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Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

191.

જો વક્ર y = f(x) ઓ સ્પર્શક જે બિંદુનો x યામ 1 હોય તે બિંદુએ અક્ષની ધન દિશા સામે $bold pi over bold 6$ માપનો ખુણો બનાવે છે તથા જે બિંદુના x - યામ અનુક્રમે 2 તથા 3 હોય તે બિંદુએ $bold pi over 3$ તથા $bold pi over bold 4$ માપના ખૂણા બનાવે છે, તો $table row bold 3 row bold integral row bold 1 end table bold f bold " bold left parenthesis bold x bold right parenthesis bold space bold f bold " bold left parenthesis bold x bold right parenthesis bold space bold dx bold space bold plus bold space table row bold 3 row bold integral row bold 2 end table bold f bold " bold left parenthesis bold x bold right parenthesis bold space bold dx bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold space$

$fraction numerator bold 3 square root of bold 3 over denominator bold 4 end fraction$
$fraction numerator begin display style bold 4 bold minus bold 3 square root of bold 3 end style over denominator begin display style bold 3 end style end fraction$
$fraction numerator bold 4 bold plus square root of bold 3 over denominator bold 3 end fraction$
$fraction numerator bold 4 bold plus bold 3 square root of bold 3 over denominator bold 3 end fraction$

fraction numerator begin display style bold 4 bold minus bold 3 square root of bold 3 end style over denominator begin display style bold 3 end style end fraction

Tips: -

આપેલ છે કે x = 1 આગળ $bold dy over bold dx bold space bold equals bold space bold tan bold space bold pi over bold 6 bold space bold equals bold space fraction numerator bold 1 over denominator square root of bold 3 end fraction$ એટલે કે $bold f bold apostrophe bold left parenthesis bold 1 bold right parenthesis bold space bold equals bold space fraction numerator bold 1 over denominator square root of bold 3 end fraction bold.$ તે જ રીતે f'(2) = $square root of bold 3$ f'(3) = 1

$table row bold 3 row bold integral row bold 1 end table bold f bold " bold left parenthesis bold x bold right parenthesis bold space bold f bold " bold left parenthesis bold x bold right parenthesis bold space bold dx bold space bold plus bold space table row bold 3 row bold integral row bold 2 end table bold f bold " bold left parenthesis bold x bold right parenthesis bold space bold dx bold space bold equals bold space table row cell bold f bold apostrophe bold left parenthesis bold 3 bold right parenthesis end cell row bold integral row cell bold f bold apostrophe bold left parenthesis bold 1 bold right parenthesis end cell end table bold t bold space bold dt bold space bold plus bold space open square brackets bold f bold apostrophe bold left parenthesis bold x bold right parenthesis close square brackets subscript bold 2 superscript bold 3 bold space bold space bold space bold left parenthesis bold f bold " bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold t bold space bold લ ે ત ાં bold space bold f bold " bold left parenthesis bold x bold right parenthesis bold dx bold space bold equals bold space bold dt bold right parenthesis bold space bold equals bold space bold 1 over bold 2 open square brackets bold t to the power of bold 2 close square brackets subscript bold f bold apostrophe bold left parenthesis bold 1 bold right parenthesis end subscript superscript bold f bold apostrophe bold left parenthesis bold 3 bold right parenthesis end superscript bold space bold plus bold space bold f bold left parenthesis bold 3 bold right parenthesis bold space bold equals bold space bold f bold apostrophe bold left parenthesis bold 2 bold right parenthesis bold space bold equals bold space bold 1 over bold 2 bold space open square brackets bold left parenthesis bold f bold apostrophe bold left parenthesis bold 3 bold right parenthesis to the power of bold 2 bold space bold minus bold space bold left parenthesis bold f bold apostrophe bold left parenthesis bold 1 bold right parenthesis to the power of bold 2 close square brackets bold space bold plus bold space bold 1 bold space bold minus bold space square root of bold 3 bold equals bold space bold 1 over bold 2 open square brackets bold 1 bold minus bold space bold 1 over bold 3 close square brackets bold space bold plus bold space bold 1 bold space bold minus bold space square root of bold 3 bold space bold equals bold space fraction numerator bold 4 bold space bold minus bold 3 square root of bold 3 over denominator bold 3 end fraction$

192.

