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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

11.

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

12. 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

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)

cot y (1 - cosⁿ cosny)

Tips: -

ધારો કે cos y = a

S = a sin y + a² sin 2y + a³ sin 3y + ... n પદ

C = a cos y + a² cos 2y + a³ cos 3y + ... n પદ

∴ C + iS = ae^iy + a² e^i2y + a³ e^i3y + ... n ∴

$bold equals bold space fraction numerator bold ae to the power of bold iy bold space end exponent bold left square bracket bold 1 bold minus bold a to the power of bold n bold space bold e to the power of bold iny bold right square bracket over denominator bold 1 bold space bold minus bold space bold ae to the power of bold iy end fraction bold equals bold space fraction numerator bold ae to the power of bold iy bold space bold left square bracket bold space bold 1 bold space bold minus bold a to the power of bold n bold space bold e to the power of bold iny bold right square bracket bold space bold left square bracket bold 1 bold minus bold ae to the power of bold minus bold iy end exponent bold right square bracket bold space over denominator bold 1 bold space bold minus bold space bold a bold space bold left parenthesis bold e to the power of bold iy bold space bold plus bold space bold e to the power of bold minus bold iy end exponent bold right parenthesis bold space bold plus bold space bold a to the power of bold 2 bold space bold e to the power of bold iy bold space bold e to the power of bold minus bold iy end exponent end fraction bold equals bold space fraction numerator bold left parenthesis bold ae to the power of bold iy bold space bold minus bold space bold a to the power of bold 2 bold right parenthesis bold space bold left square bracket bold 1 bold space bold minus bold space bold a to the power of bold n bold space bold e to the power of bold iny bold right square bracket over denominator bold 1 bold space bold minus bold space bold a bold space bold left parenthesis bold 2 bold space bold cosy bold right parenthesis bold space bold plus bold space bold a to the power of bold 2 end fraction$

$bold therefore bold space bold C bold space bold plus bold space bold iS bold space bold equals bold space bold left square bracket bold cosy bold space bold left parenthesis bold cosy bold space bold plus bold space bold i bold space bold sin bold space bold y bold right parenthesis bold space bold minus bold space bold cos to the power of bold 2 bold y bold right square bracket bold space bold times bold space fraction numerator bold left square bracket bold 1 bold space bold minus bold space bold cos to the power of bold n bold space bold y bold space bold left parenthesis bold cosny bold space bold plus bold space bold i bold space bold sin bold space bold ny bold right square bracket over denominator bold 1 bold space bold minus bold 2 bold space bold cos to the power of bold 2 bold space bold plus bold space bold cos to the power of bold 2 bold space bold y end fraction$

(a = cos, y, તથા e^iθ = cos θ + i sin θ)

$bold equals bold space fraction numerator bold i bold space bold sin bold space bold y bold space bold cos bold space bold y over denominator bold sin to the power of bold 2 bold space bold y end fraction bold space bold left square bracket bold 1 bold space bold minus bold space bold cos to the power of bold n bold ycosny bold space bold minus bold space bold i bold space bold cos to the power of bold n bold space bold ysinny bold right square bracket$

S = કાલ્પનિક ભાગ

= cot y (1 - cosⁿy cosny)

14.

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

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 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 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|

18.

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 ?

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. 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)

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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

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