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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

61.

જો $bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell bold ax bold 2 bold space bold minus bold space bold b bold comma bold space bold vertical line bold x bold vertical line bold greater-than or slanted equal to bold 1 end cell row cell begin inline style fraction numerator bold 1 over denominator bold vertical line bold x bold vertical line end fraction end style bold comma bold space bold space bold vertical line bold x bold vertical line bold greater-than or slanted equal to bold 1 end cell end table close$ એ x = 1 આગળ વિકલનીય હોય, તો a = ....... , b = ..........

$bold 1 over bold 2 bold comma bold 3 over bold 2$
$fraction numerator bold minus bold 1 over denominator bold 2 end fraction bold comma fraction numerator bold minus bold 3 over denominator bold 2 end fraction bold space$
$fraction numerator bold minus bold 1 over denominator bold 2 end fraction bold comma bold 3 over bold 2 bold space$
$bold 1 over bold 2 bold comma fraction numerator bold minus bold 3 over denominator bold 2 end fraction$

62.

bold જ ો bold space bold space bold y bold left parenthesis bold n bold right parenthesis bold space bold equals bold space bold e to the power of bold x bold space bold e to the power of bold x to the power of bold 2 end exponent bold space bold. bold. bold. bold space bold e to the power of bold x to the power of bold n end exponent bold comma bold space bold 0 bold space bold less than bold space bold x bold space bold less than bold space bold 1 bold space bold હ ો ય bold space bold ત ો bold space bold x bold space bold equals bold space bold 1 over bold 2 bold આગળ bold space bold lim with bold x bold rightwards arrow bold infinity below bold space fraction numerator bold dy bold left parenthesis bold n bold right parenthesis over denominator bold dx end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold space

63.

ધારો કે, f(x) અને g(x) એ R પર વિકલિત વિધેયો છે. જો f(2) = 0, f(4) = 10 અને g(4)= 8 તો

કોઈક x ∈(2, 4) માટે g'(x) = 4f'(x)
g'(x) > 4 f'(x), ∀x ∈(2, 4)
g(x) > f(x), ∀x ∈ (2, 4)
કોઈક x ∈(2, 4) માટે 3 g'(x) = 4f'(x)

કોઈક x ∈(2, 4) માટે g'(x) = 4f'(x)

Tips: -

ધારો કે, h(x) = g(x) - 4f(x)

h(2) = g(4) - 4 f(2) = 0 - 4.8 = -32

h(4) = g(4) - 4f(40 = 8 -4.10 = -32

વળી, h એ [2, 4] પર સતત છે તથા (2, 4) પર વિકલિત છે.

વિધેય h એ રોલના પ્રમેયની શરતનું સમાધાન કરે છે.

કોઈક x ∈(2, 4) માટે h'(x) = 0

g'(x) = 4 f'(x)

64.

જો દ્વિઘાત સમીકરણ f(x) = ax² + bx + c = 0 (a#0) ને બે એકબીજાના વ્યસ્ત ધન બીજ હોય તો

f'(1) = 0
af'(1) < 0
af'(1) > 0
af'(1) વિશે કઈ તારણ ન મળે.

65.

$Error converting from MathML to accessible text.$ જ્યાં [•] એ મહત્તમ પૂર્ણાંક ભાગ વિધેય છે.

1
-1
0
લક્ષનું અસ્તિત્વ નથી.

66.

વિધેય f એ અનૃણ સતત વાસ્તવિક વિધેય છે. જો x ≥ 1 માટે f(x) ≤ p f(x), જ્યાં p > 0 અને f(1) = 0 હોય, તો $bold left square bracket bold f bold left parenthesis square root of bold e bold right parenthesis end root bold space bold plus bold space bold f bold left parenthesis square root of bold pi bold right parenthesis bold right square bracket bold space$

∉R
R^-
R⁺
{0}

67.

bold જ ો bold space bold f bold " bold left parenthesis bold x bold right parenthesis bold space bold equals bold space fraction numerator bold cos bold left parenthesis bold log bold space bold x bold right parenthesis over denominator bold x end fraction bold comma bold space bold f bold apostrophe bold left parenthesis bold 1 bold right parenthesis bold space bold equals bold space bold 0 bold space bold space bold અન ે bold space bold y bold space bold equals bold space bold f open parentheses fraction numerator bold 2 bold x bold space bold plus bold space bold 3 over denominator bold 3 bold minus bold 2 bold x end fraction close parentheses bold space bold ત ો bold space bold dy over bold dx bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold space

$Error converting from MathML to accessible text.$
$bold space fraction numerator bold sin bold left parenthesis bold logx bold right parenthesis bold space over denominator bold cos bold space bold x end fraction$
$fraction numerator bold 1 over denominator bold left parenthesis bold 3 bold space bold minus bold space bold 2 bold x bold right parenthesis to the power of bold 2 end fraction bold space bold sin bold space open parentheses bold log open parentheses fraction numerator begin display style bold 2 bold x bold plus bold 3 end style over denominator begin display style bold 3 bold minus bold 2 bold x end style end fraction close parentheses close parentheses$
$bold space bold sin bold space open parentheses bold log open parentheses fraction numerator bold 2 bold x bold plus bold 3 over denominator bold 3 bold minus bold 2 bold x end fraction close parentheses close parentheses bold space$

68. જો $bold lim with bold h bold rightwards arrow bold 0 below bold space fraction numerator bold e to the power of bold vertical line bold x bold vertical line bold plus bold left square bracket bold x bold right square bracket end exponent bold minus bold space bold 1 over denominator bold a bold vertical line bold x bold vertical line bold space bold plus bold space bold left square bracket bold x bold right square bracket end fraction bold space$ નું અસ્તિત્વ ન હોય, તો a = ......