વક્ર $bold y bold space bold equals bold space bold x bold space bold tan bold space bold alpha bold space bold minus bold space bold 1 over bold 2 bold space fraction numerator bold x to the power of bold 2 over denominator bold u to the power of bold 2 bold space bold cos to the power of bold 2 bold space bold alpha end fraction bold comma bold space bold alpha bold space bold element of bold space open parentheses bold 0 bold comma bold pi over bold 2 close parentheses$ ના બિંદુ P આગળનો સ્પર્શક રેખા y = x + 5 ને સમાંતર છે. જો બિંદુ P નો y - યામ $bold u to the power of bold 2 over bold 4$ હોય તો α = ......

$bold pi over bold 3$
$bold pi over bold 12$
$bold pi over bold 6$
$bold pi over bold 4$

193. જો બિંદુ (0, 3) અને (5, -2)ને જોડતી રેખા એ વક્ર $bold y bold space bold equals bold space fraction numerator bold c over denominator bold x bold space bold plus bold space bold 1 end fraction$ નો સ્પર્શક હોય, તો c = .......

194. સમીકરણ x³ + 2x² + 5x + 2 cos x = 0 ને [0, 2 $bold pi$ ] માં કેટલા ઉકેલ મળે ?

195.

ધારો કે f(x) = ax³ + bx² + cx + d જ્યાં a, b, c, d ∈Rઅને 3b² < c² એ વધતું વિધેય છે અને g(x) = af (x) + bf"(x) + c². જો G(x) = g(t) dt, α ∈R તો α < x < α + 1 માટે

G(x) એ વધતું વિધેય છે.
G(x) એ ઘટતું વિધેય છે.
G(x) એ એક-એક વિધેય છે.
G(x) એ વધતું કે ઘટતું વિધેય નથી.

196.

જો વિધેય f(x) = 2x³ - 9ax² + 12a²x + 1 ને x = x₁ આગળ સ્થાનીય મહત્તમ અને x = x₂ આગળ સ્થાનીય ન્યુઅનતમ મળે જ્યાં x₂ = x₁² તો a = ........

0
2
$bold 1 over bold 4$
(A) અથવા (c)

197. R ત્રિજ્યાવાળા ગોલકની અંતર્ગત મહત્તમ ઘનફળ ધરાવતા નળાકારની ઊંચાઈ ....... થશે.

R
$fraction numerator bold 2 bold R over denominator square root of bold 3 end fraction$
$fraction numerator bold 5 bold R over denominator bold 4 end fraction$
$fraction numerator begin display style bold R end style over denominator begin display style square root of bold 3 end style end fraction$

198.

વિધેય $bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold x to the power of bold 3 over bold 2 end exponent bold space bold plus bold space bold x to the power of fraction numerator bold minus bold 3 over denominator bold 2 end fraction end exponent bold minus bold 4 bold space open parentheses bold x bold space bold plus bold space bold 1 over bold x close parentheses$ નું ન્યુનતમ મૂલ્ય ....... છે.

-10
10
0
ન મળે.

199. વિધેય $bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space fraction numerator bold tan to the power of bold 2 bold x bold space bold minus bold space bold cot to the power of bold 2 bold space bold x bold space bold plus bold space bold 1 over denominator bold tan to the power of bold 2 bold x bold space bold plus bold space bold cot to the power of bold 2 bold x bold minus bold 1 end fraction$ માટે નીચેનામાંથી કયા વિધાન સત્ય છે ?

f(x)નું વૈશ્વિક ન્યુઅનતમ મૂલ્ય -1 છે.
f(x)નું વૈશ્વિક મહત્તમ મુલ્ય $bold 5 over bold 3$ છે.
f(x)ને વૈશ્વિક ન્યુનતમ મૂલ્ય ન મળે.
f(x)ને વૈશ્વિક મહતમ મૂલ્ય ન મળે.

200. વક્ર $bold y bold space bold equals bold space bold x to the power of bold 1 over bold 3 end exponent$ (1 - cos x) ને x = 0 આગળના સ્પર્શકનું સમીકરણ .... થશે.

x = 0
x = 1
y = 0
y = 1

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Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

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