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Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

171.

વાસ્તવિક વિધેય f એ f (x, y ∈ R) સંબંધ ધરાવે છે. જો f(0) = 2 હોય તો f એ

(0, ∞) પર વધતું તથા (-∞, 0) પર ઘટતું વિધેય છે.
f વિશે કઈ શકાય નહિ.
R પર વધતું વિધેય છે.
R પર ઘટતું વિધેય છે.

R પર વધતું વિધેય છે.

Tips: -

આપેલ સમીકરણમાં x = y = 0 મૂકતાં,

$bold f bold left parenthesis bold 0 bold right parenthesis bold space bold equals bold space fraction numerator bold 4 bold f bold left parenthesis bold 0 bold right parenthesis bold space bold minus bold space bold 4 over denominator bold 6 end fraction bold. bold space bold space bold space bold space bold આથ ી bold space bold f bold left parenthesis bold 0 bold right parenthesis bold space bold equals bold space bold minus bold space bold 2 bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold left parenthesis bold 1 bold right parenthesis bold space$

આપેલ સમીકરણમાં y = 0 મૂકતાં

$bold therefore bold space bold f open parentheses bold x over bold 3 close parentheses bold space bold equals bold space fraction numerator bold 2 bold f bold left parenthesis bold x bold right parenthesis bold space bold plus bold space bold 2 bold f bold left parenthesis bold 0 bold right parenthesis bold space bold minus bold space bold 4 over denominator bold 6 end fraction bold space bold equals bold space fraction numerator bold 2 bold f bold left parenthesis bold x bold right parenthesis bold space bold minus bold space bold 4 bold space bold minus bold 4 bold space over denominator bold 6 end fraction bold space bold equals bold space fraction numerator bold f bold left parenthesis bold x bold 0 bold space bold minus bold space bold 4 over denominator bold 3 end fraction bold therefore bold f open parentheses bold x over bold 3 close parentheses bold space bold equals bold space fraction numerator bold f bold left parenthesis bold x bold right parenthesis bold space bold minus bold space bold 4 over denominator bold 3 end fraction bold. bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold આથ ી bold space bold space bold space bold space bold f bold left parenthesis bold 3 bold x bold right parenthesis bold space bold equals bold space bold 3 bold f bold left parenthesis bold x bold right parenthesis bold space bold plus bold space bold 4 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 left parenthesis bold 2 bold right parenthesis bold space bold હવ ે bold comma bold space bold f bold apostrophe bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold lim with bold h bold rightwards arrow bold 0 below bold space fraction numerator bold f bold left parenthesis bold x bold space bold plus bold space bold h bold right parenthesis bold minus bold f bold left parenthesis bold x bold right parenthesis over denominator bold h end fraction bold equals bold space bold space bold lim with bold h bold rightwards arrow bold 0 below bold space fraction numerator bold f open parentheses begin display style fraction numerator bold 3 bold x bold space bold plus bold space bold 3 bold h over denominator bold 3 end fraction end style close parentheses bold minus bold f bold left parenthesis bold x bold right parenthesis over denominator bold h end fraction bold equals bold space bold space bold lim with bold h bold rightwards arrow bold 0 below bold space fraction numerator begin display style fraction numerator bold 2 bold f bold left parenthesis bold 3 bold x bold right parenthesis bold space bold plus bold space bold 2 bold f bold left parenthesis bold 3 bold h bold right parenthesis bold space bold minus bold space bold 4 bold space over denominator bold 6 end fraction end style over denominator bold h end fraction bold equals bold space bold space bold lim with bold h bold rightwards arrow bold 0 below bold space fraction numerator bold 2 bold f bold left parenthesis bold 3 bold x bold right parenthesis bold space bold plus bold space bold 2 bold f bold left parenthesis bold 3 bold h bold right parenthesis bold space bold minus bold space bold 4 bold space bold minus bold space bold 6 bold space bold f bold left parenthesis bold x bold right parenthesis over denominator bold 6 bold h end fraction bold equals bold space bold space bold lim with bold h bold rightwards arrow bold 0 below bold space fraction numerator bold 6 bold f bold left parenthesis bold x bold right parenthesis bold space bold plus bold space bold 8 bold space bold plus bold space bold 2 bold space bold f bold left parenthesis bold 3 bold h bold right parenthesis bold space bold minus bold space bold 4 bold space bold 6 bold f bold left parenthesis bold x bold right parenthesis over denominator bold 6 bold h 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 left parenthesis bold left parenthesis bold 2 bold right parenthesis bold પરથ ી bold space bold right parenthesis bold equals bold space bold lim with bold h bold rightwards arrow bold 0 below bold space fraction numerator bold f bold left parenthesis bold 3 bold x bold right parenthesis bold space bold plus bold space bold 2 over denominator bold 3 bold h end fraction bold equals bold space bold space fraction numerator bold f bold left parenthesis bold 3 bold h bold right parenthesis bold space bold minus bold space bold f bold left parenthesis bold 0 bold right parenthesis over denominator bold 3 bold h 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 left parenthesis bold left parenthesis bold 1 bold right parenthesis bold space bold પરથ ી bold right parenthesis$

