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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

1. જો, $open vertical bar fraction numerator bold 1 bold minus bold minus bold iz over denominator bold z bold minus bold i end fraction close vertical bar bold space bold equals bold space bold 1 bold space$ તો.......

z એ શુદ્વ કાલ્પનિક સંખ્યા હોય.
P(z) એ બીજા ચરણમાં હોય.
P(z) એ ત્રીજા ચરણમાં હોય.
z એ વાસ્તવિક સંખ્યા હોય.

2. જો |z| = 1 અને z²ⁿ + 1 ≠ 0 તો $fraction numerator bold z to the power of bold n over denominator bold z to the power of bold 2 bold n end exponent bold space bold plus bold space bold 1 end fraction bold minus bold space fraction numerator bold x to the power of bold minus bold n end exponent over denominator open parentheses begin display style bold z with bold minus on top end style close parentheses to the power of bold 2 bold n end exponent bold space bold plus bold space bold 1 end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold space bold.$

3. જો x² + x + 1 = 0 નાં બે બીજ a અને b હોય, તો જેનાં બીજ a¹⁹ અને b⁷ હોય, તેવું સમીકરણ = .........

x² + x + 1 = 0
x² + x - 1 = 0
x² - x - 1 = 0
x² - x + 1 = 0

4. જો ચતુર્ઘાત સમીકરણ x⁴ + ax³ + bx² + cx + d = 0 (a, b, c, d ∈ R) નું કોઈપણ બીજ વાસ્તવિક સંખ્યા ન હોય તથા બે બીજનો સરવાળો 3 + 4 $bold i with bold hat on top$ હોય અને બાકીનાં બે બીજનો ગુણાકાર 13 + i હોય, તો b = ......... .

5. જો $bold x open parentheses bold minus bold space bold x square root of bold 3 close parentheses bold space bold plus bold space bold 1 bold space bold equals bold space bold 0 bold comma bold space$ તો $bold sum from bold n bold equals bold 1 to bold 36 of open parentheses bold x to the power of bold n bold minus bold 1 over bold x to the power of bold n close parentheses to the power of bold 2$ = ........... .

Tips: -

$bold x to the power of bold 3 bold space bold minus bold space bold x square root of bold 3 bold space bold plus bold space bold 1 bold space bold equals bold space bold 0$

$bold x bold space bold equals bold space fraction numerator square root of bold 3 bold space bold plus-or-minus bold space square root of bold 3 bold minus bold 4 end root over denominator bold 2 end fraction bold space bold equals bold space fraction numerator square root of bold 3 bold plus-or-minus bold i over denominator bold 2 end fraction$

$bold therefore bold space bold x bold space bold equals bold space fraction numerator square root of bold 3 over denominator bold 2 end fraction bold space bold plus bold space bold i over bold 2$ લેતાં, $bold x bold space bold equals bold space bold cos bold space bold pi over bold 6 bold space bold plus bold space bold i bold space bold sin bold space bold pi over bold 6$

$bold therefore bold space bold x to the power of bold 2 bold n end exponent bold space bold equals bold space bold cos bold space bold nπ over bold 3 bold space bold plus bold space bold i bold space bold sin bold space bold nπ over bold 3$ તથા $bold x to the power of bold 2 bold n end exponent bold equals bold space bold cos bold space bold nπ over bold 3 bold space bold minus bold space bold i bold space bold sin bold space bold nπ over bold 3$

$bold therefore bold space open parentheses bold x to the power of bold n bold minus bold 1 over bold x to the power of bold n close parentheses bold space bold equals bold space bold 2 to the power of bold 2 bold n end exponent bold space bold plus bold space bold 1 over bold x to the power of bold 2 bold n end exponent bold space bold minus bold space bold 2 bold space$ $bold equals bold space bold x to the power of bold 2 bold n end exponent bold space bold plus bold space bold c to the power of bold minus bold 2 bold n end exponent bold space bold minus bold space bold 2 bold space bold equals bold space bold 2 bold space bold cos bold space bold nπ over bold 3 bold minus bold 2$

હવે, $bold sum from bold n bold equals bold 1 to bold 36 of bold space open parentheses bold x to the power of bold n bold minus bold 1 over bold x to the power of bold n close parentheses to the power of bold 2 bold space bold equals bold space bold sum from bold n bold equals bold 1 to bold 36 of bold space open parentheses bold minus bold 2 bold plus bold 2 bold cos bold nπ over bold 3 close parentheses$

