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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

51.

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.

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

bold plus-or-minus bold pi over bold 2

Tips: -

bold lim with bold x bold rightwards arrow bold 0 below bold space bold equals bold space open square brackets 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 fraction numerator bold 1 over denominator bold xlog bold left parenthesis bold 1 bold plus bold b to the power of bold 2 bold right parenthesis end fraction end exponent close square brackets to the power of bold log bold left parenthesis bold 1 bold plus bold b to the power of bold 2 bold right parenthesis end exponent bold space bold equals bold space bold e to the power of bold log bold left parenthesis bold 1 bold plus bold b to the power of bold 2 bold right parenthesis end exponent bold space bold equals bold space bold 1 bold space bold plus bold space bold b to the power of bold 2 bold space bold therefore bold space bold 1 bold space bold plus bold space bold b to the power of bold 2 bold space bold plus bold space bold 2 bold b bold space bold sin to the power of bold 2 bold space bold theta bold therefore bold space bold sin bold 2 bold space bold theta bold space bold equals bold space fraction numerator bold 1 bold plus bold b to the power of bold 2 over denominator bold 2 bold b end fraction bold space bold greater-than or slanted equal to bold space bold 1 bold. bold space bold પર ં ત ુ bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold sin to the power of bold 2 bold space bold theta bold space bold less-than or slanted equal to bold space bold 1 bold therefore bold space bold sin to the power of bold 2 bold space bold theta bold space bold equals bold space bold 1 bold therefore bold space bold theta bold space bold equals bold space bold plus-or-minus bold pi over bold 2

52.

$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold g bold apostrophe bold left parenthesis bold x bold right parenthesis bold space fraction numerator bold e to the power of begin display style bold a over bold x end style end exponent bold minus bold e to the power of bold minus begin display style bold a over bold x end style end exponent over denominator bold g to the power of begin display style bold a over bold x end style end exponent bold space bold plus bold space bold e to the power of bold minus begin display style bold a over bold x end style end exponent end fraction bold space$ 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

53. 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|
$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell fraction numerator bold sin bold space bold x over denominator bold x end fraction bold comma bold space bold x bold # bold 0 bold space end cell row cell bold 0 bold space bold comma bold space bold x bold space bold equals bold space bold 0 bold space end cell end table close$