= f'(0) = 2 (f'(0) = 2)

આથી f(x) = 2x + c

∴ f એ R પર વધતું વિધેય છે.

172. જો વક્ર ax² + by² = 1 અને ax² + by² = 1 લંબચ્છેદી હોય તો

$bold 1 over bold a bold space bold minus bold space bold 1 over bold b bold space bold equals bold space fraction numerator bold 1 over denominator bold a bold apostrophe end fraction bold space bold minus bold space fraction numerator bold 1 over denominator bold b bold apostrophe end fraction$
$bold 1 over bold a bold space bold plus bold space bold 1 over bold b bold space bold equals bold space fraction numerator bold 1 over denominator bold a bold apostrophe end fraction bold space bold plus bold space fraction numerator bold 1 over denominator bold b bold apostrophe end fraction$
$bold 1 over bold a bold space bold plus bold space bold 1 over bold b bold space bold equals bold space fraction numerator bold 1 over denominator bold a bold apostrophe end fraction bold space bold minus bold space fraction numerator bold 1 over denominator bold b bold apostrophe end fraction$
$bold 1 over bold a bold space bold minus bold space bold 1 over bold b bold space bold equals bold space fraction numerator bold 1 over denominator bold a bold apostrophe end fraction bold space bold plus bold space fraction numerator bold 1 over denominator bold b bold apostrophe end fraction$

173.

વક્ર y = x³ ને ઉગમબિંદુ સિવાયના બિંદુ P₁ આગળનો સ્પર્શક વક્રને ફરીથી બિંદુ P₂ માં મળે છે. P₂ આગળનો સ્પર્શક વક્રને ફરીથી બિંદુ P₃માં મળે છે, તો આ જ રીતે આગળ વધતાં બિંદુ P₁, P₂, P₃, ....... P_n ના X-યામ ......

સ્વરિત શ્રેણીમાં હોય.
સમાંતર તેમજ સમગુણોત્તર શ્રેણીમાં હોય.
સમગુણોત્તર શ્રેણીમાં હોય.
સમાંતર શ્રેણીમાં હોય.

174.

bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold lim with bold n bold rightwards arrow bold infinity below bold space open curly brackets fraction numerator bold cos bold space bold πx bold space bold minus bold space bold x to the power of bold 2 bold n end exponent bold space bold sin bold left parenthesis bold x bold space bold minus bold 1 bold right parenthesis over denominator bold 1 bold space bold plus bold space bold x to the power of bold 2 bold n bold plus bold 1 end exponent bold space bold minus bold space bold x to the power of bold 2 bold n end exponent end fraction close curly brackets

f એ x = 1 આગળ સતત છે.
$bold lim with bold x bold rightwards arrow bold 1 below bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold 1 bold space$
$bold lim with bold x bold rightwards arrow bold 1 below bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold minus bold space bold 1 bold space$
$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table row cell bold cos bold space bold πx end cell row cell bold minus bold 1 bold comma end cell row cell fraction numerator bold sin bold left parenthesis bold x bold space bold minus bold space bold 1 bold right parenthesis over denominator bold 1 bold space bold minus bold space bold x end fraction end cell end table close table row cell bold 0 bold space bold less than bold space bold x bold space bold space bold less than bold space bold 1 end cell row cell bold x bold space bold equals bold space bold 1 end cell row cell bold x bold space bold greater than bold space bold 1 end cell end table$

175.

