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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

51.

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

52. 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|
$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell fraction numerator bold sin bold space bold x over denominator bold x end fraction bold comma bold space bold x bold # bold 0 bold space end cell row cell bold 0 bold space bold comma bold space bold x bold space bold equals bold space bold 0 bold space end cell end table close$

f(x) = |x| - |sin x|

Tips: -

$bold left parenthesis bold a bold right parenthesis bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold vertical line bold x bold vertical line bold space bold left square bracket bold x bold right square bracket bold space bold equals bold space open curly brackets table row cell bold x bold comma end cell cell bold minus bold 1 bold less-than or slanted equal to bold x bold less than bold 0 end cell row cell bold 0 bold comma end cell cell bold 0 bold less or equal than bold x bold less than bold 1 end cell row cell bold 1 bold comma end cell cell bold x bold equals bold 1 end cell end table close bold therefore bold space bold f bold space bold એ bold space bold x bold space bold equals bold space bold 0 bold space bold space bold મ ા ટ ે bold space bold વ િ કલન ી ય bold space bold નથ ી bold. bold space bold left parenthesis bold b bold right parenthesis bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold left square bracket bold x bold right square bracket bold space bold plus bold space bold left square bracket bold minus bold x bold right square bracket bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell bold 0 bold comma end cell row cell bold minus bold 1 bold comma end cell end table close table row cell bold x bold element of bold Z end cell row cell bold x bold not an element of bold Z end cell end table bold space bold therefore bold space bold f bold space bold એ bold space bold x bold space bold equals bold space bold 0 bold space bold આગળ bold space bold સતત bold space bold નથ ી bold. bold space bold left parenthesis bold c bold right parenthesis bold space bold lim with bold x bold rightwards arrow bold 0 below bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold lim with bold x bold rightwards arrow bold 0 below bold space bold sinx over bold x bold space bold equals bold space bold 1 bold space bold અન ે bold space bold f bold left parenthesis bold 0 bold right parenthesis bold space bold equals bold space bold 0 bold space bold therefore bold space bold space bold f bold space bold એ bold space bold x bold space bold equals bold space bold 0 bold space bold આગળ bold space bold સતત bold space bold નથ ી bold. bold space bold left parenthesis bold d bold right parenthesis bold space 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 space bold minus bold space bold sinx end cell row cell bold minus bold x bold space bold plus bold space bold sinx end cell end table close table row cell bold 0 bold less or equal than bold space bold x bold space bold less or equal than bold space bold 1 end cell row cell bold minus bold 1 bold space bold less or equal than bold space bold x bold space bold less or equal than bold 0 end cell end table$

∴ f એ [-1, 1] માં સતત છે. તેમજ (-1, 1)માં વિકલનીય છે.

વળી, f(-1) = f(1)

આમ, રોલના પ્રમેયની શરતોનું પાલન થાય છે.

53.

bold જ ો bold space bold space bold y bold space bold equals bold space bold cot to the power of bold minus bold 1 end exponent open parentheses fraction numerator bold log open parentheses begin display style bold e over bold x to the power of bold 2 end style close parentheses over denominator bold log bold left parenthesis bold ex to the power of bold 2 bold right parenthesis end fraction close parentheses bold space bold plus bold space bold cot to the power of bold minus bold 1 end exponent bold space open parentheses fraction numerator bold log bold left parenthesis bold ex to the power of bold 4 bold right parenthesis over denominator bold log open parentheses begin display style bold e to the power of bold 2 over bold x to the power of bold 2 end style close parentheses end fraction close parentheses bold comma bold space bold ત ો bold space bold dy over bold dx bold space bold equals bold space bold. bold. bold. bold. bold. bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold 0 bold less than bold space bold logx bold space bold less than bold 1 over bold 2

1
-1
2
0

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.

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

56.

જો y = (1+x) (1+x²) (1+x⁴) ... + log (1+x²ⁿ) તો x = 0 આગળની કિંમત ......

1
-1
2
0

57.

જો f(x) = (x+1)(x+2)(x+3)....(x + 100) અને g(x) = f(x) - (f(x))2 તો g(x) = 0 ને

એક ઉકેલ મળે.
ઉકેલ નથી.
બે ઉકેલ મળે.
ઓછામાં ઓછા ત્રણ ઉકેલ મળે.

58.

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