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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

51.

જો $fraction numerator bold 1 over denominator bold left parenthesis bold 1 bold minus bold ax bold right parenthesis bold right parenthesis bold 1 bold minus bold bx bold right parenthesis end fraction$ નું x ના ઘાતમાં વિસ્તરણ a₀ + a₁x + a₂x³ + .... હોય, તો a_n = ..... .

$fraction numerator bold b to the power of bold n bold space bold minus bold space bold a to the power of bold n over denominator bold b bold minus bold a end fraction$
$fraction numerator bold b to the power of bold n bold plus bold 1 end exponent bold minus bold n to the power of bold n bold plus bold 1 end exponent over denominator bold b bold minus bold a end fraction$
$fraction numerator bold a to the power of bold n bold plus bold 1 end exponent bold minus bold b to the power of bold n bold plus bold 1 end exponent over denominator bold b bold minus bold a end fraction$
$fraction numerator bold a to the power of bold n bold minus bold b to the power of bold n over denominator bold b bold minus bold a end fraction$

fraction numerator bold b to the power of bold n bold plus bold 1 end exponent bold minus bold n to the power of bold n bold plus bold 1 end exponent over denominator bold b bold minus bold a end fraction

Tips: -

$bold left parenthesis bold 1 bold space bold minus bold space bold ax bold right parenthesis to the power of bold minus bold 1 end exponent bold space bold left parenthesis bold 1 bold space bold minus bold space bold bx bold right parenthesis to the power of bold minus bold 1 end exponent bold space bold equals bold space bold left parenthesis bold 1 bold space bold plus bold space bold ax bold space bold plus bold space bold a to the power of bold 2 bold x to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold right parenthesis bold space bold cross times bold space bold left parenthesis bold 1 bold space bold plus bold space bold bx bold space bold plus bold space bold b to the power of bold 2 bold x to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold right parenthesis bold space$

a_n = xⁿ = નો સહગુણક

$bold equals bold space bold b to the power of bold n bold space bold plus bold space bold ab to the power of bold b bold minus bold 1 end exponent bold space bold plus bold space bold a to the power of bold 2 bold b to the power of bold n bold minus bold 2 end exponent bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold a to the power of bold n bold space bold equals bold space bold b to the power of bold n bold space open parentheses bold 1 bold plus bold a over bold b bold plus bold a to the power of bold 2 over bold b to the power of bold 2 bold plus bold. bold. bold. bold plus bold a to the power of bold n over bold b to the power of bold n close parentheses bold space bold equals bold space bold b to the power of bold n bold space open parentheses fraction numerator open parentheses begin display style bold a over bold b end style close parentheses to the power of bold n bold plus bold 1 end exponent bold minus bold 1 over denominator begin display style bold a over bold b end style bold minus bold 1 end fraction close parentheses bold space bold equals bold space bold b to the power of bold n bold space open parentheses fraction numerator bold a to the power of bold n bold plus bold 1 end exponent bold minus bold b to the power of bold n bold plus bold 1 end exponent over denominator bold a bold minus bold b end fraction close parentheses bold space bold b over bold b to the power of bold n bold plus bold 1 end exponent bold space bold equals bold space fraction numerator bold b to the power of bold n bold plus bold 1 end exponent bold minus bold a to the power of bold n bold plus bold 1 end exponent over denominator bold b bold minus bold a end fraction$

52.

પ્રાકૃતિક સંખ્યાઓ m, n માટે જો (1-y)^m(1+y)ⁿ = 1 + a₁y + a₂y + ... અને a₁ = 10, તો (m ,n) = ...... .

(20,45)
(35,45)
(45,35)
(35,20)

53. જો $table row bold lim row cell bold x bold rightwards arrow bold infinity end cell end table bold space open parentheses bold 1 bold plus bold a over bold x bold plus bold b over bold x to the power of bold 2 close parentheses to the power of bold 2 bold x end exponent bold space bold equals bold space bold e to the power of bold 2 bold comma$ તો a અને b ની કિંમતો ....... છે.

a = 1, b ∈ R
b = 2, a ∈ R
a, b ∈ R
a = 1, b = 2

54. (1 + x) (1 - x)ⁿ ના વિસ્તરણમાં xⁿ નો સહગુણક ...... છે.

n-1
(-1)ⁿ (1 - n)
(-1)^n-1 n
(-1)ⁿ (n + 1)

55. જો (a-b)ⁿ, n ≥ 5 ના દ્વિપદી વિસ્તરણમાં પાંચમા અને છઠ્ઠા પદોનો સરવાળો 0 હોય, તો $bold a over bold b$

$fraction numerator bold n bold minus bold 5 over denominator bold 6 end fraction$
$fraction numerator bold 5 over denominator bold n bold minus bold 4 end fraction$
$fraction numerator bold 6 over denominator bold n bold minus bold 5 end fraction$
$fraction numerator bold n bold plus bold 4 over denominator bold 5 end fraction$

56. જો x > 0, તો $scriptbase bold 27 over bold 5 end scriptbase presubscript bold left parenthesis bold 1 bold plus bold x bold right parenthesis end presubscript$ ના વિસ્તરણમાં પ્રથમ ઋણ પદ ......... છે.

7 મું પદ
5 મું પદ
8 મું પદ
6 ઠ્ઠું પદ

57.

જો x એટલો નાનો હોય કે જેથી x³ અને x ની 3 કરતાં મોટી ઘાતવાળાં પદો અવગણી શકાય, તો $fraction numerator bold left parenthesis bold 1 bold plus bold x bold right parenthesis to the power of bold 3 over bold 2 end exponent bold minus open parentheses bold 1 bold plus begin display style bold x over bold 2 end style close parentheses to the power of bold 3 over denominator bold left parenthesis bold 1 bold minus bold x bold right parenthesis to the power of bold 1 over bold 2 end exponent end fraction$ નું લગભગ મૂલ્ય ...... છે.

$bold minus bold 3 over bold 8 bold space bold x to the power of bold 2$
$bold x over bold 2 bold minus bold 3 over bold 8 bold x to the power of bold 2$
$bold 1 bold space bold minus bold space bold 3 over bold 8 bold space bold x to the power of bold 2$
$bold 3 bold x bold space bold plus bold space bold 3 over bold 8 bold space bold x to the power of bold 2$

58.

(1 + αx)⁴ અને (1 - αx)⁶ ના વિસ્તરણમાં મધ્યમ પદોના સહગુણકો સમાન હોય, તો શૂન્યેતર α = ....... .

$fraction numerator bold minus bold 3 over denominator bold 10 end fraction$
$fraction numerator bold minus bold 5 over denominator bold 3 end fraction$
$bold 3 over bold 5$
$bold 10 over bold 3$

59. $bold left parenthesis square root of bold 3 bold space bold plus bold space root index bold 8 of bold 5 bold right parenthesis to the power of bold 256$ માં પૂર્ણાંક પદોની સંખ્યા ....... છે.

60.

જો $open parentheses bold ax to the power of bold 2 bold plus bold 1 over bold bx close parentheses to the power of bold 11$ માં x⁷ નો સહગુણક અને $open parentheses bold ax bold minus bold 1 over bold bx to the power of bold 2 close parentheses to the power of bold 11$ માં x^-7 નો સહગુણક સમાન હોય, તો

a-b=1
a+b=1
ab=1
a=b

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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

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