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CBSE

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Mathematics

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

81.

bold જ ો bold space bold f bold " bold left parenthesis bold 0 bold right parenthesis bold space bold equals bold space bold 4 bold comma bold space bold ત ો bold space bold lim with bold x bold rightwards arrow bold 0 below bold space fraction numerator bold 2 bold f bold left parenthesis bold x bold right parenthesis bold space bold minus bold space bold 3 bold space bold left parenthesis bold 2 bold x bold right parenthesis bold space bold plus bold space bold f bold left parenthesis bold 4 bold x bold right parenthesis bold space over denominator bold x to the power of bold 2 end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold.

-6
6
-12
12

82. જો f(x) = (x - 1) (x - 2) (x - 3) (x - 4) , તો f'(x) = 0 ના બીજ પૈકી ....

ત્રણે બીજ ઋણ હોય.
ત્રણે બીજ ધન હોય.
એક ઋણ, બે ધન હોય.
બે અવાસ્તવિક સંકર સંખ્યા હોય.

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

$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell fraction numerator bold sin bold space bold x over denominator bold x end fraction bold comma bold space end cell row cell bold 1 bold comma end cell end table close table row cell bold x bold space bold # bold space bold 0 bold space end cell row cell bold x bold space bold equals bold space bold 0 bold space end cell end table bold space bold space$
$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell bold 1 over bold 2 bold minus bold x bold comma end cell cell bold x bold space bold less than bold space bold 1 over bold 2 end cell row cell open parentheses bold 1 over bold 2 bold minus bold x close parentheses to the power of bold 2 bold comma bold space end cell cell bold x bold space bold greater-than or slanted equal to bold space bold 1 over bold 2 end cell end table close$
f(x) |x|
f(x) = x |x|

84.

f: [0,4] → R એ વિકલનીય વિધેય છે. જો અંતરાલ (0, 4) માં a, b એવા મળે કે જેથી
(f(4)2 - f(0))² = k f'(a) f(b) તો k = .....

1
4
8
શક્ય નથી.

85. વિધેય f એ R પર વિકલનીય છે. જો f(1) = -2 અને f(x) ≥ 2, ∀ x ∈ [1, 6] તો,

f(6)≥8
f(6) = 5
f(6)< 5
f(6)<2

86.

bold જ ો bold space bold lim with bold n bold rightwards arrow bold infinity below bold space fraction numerator bold left parenthesis bold 1 to the power of bold a bold space bold plus bold space bold 2 to the power of bold a bold space bold plus bold space bold. bold. bold. bold. bold space bold plus bold space bold n to the power of bold a bold right parenthesis bold space over denominator bold left parenthesis bold n bold space bold plus bold space bold 1 bold right parenthesis to the power of bold a bold minus bold 1 end exponent bold space bold left square bracket bold left parenthesis bold na bold space bold plus bold 1 bold right parenthesis bold space bold plus bold space bold left parenthesis bold na bold plus bold 2 bold right parenthesis bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold left parenthesis bold na bold plus bold n bold right parenthesis bold right square bracket bold space end fraction bold space bold equals bold space bold 1 over bold 60 bold space bold ત ો bold space bold a bold space bold equals bold space bold. bold. bold. bold. 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 space bold space bold space bold left parenthesis bold a bold # bold minus bold 1 bold right parenthesis bold space

$fraction numerator bold minus bold 15 over denominator bold 2 end fraction bold space$
$fraction numerator bold minus bold 17 over denominator bold 2 end fraction$
5
7

