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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

1. 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 = ......... .

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

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

4. 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.

6. 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 $bold sum from bold n bold equals bold 1 to bold 36 of open parentheses bold x to the power of bold n bold minus bold 1 over bold x to the power of bold n close parentheses to the power of bold 2$ = ........... .

7. 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

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

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

x² + x - 3 = 0

Tips: -

બીજોનો સરવાળો

bold alpha bold plus bold beta bold space bold equals bold space bold left parenthesis bold a 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 4 bold space bold plus bold space bold a to the power of bold minus bold 4 end exponent 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 bold right parenthesis bold space bold left parenthesis bold a to the power of bold 2 bold space bold plus bold space bold a to the power of bold 5 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 bold right parenthesis

bold equals bold space bold left parenthesis 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 9 bold space bold plus bold space bold a to the power of bold 10 bold space bold plus bold space bold a to the power of bold 12 bold right parenthesis bold space bold plus bold space bold left parenthesis bold a to the power of bold 2 bold space bold plus bold space bold a to the power of bold 5 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 7 bold space bold plus bold space bold a to the power of bold 8 bold space bold plus bold space bold a to the power of bold 11 bold right parenthesis bold equals bold space bold a 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 3 bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold a to the power of bold 12 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 space bold space bold space bold left parenthesis bold a to the power of bold 12 bold space bold equals bold space bold 1 bold right parenthesis bold space bold equals bold space fraction numerator bold a bold left parenthesis bold 1 bold minus bold a to the power of bold 12 bold right parenthesis over denominator bold 1 bold minus bold a end fraction bold space bold equals bold space fraction numerator bold a bold left parenthesis bold 1 bold minus bold a to the power of bold 12 bold space bold a to the power of bold minus bold 1 end exponent bold right parenthesis over denominator bold 1 bold minus bold a end fraction bold space bold equals bold space fraction numerator bold a bold left parenthesis bold 1 bold minus bold a to the power of bold minus bold 1 end exponent bold right parenthesis over denominator bold 1 bold minus bold a end fraction bold space bold equals 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 left parenthesis bold a to the power of bold 13 bold space bold equals bold space bold 1 bold right parenthesis

બીજોનો ગુણાકાર

$bold alpha bold space bold beta bold space bold equals bold space bold left parenthesis bold a bold space bold plus bold space bold a to the power of bold 3 bold space bold plus bold space bold a to the power of bold 4 bold space bold plus bold space bold a to the power of bold minus bold 4 end exponent bold space bold plus bold space bold a to the power of bold minus bold 3 end exponent bold space bold plus bold space bold d to the power of bold minus bold 1 end exponent bold right parenthesis bold space bold left parenthesis bold a to the power of bold 2 bold space bold plus bold space bold a to the power of bold 5 bold space bold plus bold space bold a to the power of bold 6 bold space bold plus bold space bold a to the power of bold minus bold 5 end exponent bold space bold plus bold space bold a to the power of bold minus bold 2 end exponent bold right parenthesis bold space$

bold equals bold space bold left parenthesis 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 9 bold space bold plus bold space bold a to the power of bold 10 bold space bold plus bold space bold a to the power of bold 12 bold right parenthesis bold space bold left parenthesis bold a to the power of bold 2 bold space bold plus bold space bold a to the power of bold 5 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 7 bold space bold plus bold space bold a to the power of bold 8 bold space bold plus bold space bold a to the power of bold 11 bold right parenthesis bold equals bold space bold a to the power of bold 3 bold space bold left parenthesis bold 1 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 2 bold space bold plus bold space bold a to the power of bold 8 bold space bold plus bold space bold a to the power of bold 9 bold space bold plus bold space bold a to the power of bold 11 bold right parenthesis bold space bold left parenthesis bold 1 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 5 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 9 bold right parenthesis bold space bold equals bold space bold a to the power of bold 3 bold space bold left square bracket bold 1 bold space bold plus bold space bold a to the power of bold 2 bold space bold plus bold space bold 2 bold space bold left parenthesis bold a to the power of bold 3 bold space bold plus bold space bold a to the power of bold 5 bold space bold plus bold space bold a to the power of bold 7 bold space bold plus bold space bold a to the power of bold 9 bold space bold plus bold space bold a to the power of bold 15 bold space bold plus bold space bold a to the power of bold 17 bold right parenthesis bold space bold plus bold space bold a to the power of bold 4 bold space bold plus bold space bold 3 bold space bold left parenthesis bold a to the power of bold 6 bold space bold plus bold space bold a to the power of bold 8 bold space bold plus bold space bold a to the power of bold 11 bold space bold plus bold space bold a to the power of bold 12 bold space bold plus bold space bold a to the power of bold 14 bold right parenthesis bold space bold plus bold space bold left parenthesis bold a to the power of bold 2 bold space bold plus bold space bold a to the power of bold 25 bold space bold plus bold space bold a to the power of bold 18 bold space bold plus bold space bold a to the power of bold 20 bold right parenthesis bold right square bracket bold equals bold space bold 3 bold space bold left parenthesis bold a 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 3 bold space bold plus bold space bold. bold. bold. bold space bold a to the power of bold 12 bold right parenthesis bold space bold equals bold space bold 3 bold left parenthesis bold minus bold 1 bold right parenthesis bold space bold equals bold space bold minus bold 3 bold space

∴ માંગેલ દ્વિઘાત સમીકરણ x² - (-1)x - 3 = 0

x² + x - 3 = 0

10. 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.

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Gujarati JEE Mathematics : સંકર સંખ્યાઓ

Multiple Choice Questions

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