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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

71. જો a + b + c હોય તો (0, 1) અંતરાલમાં સમીકરણ 3ax² + 2bx + c = 0 ને ......

એક વસ્તવિક અને એક સંકર બીજ મળે.
એક પણ બીજ ન મળે.
બે અવાસ્તવિક સંકર બીજ મળે.
ઓછામાં ઓછું એક વાસ્તવિક બીજ મળે.

72.

નીચેનામાંથી કયું વિધેય અંતરાલ [0,1] પર મધ્યકમાન પ્રમેયની શરતોનું પાલન કરતું નથી ?

f(x) = (x² - 3x + 2) |x - 1|
f(x) = |2x - 1|
$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell bold x bold comma bold space end cell row cell bold 1 over bold 2 open parentheses bold 1 over bold 2 bold plus bold x close parentheses bold comma end cell end table close table row cell bold x bold less than bold 1 over bold 2 end cell row cell bold x bold greater-than or slanted equal to bold 1 over bold 2 end cell end table$
$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell fraction numerator bold sin bold space bold x over denominator bold x end fraction bold comma end cell row cell bold 1 bold comma end cell end table close table row cell bold x bold # bold 0 end cell row cell bold x bold equals bold 0 end cell end table$

73.

bold જ ો bold space bold D bold space bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open vertical bar table row cell bold tan bold space bold x end cell cell bold tan bold left parenthesis bold x bold space bold plus bold space bold h bold right parenthesis end cell cell bold tan bold left parenthesis bold x bold space bold plus bold space bold 2 bold h bold right parenthesis end cell row cell bold tan bold left parenthesis bold x bold space bold plus bold space bold 2 bold h bold right parenthesis end cell cell bold tan bold space bold x end cell cell bold tan bold left parenthesis bold x bold space bold plus bold space bold h bold right parenthesis end cell row cell bold tan bold left parenthesis bold x bold space bold plus bold space bold h bold right parenthesis end cell cell bold tan bold left parenthesis bold x bold space bold plus bold space bold 2 bold h bold right parenthesis end cell cell bold tan bold space bold x end cell end table close vertical bar bold space bold ત ો bold space bold lim with bold h bold rightwards arrow bold 0 below bold space fraction numerator bold D open parentheses begin display style bold pi over bold 3 end style close parentheses over denominator bold h to the power of bold 2 square root of bold 3 end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold space

74.

f : R → R એ n ઘાતવાળું બહુપદી વિધેય છે. જ્યાં f(x) = 0 ને n ભિન્ન વાસ્તવિક ઉકેલો છે. સમીકરણ [f'(x)]² - f(x0 f"(x) = 0 ને કેટલા વાસ્તવિક ઉકેલ મળે ?

1
2
n
એક પણ નહિ.

એક પણ નહિ.

Tips: -

ધારો કે, f(x) = (x - x₁) (x - x₂) ..... (x - x_n) તથા h(x) = $bold equals fraction numerator bold f bold apostrophe bold left parenthesis bold x bold right parenthesis over denominator bold f bold apostrophe bold left parenthesis bold x bold right parenthesis end fraction$

∴ logf(x) = log(x - x₁) + log(x - x₂) + ..... + log(x - x_n)

$bold therefore bold space bold h bold left parenthesis bold x bold right parenthesis bold space bold equals bold space fraction numerator bold f bold apostrophe bold left parenthesis bold x bold right parenthesis over denominator bold f bold left parenthesis bold x bold right parenthesis end fraction bold space bold equals bold space fraction numerator bold 1 over denominator bold x bold space bold minus bold x subscript bold 1 end fraction bold space bold plus bold space fraction numerator bold 1 over denominator bold x bold minus bold x subscript bold 2 end fraction bold space bold plus bold space bold. bold. bold. bold. bold. bold. bold space bold plus bold space fraction numerator bold 1 over denominator bold x bold space bold minus bold space bold x subscript bold n end fraction bold space bold h bold apostrophe bold left parenthesis bold x bold right parenthesis bold space bold equals bold space fraction numerator bold f bold left parenthesis bold x bold 0 bold f bold " bold left parenthesis bold x bold right parenthesis bold space bold minus bold space bold left square bracket bold f bold apostrophe bold left parenthesis bold x bold right parenthesis bold right square bracket to the power of bold 2 over denominator bold left square bracket bold f bold left parenthesis bold x bold right parenthesis bold right square bracket to the power of bold 2 end fraction bold space bold equals bold space bold minus bold space open square brackets fraction numerator bold 1 over denominator bold left parenthesis bold x bold space bold minus bold space bold x subscript bold 1 bold right parenthesis to the power of bold 2 end fraction bold plus fraction numerator bold 1 over denominator bold x bold minus bold x subscript bold 2 bold right parenthesis to the power of bold 2 end fraction bold space bold plus bold space bold. bold. bold. bold. bold plus bold space fraction numerator bold 1 over denominator bold left parenthesis bold x bold space bold minus bold space bold x subscript bold n bold right parenthesis to the power of bold 2 end fraction close square brackets bold space bold less than bold space bold 0 bold comma bold space bold for all bold space bold x$