$bold equals bold space bold minus bold 2 bold space bold cross times bold space bold 36 bold space bold plus bold space bold 2 bold space open square brackets bold cos bold pi over bold 3 bold plus bold space bold cos bold space fraction numerator bold 2 bold pi over denominator bold 3 end fraction bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold cos bold space fraction numerator bold 36 bold pi over denominator bold 3 end fraction close square brackets bold equals bold space bold minus bold 72 bold space bold plus bold space bold 2 bold space fraction numerator open square brackets bold cos open parentheses begin display style bold pi over bold 3 end style bold plus begin display style fraction numerator bold 35 bold pi over denominator bold 6 end fraction end style close parentheses bold space bold sin bold space open parentheses begin display style fraction numerator bold 36 bold space bold pi over denominator bold 3 bold cross times bold 2 end fraction end style close parentheses close square brackets over denominator bold sin bold space begin display style bold pi over bold 6 end style end fraction bold equals bold space bold minus bold 72 bold space bold plus bold space bold 0 bold space open parentheses bold cos bold space bold alpha bold space bold plus bold space bold cos bold space bold left parenthesis bold alpha bold space bold plus bold space bold beta bold right parenthesis bold space bold plus bold space bold. bold. bold. bold. bold. bold. bold. bold space bold n bold space bold પદ bold space bold સ ુ ધ ી bold space bold equals bold space fraction numerator bold sin begin display style bold nβ over bold 2 end style over denominator bold sin begin display style bold beta over bold 2 end style end fraction bold space bold cos bold space open parentheses bold alpha bold space bold plus fraction numerator bold left parenthesis bold n bold minus bold 1 bold right parenthesis bold space bold beta over denominator bold 2 end fraction close parentheses close parentheses bold equals bold space bold minus bold 72 bold space$

6. જો α, β એ x²-2x + 2 = 0 નાં બીજ હોય, તો αⁿ + βⁿ = ......... .

$bold 2 to the power of begin inline style bold n over bold 2 end style end exponent bold space bold cos bold nπ over bold 4$
$bold 2 to the power of begin inline style bold n over bold 2 end style bold minus bold 1 end exponent bold space bold cos bold nπ over bold 4$
0
$bold 2 to the power of begin inline style bold n over bold 2 end style bold plus bold 1 end exponent bold space bold cos bold space bold nπ over bold 4$

$bold log subscript begin inline style bold 1 over bold 2 end style end subscript bold space open parentheses fraction numerator bold vertical line bold z bold minus bold 1 bold vertical line bold plus bold 4 over denominator bold 3 bold vertical line bold z bold space bold minus bold space bold 1 bold vertical line bold minus bold 1 end fraction close parentheses bold space bold greater than bold space bold 1$ (જ્યાં $bold vertical line bold z bold space bold minus bold space bold 1 bold vertical line bold space bold not equal to bold space bold 2 over bold 3 bold right parenthesis$ અસમતાનું પાલન કરતી સંકર સંખ્યાનો બિંદુગણ ......

વર્તુળનો બહારનો ભાગ.
વર્તુળ છે.
વર્તુળની અંદરનો ભાગ.
રેખા છે.

z₁ અને z₂ ભિન્ન સંકર સંખ્યાઓ છે તથા |z₁| = |z₂|. જો z₁ નો વાસ્તવિક ભાગ ધન સંખ્યા હોય તથા z₂ નો કાલ્પનિક ભાગ ઋણ સંખ્યા હોય, તો $fraction numerator bold z subscript bold 1 bold space bold plus bold space bold z subscript bold 2 over denominator bold z subscript bold 2 bold space bold minus bold space bold z subscript bold 2 end fraction$ એ ...... છે. $open parentheses fraction numerator bold z subscript bold 1 bold space bold plus bold space bold z subscript bold 2 over denominator bold z subscript bold 1 bold space bold minus bold space bold z subscript bold 2 end fraction bold space bold not equal to bold space bold 0 close parentheses$

વાસ્તવિક અને ધન
શુદ્વ કાલ્પનિક સંખ્યા
વાસ્તવિક અને ઋણ
શૂન્ય સંખ્યા

f(z)ને z-i વડે ભાગીએ તો શેષ i મળે છે તથા જો z + i વડે ભાગીએ તો શેષ 1 + i મળે છે. જો f(z) ને z² + 1 વડે ભાંગીએ તો મળતી શેષ ......

0
$bold 1 over bold 2 bold left parenthesis bold iz bold plus bold 1 bold plus bold 2 bold i bold right parenthesis$
$bold 1 over bold 2 bold left parenthesis bold italic i bold italic z bold plus bold 1 bold right parenthesis$
iz+1+i

10.

ધારો કે $bold a bold space bold equals bold space fraction numerator bold 12 bold pi over denominator bold e to the power of bold 13 end fraction bold. bold space bold alpha bold space bold plus bold space bold a bold space bold plus bold space bold a to the power of bold 3 bold space bold plus bold space bold a to the power of bold 4 bold space bold plus bold space bold a to the power of bold minus bold 4 end exponent bold space bold space bold plus bold space bold a to the power of bold minus bold 3 end exponent bold space bold plus bold space bold a to the power of bold minus bold 1 end exponent$ તથા $bold beta bold space bold equals bold space bold a to the power of bold 2 bold space bold plus bold space bold a to the power of bold 2 bold space bold plus bold space bold a to the power of bold 6 bold space bold plus bold space bold a to the power of bold minus bold 6 end exponent bold space bold plus bold space bold a to the power of bold minus bold 5 end exponent bold space bold plus bold space bold a to the power of bold minus bold 2 end exponent$ જેનાં બીજ હોય તેવું દ્વિઘાત સમીકરણ ......... છે.

x² + x + 3 = 0
x² - x + 2 = 0
x² + x - 3 = 0
x³ - x - 3 = 0

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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

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