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Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

131. વક્રો x²y = 1 અને x³ = a⁵y પરસ્પર કાટખૂણે છેદે તો a⁶ = ......

132. સમીકરણ g^x-8 + 2x - 17 = 0 ને .............

એક વાસ્તવિક બીજ મળે.
બે વાસ્તવિક બીજ મળે.
આઠ વાસ્તવિક બીજ મળે.
અનંત ઉકેલો મળે.

133.

ધારો કે, f(x) એ વિકલનીય વિધેય છે અને G એ f(x)નો આલેખ છે. જો P(a, f(a)) એ G પર (0,0) થી સૌથી નજીકનું બિંદુ હોય, તો f(a) f'(a) = ........

-a
a
-1
1

134.

bold lim with bold x bold rightwards arrow bold 0 below bold space fraction numerator bold x to the power of bold 2 bold integral subscript bold 0 superscript bold x bold space square root of bold t to the power of bold 3 bold space bold plus bold space bold 1 end root bold dt bold space bold space over denominator bold integral subscript bold 0 superscript bold x bold space bold t to the power of bold 2 bold space bold costdt end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space

Tips: -

bold lim with bold x bold rightwards arrow bold 0 below bold space fraction numerator bold x to the power of bold 2 bold integral subscript bold 0 superscript bold x bold space square root of bold t to the power of bold 3 bold space bold plus bold space bold 1 end root bold dt bold space bold space over denominator bold integral subscript bold 0 superscript bold x bold space bold t to the power of bold 2 bold space bold costdt end fraction 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 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 space bold space bold space bold space bold space bold space open parentheses bold 0 over bold 0 bold સ ્ વર ૂ પ bold space close parentheses equals bold lim with bold x bold rightwards arrow bold 0 below bold space fraction numerator bold x to the power of bold 2 square root of bold x bold 3 bold space bold plus bold space bold 1 end root bold space bold plus bold space bold 2 bold x bold integral subscript bold 0 superscript bold x square root of bold t to the power of bold 3 bold space bold plus bold space bold 1 end root bold dt over denominator bold x to the power of bold 2 bold cosx end fraction equals space bold lim with bold x bold rightwards arrow bold 0 below bold space fraction numerator square root of bold x to the power of bold 3 bold space bold plus bold space bold 1 end root over denominator bold cosx end fraction bold space bold 2 bold lim with bold x bold rightwards arrow bold 0 below bold space fraction numerator bold integral subscript bold 0 superscript bold x square root of bold t to the power of bold 3 bold space bold plus bold space bold 1 end root bold dt over denominator bold xcosx end fraction 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 space bold space bold space bold space bold space open parentheses bold 0 over bold 0 bold સ ્ વર ૂ પ bold space close parentheses equals space bold 1 bold space bold plus bold space bold 2 bold space bold lim with bold x bold rightwards arrow bold 0 below bold space bold space fraction numerator square root of bold x to the power of bold 3 bold space bold plus bold space bold 1 bold space end root over denominator bold cosx bold space bold minus bold space bold xsinx end fraction bold space bold equals bold space bold 1 bold space bold plus bold space bold 2 bold space bold equals bold space bold 3

135.

વિધેય f એ દ્વિતિય વિકલન ધરાવે છે. તથા f"(t) < 0, ∀t ∈R. જો x, y ∈ R માટે, f'(y) + x < f(y + 1) હોય, તો .....

f(y) < x
f(y) < x
f(y) = x
f(y) > x

136.

વક્ર y² - x² = 1 પર બિંદુ P એવું છે કે જેનો x યામ n હોય, જ્યાં n ∈ N. જો dn એ બિંદુ Pરેખા y = x પરનું અંતર દર્શાવે, તો $bold lim with bold n bold rightwards arrow bold infinity below bold left parenthesis bold nd subscript bold n bold right parenthesis$ = .......

$bold 2 square root of bold 2 bold space bold space$
$fraction numerator bold 1 over denominator bold 2 square root of bold 2 end fraction bold space$
$bold 1 over bold 2$
1

137.

જો વક્ર y = x cos x તથા y = ના કોઈ પણ બિંદુ (x, y) આગળના સ્પર્શકો X - અક્ષને સમાંતર હોય, તો x એ અનુક્રમે ....... નાં બીજ થશે.

tan x = x, cot x = x
sin x = x, tan x = x
cot x =x, sec x = x
cot x = x, tan x = x

138.

ધારો કે, f(x) = x² - bx + c, જ્યાં b, c એ અયુગ્મ પ્રાકૃતિક સંખ્યા તથા f(x) = 0 નાં બંને બીજ અવિભાજ્ય સંખ્યાઓ છે. જો b + c = 35 હોય, તો f(x) નું ન્યુનતમ મૂલ્ય ....... છે.

$bold 81 over bold 4$
$bold minus fraction numerator begin display style bold 183 end style over denominator begin display style bold 4 end style end fraction bold space$
$fraction numerator begin display style bold 173 end style over denominator begin display style bold 16 end style end fraction$
$bold minus fraction numerator begin display style bold 81 end style over denominator begin display style bold 4 end style end fraction$

139. $bold ધ ા ર ો bold space bold ક ે bold space bold F bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell open parentheses bold 1 bold space bold plus bold space bold x bold space bold plus bold space fraction numerator bold f bold left parenthesis bold x bold right parenthesis over denominator bold x end fraction close parentheses to the power of bold 1 over bold x end exponent bold comma bold space end cell row cell bold e to the power of bold 3 bold space bold comma bold space end cell end table close table row cell bold x bold space bold # bold space bold 0 end cell row cell bold x bold space bold equals bold space bold 0 bold space end cell end table bold space bold space bold space bold space bold space bold અન ે bold space bold space bold G bold left parenthesis bold x bold right parenthesis bold space bold equals open curly brackets table attributes columnalign left end attributes row cell open parentheses bold 1 bold space bold plus bold space fraction numerator bold f bold left parenthesis bold x bold right parenthesis over denominator bold x end fraction close parentheses to the power of bold 1 over bold x end exponent bold comma bold space end cell row cell bold k bold space bold comma bold space end cell end table close table row cell bold x bold space bold # bold space bold 0 end cell row cell bold x bold space bold equals bold space bold 0 bold space end cell end table$

જ્યાં f(x) એ વાસ્તવિક વિધેય છે. જો F(x) એ x = 0 આગળ સતત હોય, તો G(x) ને x = 0 આગળ સતત થવા માટે logk ની કિંમત ...... છે.

140.

વર્તુળ અને ચોરસની પરિમિતિનો સરવાળો અચળ છે. તેમના ક્ષેત્રફળનો સરવાળો ન્યુનતમ હોય, ત્યારે ચોરસની બાજુ તથા વર્તુળની ત્રિજ્યાનો ગુણોત્તર ...... છે.

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Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

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