f : R → R વિકલનીય વિધેય છે. જો f(y) f(x - y) = f(x), ∀x, y ∈ R અને f'(0) = p, f'(5) = q, p, q # 0 તો f'(-5) = ....... from Mathematics લક્ષ-સાતત્ય અને વિકલન

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Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

121.

f(x) = sin x + cos x, 0  ≤ x  ≤ 2bold pi એ ...... અંતરાલમાં ચુસ્ત ઘટતું વિધેય છે. 

  • open parentheses fraction numerator bold 5 bold pi over denominator bold 4 end fraction bold comma bold 2 bold pi close parentheses
  • bold left parenthesis bold 0 bold comma bold space bold 2 bold pi bold right parenthesis
  • open parentheses bold 0 bold comma bold pi over bold 4 close parentheses
  • open parentheses bold pi over bold 4 bold comma bold pi over bold 4 close parentheses

122. bold જ ો bold space bold 3 bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold minus bold space bold 2 bold f open parentheses bold 1 over bold x close parentheses bold space bold equals bold space bold x bold comma bold space bold ત ો bold space bold f bold apostrophe bold left parenthesis bold 2 bold right parenthesis bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space
  • bold 1 over bold 2
  • bold 2 over bold 7
  • bold 7 over bold 2
  • 2


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123.
f : R → R વિકલનીય વિધેય છે. જો f(y) f(x - y) = f(x), ∀x, y ∈ R અને f'(0) = p, f'(5) = q, p, q # 0 તો f'(-5) = .......
  • q

  • bold p over bold q bold space
  • bold p to the power of bold 2 over bold q bold space
  • bold q over bold p

C.

bold p to the power of bold 2 over bold q bold space

Tips: -

f(y) f(x - y) = f(x) નું x પ્રત્યે વિકલન કરતાં, f(y) f'(x - y) = f'(x)

x = y મૂકતાં, f(x) f'(0) = f'(x)


∴ f'(x) = p f(x)                                                       (1)

bold therefore bold space bold integral bold space fraction numerator bold f bold apostrophe bold left parenthesis bold x bold right parenthesis over denominator bold f bold left parenthesis bold x bold right parenthesis end fraction bold space bold dx bold space bold equals bold space bold integral bold space bold p bold space bold dx bold space bold plus bold space bold c


∴ logef(x) = px + c                                                (2)

સમીકરણ (1) માં x = 0 મૂકતાં, f'(0) = p f(0). આથી p = pf(0).  આથી f(0) = 1

∴ loge f(0) = c

સમીકરણ (2) પરથી, આથી loge 1 = c. તેથી c = 0


∴ logef(x) = px


∴ f(x) = epx. આથી f'(x) = pepx


∴ f'(5) = pe5p = q. આથી e5p =


∴ f'(-5) = pe-5pfraction numerator bold p over denominator bold q bold divided by bold p end fraction bold space bold equals bold space bold p to the power of bold 2 over bold q


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124.
અરિક્ત ગણ A, B માટે  f : A → B અને g : B → A એવાં વિધેય છે જ્યાં f(g(x)) = x, ∀ x ∈ B. નીચેનામાંથી કયા વિધાન સત્ય (T) અને મિથ્યા (F) છે ? 

(1) વિધેય f એક-એક વિધેય છે. 
(2) વિધેય f વ્યાત્પ વિધેય છે. 
(3) વિધેય g એક-એક વિધેય છે. 
(4) વિધેય g વ્યાપ્ત વિધેય છે.
  • FTFT

  • TTFF 

  • TFTT

  • FTTF


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125.
જો વક્ર xy + ax + by = 0 ને (1, 1) આગળનો સ્પર્શક X-અક્ષ સાથે tan-1 2 માપનો ખૂણો બનાવે, તો fraction numerator bold a bold space bold plus bold space bold b bold space over denominator bold ab bold space end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold space
  • 0

  • 1

  • bold 1 over bold 2
  • fraction numerator bold minus bold 1 over denominator bold 2 end fraction

126. bold lim with bold n bold rightwards arrow bold infinity below bold space open parentheses root index bold 3 of bold n to the power of bold 2 bold space bold minus bold space bold n to the power of bold 3 bold space end root bold plus bold space bold n close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space
  • bold minus bold 1 over bold 3
  • bold 2 over bold 3
  • bold minus bold 2 over bold 3
  • fraction numerator begin display style bold 1 end style over denominator begin display style bold 3 end style end fraction

127.
જો a < b < c, f(x) એ (a, c) પર ચુસ્ત રીતે વધતું વિધેય હોય અને f(x) એ [a, c] પર સતત હોય તો .......
  • (b - c) f(a) + (c - b) f(b) > (c - a) f(c)

  • (b - a) f(c) + (c - b) f(a) > (c - a) f(b) 

  • (b - a) f(c) + (c - b) f(a) < (c - a) f(b) 

  • (b - c) f(a) + (c - b) f(b) > (c - a) f(c) 


128.
R ત્રિજ્યાવાળા વર્તુળમાં અંતર્ગત ત્રિકોણની બાજુનો શૂન્યેત્તર વૃદ્ધિદર એ તેન સામેની બાજુના ખૂણાના વૃદ્દિદર કરતા Rગણો છે. આ ખૂણાનું માપ ..... થાય. 
  • bold pi over bold 2
  • bold pi over bold 3
  • bold pi over bold 4
  • bold pi over bold 6

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129.

વિધેય f : (0, ∞) → (0, ∞) માટે,

(1) f(ab) = f(a) f(b) અને
(2) bold lim with bold x bold rightwards arrow bold infinity below f(x) = c, (જ્યાં ક # 0) પ્રકારનું છે. f(4) = ....

  • 1

  • 2

  • 3

  • 4


130. bold જ ો bold space bold v bold space bold e to the power of bold u over bold v to the power of bold 3 end exponent bold space bold equals bold space bold 1 bold space bold ત ો bold space bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold.
  • bold v bold space fraction numerator bold d to the power of bold 2 bold u over denominator bold dv to the power of bold 2 end fraction bold space bold plus bold space bold 2 bold space bold du over bold dv bold space bold plus bold space bold 3 bold v to the power of bold 2 bold space bold equals bold space bold 0 bold space
  • bold v bold space fraction numerator bold d to the power of bold 2 bold u over denominator bold dv to the power of bold 2 end fraction bold space bold plus bold space bold 2 bold space bold du over bold dv bold space bold equals bold space bold space bold 3 bold v to the power of bold 2 bold space
  • bold v bold space fraction numerator bold d to the power of bold 2 bold u over denominator bold dv to the power of bold 2 end fraction bold space bold minus bold space bold 2 bold space bold du over bold dv bold space bold plus bold space bold 3 bold v to the power of bold 2 bold space bold equals bold space bold 0 bold space
  • fraction numerator bold d to the power of bold 2 bold u over denominator bold dv to the power of bold 2 end fraction bold space bold minus bold space bold 2 bold space bold du over bold dv bold space bold equals bold space bold 3 bold v to the power of bold 2 bold space

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