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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.

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

14.

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

15. 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)

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

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

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)

Tips: -

$open vertical bar bold z bold minus bold 4 over bold z close vertical bar bold space bold greater or equal than bold space open vertical bar bold vertical line bold z bold vertical line bold space bold minus fraction numerator bold 4 over denominator bold vertical line bold z bold vertical line end fraction close vertical bar bold therefore bold space open vertical bar bold vertical line bold z bold vertical line bold space bold minus fraction numerator bold 4 over denominator bold vertical line bold z bold vertical line end fraction close vertical bar bold space bold less or equal than bold space open vertical bar bold z bold minus bold 4 over bold 2 close vertical bar bold therefore bold space open vertical bar bold vertical line bold z bold vertical line bold minus fraction numerator bold 4 over denominator bold vertical line bold z bold vertical line end fraction close vertical bar bold space bold less or equal than bold space bold 2 bold therefore bold space bold minus bold 2 bold space bold less or equal than bold space bold vertical line bold italic z bold vertical line bold space bold minus fraction numerator bold 4 over denominator bold vertical line bold z bold vertical line end fraction bold space bold less or equal than bold space bold 2$

∴ -2 ≤ |z| - 4 ≥ 0 અને |z|² - 2|z| - 4 ≤0

(|z| + 1)² ≥ 5 તથા (|z|-1)² ≤ 5

∴ |z|+1 ≥ $square root of bold 5$ અથવા -(|z|-1) ≥ $square root of bold 5$ તથા $bold minus square root of bold 5 bold space bold less or equal than bold space bold vertical line bold z bold vertical line bold space bold minus bold 1 bold space bold less or equal than bold space square root of bold 5$

∴ |z| ≥ $square root of bold 5$ -1 અથવા -|z| ≥ $square root of bold 5 bold space bold plus bold space bold 1$ તથા 1 - $square root of bold 5 bold space bold less or equal than bold space bold vertical line bold italic z bold vertical line bold space bold less or equal than bold space square root of bold 5 bold space bold plus bold space bold 1$

$square root of bold 5 bold space bold minus bold space bold 1 bold space bold less or equal than bold space bold vertical line bold z bold vertical line bold space bold less or equal than bold space square root of bold 5 bold space bold plus bold space bold 1$

∴ મહત્તમ |z| - ન્યુનતમ |z| = $bold left parenthesis bold space square root of bold 5 bold space bold plus bold space bold 1 bold right parenthesis bold space bold minus bold space bold left parenthesis square root of bold 5 bold space bold minus bold space bold 1 bold right parenthesis bold space bold equals bold space bold 2$

∴ |z| નાં મહત્તમ તથા ન્યુનતમ મૂલ્યો વચ્ચેનો તફાવત 2 થશે.

18. 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|

19. 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.

20.

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

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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

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