Tips: -

bold આપ ે લ bold space bold લક ્ ષ bold space bold equals bold space bold lim with bold n bold rightwards arrow bold infinity below bold space fraction numerator begin display style bold sum from bold r bold equals bold 1 to bold n of end style bold r to the power of bold a over denominator bold left parenthesis bold n bold space bold plus bold space bold 1 bold right parenthesis bold space open curly brackets bold n to the power of bold 2 bold a bold space bold plus bold space begin display style fraction numerator bold n bold left parenthesis bold n bold plus bold 1 bold right parenthesis over denominator bold 2 end fraction end style close curly brackets end fraction bold equals bold space bold lim with bold n bold rightwards arrow bold infinity below bold space fraction numerator begin display style bold sum from bold r bold minus bold 1 to bold n of end style bold r to the power of bold a over denominator bold na bold minus bold 1 bold. bold n bold 2 open parentheses bold 1 bold plus begin display style bold 1 over bold n end style close parentheses to the power of bold a bold minus bold 1 end exponent bold space open curly brackets bold a bold space bold plus bold space begin display style bold 1 over bold 2 end style bold space bold plus bold space begin display style fraction numerator bold 1 over denominator bold 2 bold n end fraction end style close curly brackets end fraction bold equals bold space bold lim with bold n bold rightwards arrow bold infinity below bold space fraction numerator begin display style bold 1 over bold n end style begin display style bold sum from bold r bold equals bold 1 to bold n of end style open parentheses begin display style bold r over bold n end style close parentheses to the power of bold a over denominator open parentheses bold 1 bold plus begin display style bold 1 over bold n end style close parentheses to the power of bold a bold minus bold 1 end exponent bold space open curly brackets bold a bold space bold plus bold space begin display style bold 1 over bold 2 end style bold space bold plus bold space begin display style fraction numerator bold 1 over denominator bold 2 bold n end fraction end style close curly brackets end fraction bold therefore bold space bold 1 over bold 60 bold space bold equals bold space fraction numerator begin display style bold integral from bold 0 to bold 1 of end style bold x to the power of bold a bold space bold dx over denominator bold a bold plus begin display style bold 1 over bold 2 end style end fraction bold space bold equals bold space fraction numerator bold 1 over denominator bold left parenthesis bold a bold space bold plus bold space bold 1 bold right parenthesis bold space open parentheses bold a bold plus bold space begin display style bold 1 over bold 2 end style close parentheses end fraction bold therefore bold space bold left parenthesis bold 2 bold a bold space bold plus bold space bold 1 bold right parenthesis bold space bold left parenthesis bold a bold space bold plus bold space bold 1 bold right parenthesis bold space bold equals bold space bold 120 bold space bold therefore bold space bold 2 bold a to the power of bold 2 bold space bold plus bold space bold 3 bold a bold space bold minus bold space bold 119 bold space bold equals bold space bold 0 bold space bold આથ ી bold space bold space bold space bold space bold left parenthesis bold 2 bold a bold space bold plus bold space bold 17 bold right parenthesis bold space bold left parenthesis bold a bold space bold minus bold 7 bold right parenthesis bold space bold equals bold space bold 0 bold space bold આથ ી bold space bold space bold a bold space bold equals bold space bold 7 bold space bold અથવ ા bold space bold a bold space bold equals bold space fraction numerator bold minus bold 17 over denominator bold 2 end fraction bold પર ં ત ુ bold space bold જ ો bold space bold a bold space bold greater than bold space bold minus bold 1 bold space bold હ ો ય bold comma bold space bold ત ો bold space bold જ bold space bold integral from bold 0 to bold 1 of bold x to the power of bold a bold space bold dx bold space bold ન ુ bold space bold અસ ્ ત િ ત ્ વ bold space bold છ ે bold. bold space bold આથ ી bold space bold a bold space bold equals bold space bold 7 bold space

87.

bold lim with bold x bold rightwards arrow bold 0 below bold space fraction numerator bold xtan bold space bold 2 bold x bold space bold minus bold space bold 2 bold space bold x bold space bold tan bold space bold x over denominator bold left parenthesis bold 1 bold minus bold cos bold 2 bold x bold right parenthesis to the power of bold 2 end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold.

$bold 1 over bold 2$
-1
2
-2

88.

bold જ ો bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals open curly brackets table attributes columnalign left center end attributes row cell fraction numerator bold a bold vertical line bold x to the power of bold 2 bold space bold minus bold space bold x bold space bold 2 bold vertical line over denominator bold 2 bold space bold plus bold space bold x bold space bold minus bold space bold x to the power of bold 2 end fraction bold comma end cell cell bold x bold space bold less-than or slanted equal to bold space bold 2 end cell row cell bold b bold space bold comma end cell cell bold x bold space bold equals bold space bold 2 end cell row cell fraction numerator bold x bold space bold minus bold space bold left square bracket bold x bold right square bracket over denominator bold x bold minus bold 2 end fraction bold comma end cell cell bold x bold space bold greater than bold space bold 2 end cell end table close bold space bold એ bold space bold x bold space bold equals bold space bold 2 bold space bold આગળ bold space bold સતત bold space bold હ ો ય bold comma bold space bold ત ો bold space bold left parenthesis bold a bold comma bold space bold b bold right parenthesis bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold space

(1, 1)
(1, 2)
(2, 1)
(2, 2)

89.

bold જ ો bold space bold lim with bold x bold rightwards arrow bold 0 below bold space fraction numerator bold sin bold space bold left parenthesis bold space bold sin bold space bold x bold right parenthesis bold space bold minus bold space bold sin bold space bold x over denominator bold ax to the power of bold 3 bold space bold plus bold space bold bx to the power of bold 5 bold space bold plus bold space bold c end fraction bold space bold equals bold space fraction numerator bold minus bold 1 over denominator bold 12 end fraction bold space bold ત ો bold space

a = -2, b ∈ R, c = 0
a = 1, b ∈ R, c = 0
a = -1, B ∈ R, c = 0
a = 2, b ∈ R, c = 0

90.

bold જ ો 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 integral from bold 0 to bold x of bold left parenthesis bold 1 bold space bold plus bold space bold 1 bold vertical line bold 1 bold space bold minus bold space bold t bold vertical line bold space bold dt bold comma bold space bold space end cell row cell bold 5 bold x bold space bold minus bold space bold 7 bold comma bold space bold space bold space bold space bold space bold space bold space bold space end cell end table close table row cell bold x bold space bold greater than bold space bold 2 end cell row cell bold x bold less-than or slanted equal to bold space bold 2 end cell end table bold space bold ત ો bold space

f એ x = 2 આગળ સતત છે પરંતુ વિકલનીય નથી.
f એ x = 2 આગળ સતત નથી
f એ R પર વિકલનીય છે.
$bold lim with bold x bold rightwards arrow bold 2 subscript bold plus below$ f(x) નું અસ્તિત્વ નથી

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