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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

81.

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

82.

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

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.

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

85.

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

86.

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.

Tips: -

મધ્ય કમાન પ્રમેય પરથી a ∈(0,4) એવો મળે કે જેથી $fraction numerator bold f bold left parenthesis bold 4 bold right parenthesis bold space bold minus bold space bold f bold left parenthesis bold 0 bold right parenthesis over denominator bold 4 bold space bold minus bold space bold 0 end fraction bold space bold equals bold space bold f bold apostrophe bold left parenthesis bold a bold right parenthesis$

∴ f(4) - f(0) = 4f'(a)

વળી વિધેય f એ [0,4] પર સતત હોવાથી તે ન્યુનત્તમ અને મહત્તમ વચ્ચેની બધી જ કિંમત ધારણ કરશે.

હવેેે, $fraction numerator bold f bold left parenthesis bold 4 bold right parenthesis bold space bold minus bold space bold f bold left parenthesis bold 0 bold right parenthesis over denominator bold 4 bold space bold minus bold space bold 0 end fraction bold element of bold space bold left parenthesis bold f bold left parenthesis bold 0 bold right parenthesis bold space bold plus bold space bold f bold left parenthesis bold 4 bold right parenthesis bold right parenthesis$

કોઈક b ∈ (0, 4) માટે $fraction numerator bold f bold left parenthesis bold 4 bold right parenthesis bold space bold minus bold space bold f bold left parenthesis bold 0 bold right parenthesis over denominator bold 4 bold space bold minus bold space bold 0 end fraction bold space bold equals bold f bold left parenthesis bold b bold right parenthesis bold space$

∴ f(4) + f(0) = 2 f(b)

(1) અને (2) પરથી, (f(4))2 - (f(0))2 = 8 f'(a)f(b)

∴ k = 8

87.

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