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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

51.

bold જ ો bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table row cell bold minus bold x bold minus bold pi over bold 2 bold comma end cell cell bold x bold less-than or slanted equal to bold minus bold pi over bold 2 end cell row cell bold minus bold cos bold space bold x bold comma end cell cell bold minus bold pi over bold 2 bold less than bold x bold less-than or slanted equal to bold 0 end cell row cell bold x bold minus bold 1 bold comma end cell cell bold 0 bold less than bold x bold less-than or slanted equal to bold 1 end cell row cell bold log bold space bold x bold comma end cell cell bold x bold greater than bold 1 end cell end table close bold space bold ત ો bold space

f(x) એ x = $bold minus bold 3 over bold 2$ , 1આગળ વિકલનીય છે.
f(x) એ x = $bold minus bold pi over bold 2$ આગળ સતત છે.
f(x) એ x = 0 આગળ વિકલનીય નથી.
આપેલ તમામ

52.

ધારો કે [0,1] માં વિધેય f ના દ્વિતિય વિકલિતનું અસ્તિત્વ છે અને |f"(x)| ≤ 1, ∀ x ∈[0,1]. જો f(0) = f(1) હોય તો, અંતરાલ (0,1) માં

|f'(x)| > 1
|f'(x)|>1
|f''(x)| < 1
|f(x)|<1

53.

$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold g bold apostrophe bold left parenthesis bold x bold right parenthesis bold space fraction numerator bold e to the power of begin display style bold a over bold x end style end exponent bold minus bold e to the power of bold minus begin display style bold a over bold x end style end exponent over denominator bold g to the power of begin display style bold a over bold x end style end exponent bold space bold plus bold space bold e to the power of bold minus begin display style bold a over bold x end style end exponent end fraction bold space$ જ્યાં g' એ વિધેય g નું વિકલિત છે તથા સતત વિધેય છે અને a > 0. જો $bold lim with bold x bold rightwards arrow bold 0 below bold space bold f bold left parenthesis bold x bold right parenthesis$ નું અસ્તિત્વ હય તો

g(x) = a₀ + a₁x + a₂x² + ..... + a_nx^-n જ્યાં a_i ∈ R, i = 1,2 ....n તથા a₁ # 0 a_n # 0
g(x) = log (1 + x)
g એવું વિધેય હોય જ્યાં g'(0) = 0
g(x0 = sin x

54.

bold જ ો bold space bold lim with bold x bold rightwards arrow bold 0 below bold space bold left square bracket bold 1 bold plus bold xlog bold left parenthesis bold 1 bold plus bold b to the power of bold 2 bold right parenthesis bold right square bracket to the power of bold 1 over bold x end exponent bold space bold equals bold space bold 2 bold space bold b bold space bold sin to the power of bold 2 bold space bold theta bold comma bold space bold b bold space bold greater than bold space bold 0 bold space bold અન ે bold space bold theta bold space bold element of bold space bold left square bracket bold minus bold pi bold comma bold pi bold right square bracket bold space bold ત ો bold space bold theta bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold.

$bold plus-or-minus bold pi over bold 2$
$bold plus-or-minus bold pi over bold 4$
$bold plus-or-minus bold pi over bold 3$
$bold plus-or-minus bold pi over bold 6$

55. નીચેનામાંથી કયું વિધેય અંતરાલ [-1, 1] માં રોલના પ્રમેયની શરતોનું પાલન કરે છે.

f(x) = [x] + [-x]
f(x) = |x| [x]
f(x) = |x| - |sin x|
$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell fraction numerator bold sin bold space bold x over denominator bold x end fraction bold comma bold space bold x bold # bold 0 bold space end cell row cell bold 0 bold space bold comma bold space bold x bold space bold equals bold space bold 0 bold space end cell end table close$