∴ f(x) f"(x) - [f'(x)]² < 0 ∀x

∴ માંગેલ સમીકરણને એક પણ વાસ્તવિક ઉકેલ મળે નહિ.

75.

fraction numerator bold d to the power of bold 2 bold x over denominator bold dy to the power of bold 2 end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold.

$bold space bold space open parentheses fraction numerator bold d to the power of bold 2 bold x over denominator bold dy to the power of bold 2 end fraction close parentheses to the power of bold minus bold 1 end exponent bold space bold space$
$open parentheses fraction numerator bold d to the power of bold 2 bold x over denominator bold dy to the power of bold 2 end fraction close parentheses open parentheses bold dy over bold dx close parentheses to the power of bold minus bold 2 end exponent$
$bold space bold minus open parentheses fraction numerator bold d to the power of bold 2 bold x over denominator bold dy to the power of bold 2 end fraction close parentheses open parentheses bold dy over bold dx close parentheses to the power of bold minus bold 3 end exponent bold space bold space$
$bold minus open parentheses fraction numerator bold d to the power of bold 2 bold x over denominator bold dy to the power of bold 2 end fraction close parentheses to the power of bold minus bold 1 end exponent bold space open parentheses bold dy over bold dx close parentheses to the power of bold minus bold 3 end exponent$

76.

સમીકરણ e^x sin x = 1 નાં બે બીજની વચ્ચે સમીકરણ e^x cos x = -1 ને

0< x < $fraction numerator bold 3 bold pi over denominator bold 2 end fraction$ હોય તેવું બીજ મળે.
$fraction numerator bold 3 bold pi over denominator bold 2 end fraction$ < x < 2 હોય તેવું બીજ મળે.
ઓછામાં ઓછું એક વાસ્તવિક બીજ મળે.
એક પણ બીજ પણ ના મળે.

77.

bold જ ો bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold product from bold 2 bold equals bold 1 to bold 18 of bold space bold left parenthesis bold space bold x bold space bold minus bold space bold r bold right parenthesis to the power of bold r to the power of bold 2 bold left parenthesis bold 2016 bold minus bold r bold right parenthesis end exponent bold comma bold space bold ત ો bold space fraction numerator bold f bold apostrophe bold left parenthesis bold 2016 bold right parenthesis over denominator bold f bold apostrophe bold left parenthesis bold 2016 bold right parenthesis end fraction bold space bold equals bold space bold. bold. bold. bold. bold space

0
અવિભાજ્ય સંખ્યા
અયુગ્મ વિભાજ્ય સંખ્યા
અસંમેય સંખ્યા

78.

bold જ ો bold space bold space bold x bold space bold equals bold space bold a bold space bold cos bold space bold t bold space square root of bold cos bold space bold 2 bold space bold t end root bold space bold અન ે bold space bold y bold space bold equals bold space bold a bold space bold sin bold space bold t bold space square root of bold cos bold space bold 2 bold space bold t end root bold space bold left parenthesis bold space bold a bold space bold greater than bold space bold 0 bold right parenthesis bold space bold ત ો bold space open vertical bar table row cell fraction numerator bold 1 bold space bold plus bold space open parentheses begin display style bold dy over bold dx end style close parentheses to the power of bold 2 over denominator begin display style fraction numerator bold d to the power of bold 2 bold y over denominator bold dx to the power of bold 2 end fraction end style end fraction end cell end table close vertical bar subscript bold t bold space bold equals bold space bold pi over bold 6 end subscript bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold.

$bold a square root of bold 2 bold space$
$fraction numerator square root of bold 2 over denominator bold 3 bold a end fraction bold space$
$fraction numerator square root of bold 2 bold a end root over denominator bold 3 end fraction$
$fraction numerator begin display style bold a end style over denominator begin display style bold 3 end style end fraction$