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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

141.

જો f(x)= અને g(x) = , 0<x<1, તો આ અંતરાલમાં

f(x) વધતું વિધેય છે અને g(x) ઘટતું વિધેય છે.
g(x) વધતું વિધેય છે અને f(x) ઘટતું વિધેય છે.
f(x) અને g(x) બંને વધતાં વિધેય છે.
f(x) અને g(x) બંને ઘટતાં વિધેય છે.

142.

વક્ર y = [| sinx | + | cosx |] અને x² + y² = 5 વચ્ચેના ખૂણાનું માપ ........ છે. જ્યાં [.] એ પૂર્ણાંક ભાગ વિધેય છે.

$bold cos to the power of bold minus bold 1 end exponent open parentheses fraction numerator bold 2 over denominator square root of bold 5 end fraction close parentheses$
$bold tan to the power of bold minus bold 1 end exponent bold space bold 3 bold space$
$bold pi over bold 6 bold space$
$bold pi over bold 2$

143.

O કેન્દ્રવાળો ઉપવલય છે $bold x to the power of bold 2 over bold y to the power of bold 2 bold space bold plus bold space bold y to the power of bold 2 over bold b to the power of bold 2 bold space bold equals bold space bold 1$ પર P કોઈ એક બિંદુ છે. કેન્દ્ર 0 માંથી દોરેલ P પરના સ્પર્શક પર દોરેલ લંબનો લંબપાદ N છે. ધારો કે A_max એ ∆OPN નું મહત્તમ ક્ષેત્રફળ છે. $fraction numerator bold a to the power of bold 2 bold space bold minus bold space bold b to the power of bold 2 over denominator bold A subscript bold max end fraction$ ની કિંમત .....

144.

વક્ર 4x² + a²y² = 4a², 4 < a² < 8 પરનું ......... બિંદુ એ (0, -2) થી સૌથી દૂરનું બિંદુ છે.

(0, 2)
(0, -2)
(0, 1)
(0, -1)

145. $bold વ િ ધ ે ય bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table row cell bold left parenthesis bold x bold space bold plus bold space bold 1 bold right parenthesis bold 3 bold comma bold space end cell row cell bold x to the power of bold 2 over bold 3 end exponent bold minus bold 1 bold comma bold space end cell row cell bold minus bold left parenthesis bold x bold space bold minus bold 1 bold right parenthesis end cell end table close bold space bold space bold space table attributes columnalign left end attributes row cell bold minus bold 2 bold space bold less than bold x bold space bold less-than or slanted equal to bold space bold minus bold space bold 1 end cell row cell bold minus bold 1 bold space bold less than bold space bold x bold space bold less-than or slanted equal to bold space bold 1 end cell row cell bold 1 bold space bold less than bold space bold X bold less than bold space bold 2 end cell end table$

ને કેટલાં બિંદુઓએ સ્થાનીય મહત્તમ ન્યુનત્તમ મૂલ્યોનું અસ્તિત્વ હોય ?

146.

વક્ર $bold x bold space bold equals bold space bold a bold space square root of bold cos bold 2 bold theta end root bold space bold cosθ bold comma bold space bold y bold space bold equals bold space bold a square root of bold cos bold 2 bold theta end root bold space bold sinθ bold space bold ન ો bold space bold theta bold space bold equals bold pi over bold 6 bold space$ આગળનો સ્પર્શક .........

રેખા y = x ને સમાંતર છે.
રેખા x + y = 1 ને સમાંતર છે.
X-અક્ષને સમાંતર છે.
Y-અક્ષને સમાંતર છે.

X-અક્ષને સમાંતર છે.

Tips: -

bold dx over bold dθ bold space bold equals bold space bold minus bold space bold alpha bold space square root of bold cos bold 2 bold theta end root bold space bold sinθ bold space bold plus bold space open parentheses fraction numerator bold minus bold acosθ bold space bold sin bold 2 bold theta over denominator square root of bold cos begin display style bold 2 end style begin display style bold theta end style end root end fraction close parentheses bold equals bold space bold minus bold space bold a bold space fraction numerator bold left parenthesis bold cos bold 2 bold theta bold space bold sinθ bold space bold plus bold space bold cosθ bold space bold sin bold 2 bold theta bold right parenthesis over denominator square root of bold cos begin display style bold 2 end style begin display style bold theta end style end root end fraction bold space bold equals bold space fraction numerator bold minus bold acos bold 3 bold theta over denominator square root of bold cos begin display style bold 2 end style begin display style bold theta end style end root end fraction bold dy over bold dθ bold space bold equals bold space bold a bold space square root of bold cos bold 2 bold theta end root bold space bold cosθ bold space open parentheses fraction numerator bold minus bold asinθ bold space bold sin bold 2 bold theta over denominator square root of bold cos begin display style bold 2 end style begin display style bold theta end style end root end fraction close parentheses bold space bold equals bold space fraction numerator begin display style bold acos bold 3 bold theta end style over denominator begin display style square root of bold cos begin bold style 2 end style begin bold style theta end style end root end style end fraction bold therefore bold space fraction numerator begin display style bold dy end style over denominator begin display style bold dx end style end fraction bold equals bold space bold cot bold 3 bold theta bold space bold space bold space bold આથ ી bold space open parentheses fraction numerator begin display style bold dy end style over denominator begin display style bold dx end style end fraction close parentheses subscript bold theta bold equals bold pi over bold 6 end subscript bold equals bold 0 bold સ ્ પર ્ શક bold space bold X bold minus bold અક ્ ષન ે bold space bold સમ ાં તર bold space bold છ ે bold. bold space

147.

જો α β અને γ એ સમીકરણ x³ + x² - 5x - 1 = 0 નાં બીજ હોય, તો [α] + [β] + [γ] = ........

2
-3
4
5

148.

જો a, b > 0 તો y = $fraction numerator bold b to the power of bold 2 over denominator bold a bold minus bold x end fraction bold space bold plus bold space bold a to the power of bold 2 over bold x$ , 0 < x < a નું ન્યુનતમ મૂલ્ય ...... થાય.

$fraction numerator bold ab over denominator bold a bold plus bold b end fraction$
$bold 1 over bold a bold plus bold 1 over bold b$
$fraction numerator bold left parenthesis bold a bold plus bold b bold right parenthesis to the power of bold 2 over denominator bold a end fraction$
$fraction numerator begin display style bold a bold space bold plus bold space bold b end style over denominator begin display style bold a end style end fraction$

149.

bold વ િ ધ ે ય bold space bold g bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold tanx bold. bold space bold ex bold space bold e to the power of bold x to the power of bold 2 end exponent bold space fraction numerator bold log bold left parenthesis bold pi bold space bold plus bold space bold x bold right parenthesis over denominator bold log bold left parenthesis bold e bold space bold plus bold space bold x bold right parenthesis bold space end fraction bold left parenthesis bold x bold greater-than or slanted equal to bold 0 bold right parenthesis bold space bold એ

$open square brackets 0 comma straight pi over e close square brackets$ પર વધતું તથા $open square brackets bold pi over bold e bold comma bold infinity close square brackets$ પર ઘટતું વિધેય છે.
$open square brackets bold 0 bold comma bold pi over bold e close square brackets$ પર ઘટતું તથા $open square brackets bold pi over bold e bold comma bold infinity close square brackets$ પર વધતું વિધેય છે.
[0, ∞]પર વધતું વિધેય છે.
[0, ∞]પર ઘટતું વિધેય છે.