space bold equals bold space bold italic z to the power of bold 2

શરતનું પાલન કરતી કેટલી સંકર સંખ્યાઓ મળે ?

15.

જો w એ 1 નું ઘનમૂળ હોય તો 2 (1 + w) (1 + w²) + 3 (2w + 1) + ... + (n+1) (nw²+1) = ......... (w ≠1)

$fraction numerator bold n to the power of bold 2 bold left parenthesis bold n bold plus bold 1 bold right parenthesis to the power of bold 2 over denominator bold 4 end fraction$ +n
$fraction numerator bold n to the power of bold 2 bold left parenthesis bold n bold plus bold 1 bold right parenthesis to the power of bold 2 over denominator bold 4 end fraction$
$fraction numerator bold n to the power of bold 2 bold left parenthesis bold n bold plus bold 1 bold right parenthesis to the power of bold 2 over denominator bold 4 end fraction bold minus bold n$
$fraction numerator bold n to the power of bold 2 bold left parenthesis bold n bold plus bold 1 bold right parenthesis to the power of bold 2 over denominator bold 2 end fraction bold plus bold n$

16. જો z એ સંકર સંખ્યા હોય તથા $bold vertical line bold z bold vertical line bold space bold greater or equal than bold space bold 2 bold space bold ત ો bold space open vertical bar bold z bold plus bold 1 over bold 2 close vertical bar$ તો ની ન્યુનતમ કિંમત

અંતરાલ (1, 2) માં છે,
5/2 થી વધુ હોય.
3/2 થી વધુ તથા 5/2 થી ઓછી હોય.
5/2 હોય.

17. જો $open vertical bar bold z bold minus bold 4 over bold z close vertical bar bold space bold equals bold space bold 2$ હોય, તો |z| નાં મહત્તમ તથા ન્યુનતમ મૂલ્યો વચ્ચેનો તફાવત ......... છે. (z≠0)

18.

વાસ્તવિક સહગુણકવાળી બહુપદી f(x) = x⁴ + ax³ + bx³ + cx + d માટે f(2i) = f(2+i) = 0 હોય તો a + b + c + d = .......

19. શૂન્યેતર ભિન્ન સંકર સંખ્યાઓ z અને w માટે જો |z|² w-|w|² z = z - w તો ......

$bold zw bold space bold equals bold space bold 1$
$bold z bold w with bold minus on top bold space bold equals bold space bold 1$
$bold z bold space bold equals bold space bold w with bold minus on top$
z = -w

20. જો z ≠ 0, |zⁱ| ની મહત્તમ સીમા ........ થશે.

$bold e to the power of bold pi$
$bold e to the power of bold minus bold pi end exponent$
1
|z|

bold e to the power of bold pi

Tips: -

ધારો કે, $bold z bold space bold equals bold space bold re to the power of bold iθ bold comma bold space bold minus bold pi bold space bold less than bold space bold theta bold space bold pi bold space bold. bold space$

$bold therefore bold space bold z to the power of bold i bold space bold equals bold space bold r to the power of bold i bold space bold e to the power of bold minus bold theta end exponent bold space bold equals bold space bold e to the power of bold log bold space bold r to the power of bold i end exponent bold space bold. bold space bold e to the power of bold minus bold theta end exponent bold equals bold space bold e to the power of bold left parenthesis bold log bold space bold r bold space bold minus bold theta bold right parenthesis end exponent bold equals bold space bold left parenthesis bold cos bold left parenthesis bold log bold space bold r bold right parenthesis bold space bold plus bold space bold i bold space bold sin bold space bold left parenthesis bold log bold space bold r bold right parenthesis bold right parenthesis bold space bold eθ bold therefore bold space bold left parenthesis bold cos bold space bold left parenthesis bold log bold space bold r bold right parenthesis bold space bold plus bold space bold i bold space bold sin bold space bold left parenthesis bold log bold space bold r bold right parenthesis bold right parenthesis bold space bold e to the power of bold minus bold theta end exponent bold therefore bold space bold vertical line bold z to the power of bold i bold vertical line bold space bold equals bold space bold vertical line bold e to the power of bold minus bold theta end exponent bold vertical line bold space bold less or equal than bold space bold e to the power of bold pi bold space bold space bold space bold space bold space bold space bold space bold space bold space bold minus bold pi bold space bold less than bold theta bold space bold less or equal than bold space bold pi bold comma bold space bold minus bold pi bold space bold less than bold space bold minus bold theta bold space bold less or equal than bold space bold pi$

∴ |zⁱ| ની મહત્તમ સીમા $bold e to the power of bold pi$ થાય.

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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

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