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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

11.

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 ?

12. 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.

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. 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

bold z bold w with bold minus on top bold space bold equals bold space bold 1

Tips: -

$bold vertical line bold z bold vertical line to the power of bold 2 bold space bold w bold space bold minus bold space bold vertical line bold w bold vertical line to the power of bold 2 bold space bold z bold space bold equals bold space bold minus bold w$ ની પુન:ગોઠવણી કરતાં,
$bold z over bold w bold space bold equals bold space fraction numerator bold 1 bold plus bold vertical line bold z bold vertical line to the power of bold 2 over denominator bold 1 bold plus bold vertical line bold w bold vertical line to the power of bold 2 end fraction$ ... (1)

∴ $bold z over bold w$ એ વાસ્તવિક સંખ્યા થશે. ધારો કે $bold z over bold w bold space bold equals bold space bold k$ જ્યાં k ∈ R

$bold therefore bold space bold kw over bold w bold space bold equals bold space fraction numerator bold 1 bold plus bold k to the power of bold 2 bold vertical line bold w bold vertical line to the power of bold 2 over denominator bold 1 bold plus bold vertical line bold w bold vertical line to the power of bold 2 end fraction$ ((1)પરથી)

$bold therefore bold space bold k bold space bold equals bold space bold k bold vertical line bold w bold vertical line to the power of bold 2 bold space bold left parenthesis bold k bold minus bold 1 bold right parenthesis bold space bold therefore bold space bold left parenthesis bold k bold vertical line bold w bold vertical line to the power of bold 2 bold minus bold 1 bold right parenthesis bold left parenthesis bold k bold minus bold 1 bold right parenthesis bold space bold equals bold space bold 0$

∴ k = 1 અથવા $bold k bold space bold equals bold space fraction numerator bold 1 over denominator bold vertical line bold w bold vertical line to the power of bold 2 end fraction$

∴ $bold z over bold w$ = 1 અથવા $bold z over bold w bold space bold equals bold space fraction numerator bold 1 over denominator bold vertical line bold w bold vertical line to the power of bold 2 end fraction bold space bold equals bold space fraction numerator bold 1 over denominator bold w begin display style bold w with bold minus on top end style end fraction$

$bold z bold space bold w with bold minus on top bold space bold space bold equals bold space bold space bold 1 bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold left parenthesis bold z bold space bold not equal to bold w bold comma bold space bold w bold not equal to bold space bold o bold right parenthesis$

15. 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)

16.

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

17.

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

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

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

19. 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|

20.

$open vertical bar fraction numerator bold z bold minus bold 12 over denominator bold z bold minus bold 8 bold i end fraction close vertical bar bold space bold equals bold space bold 5 over bold 3$ 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.

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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

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