જો વક્ર xy + ax + by = 0 નો (1,1) આગળનો સ્પર્શક X-અક્ષ સાથે tan^-1 2 માપનો ખૂણો બનાવે તો (a, b)

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

176. જો વક્ર $open parentheses bold x over bold a close parentheses to the power of bold n bold space bold plus bold space open parentheses bold y over bold b close parentheses to the power of bold n$ = 2 એ રેખા $bold x over bold a bold space bold plus bold space bold y over bold b$ = 2 ને સ્પર્શે તો n = ......

માત્ર 2
માત્ર 3
માત્ર 4
કોઈપણ શુન્યેત્તર વાસ્તવિક સંખ્યાં

177.

bold જ ો bold space bold P bold equals bold space bold lim with bold y bold rightwards arrow bold 0 below bold space bold 1 over bold y bold space table row bold pi row bold integral row bold 0 end table bold tan bold space bold left parenthesis bold y bold space bold sin bold space bold x bold right parenthesis bold space bold dx bold space bold અન ે bold space bold Q bold space bold equals bold space bold lim with bold y bold rightwards arrow bold 0 below bold space table row bold pi row bold integral row bold 0 end table open parentheses bold 1 bold minus bold x over bold n close parentheses to the power of bold n bold space bold e to the power of bold x over bold 3 end exponent bold space bold dx bold comma bold space bold ત ો bold space

3Q = 4P
3P = 4Q
p = 2Q = 5
2(P + Q) = 7

178.

જો f : R → R એ એવું બહુપદી વિધેય હોય, જ્યાં f(2x) = f'(x) f"(x) તો ...... (ધાત n > 2)

f(3) = 12
f એક-એક છે પરંતુ વ્યપ્ત નથી.
f એક - એક અને વ્યાપ્ત છે.
f(x) = x ને ત્રણ ભિન્ન વાસ્તવિક ઉકેલ મળે.

179.

જો વક્ર ay² = x³ ના બિંદુ P(at², at³) આગળનો સ્પર્શ વક્રને ફરીથી વક્રના બિંદુ Q(t') આગળ મળે તો t' = ........

$bold minus bold t over bold 2$
$bold space bold t over bold 2$
2t
-t

180.

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 row cell bold e to the power of bold x bold comma end cell cell bold 0 bold space bold less than bold space bold x bold space bold less-than or slanted equal to bold space bold 1 end cell row cell bold 2 bold space bold minus bold space bold e to the power of bold x bold minus bold 1 end exponent bold comma end cell cell bold 1 bold space bold less than bold space bold x bold space bold less-than or slanted equal to bold space bold 2 end cell row cell bold x bold space bold minus bold space bold e bold comma end cell cell bold 2 bold space bold less than bold space bold x bold space bold less-than or slanted equal to bold space bold 3 end cell end table close bold space bold અન ે bold space bold g bold left parenthesis bold x bold right parenthesis bold space bold equals bold space table row bold x row bold integral row bold 0 end table bold space bold f bold left parenthesis bold t bold right parenthesis bold space bold dt bold space bold ત ો bold comma

g ને મહત્તમ મૂલ્ય ન મળે.
g ને ન્યુનત્તમ મૂલ્ય ન મળે.
x = e આગળ g ને સ્થાનીય ન્યુનત્તમ તથા x = 1 + log આગળ g ને સ્થાનીય મહત્તમ મૂલ્ય મળે છે.
x = 1 આગળ g ને સ્થાનીય મહત્તમ તથા x = 2 આગળ g ને સ્થાનીય ન્યુનત્તમ મૂલ્ય મળે છે.

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Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

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