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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

131.

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

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

Tips: -

વક્ર y² - x² = 1 પરનું બિંદુ P (n, $square root of bold n to the power of bold 2 bold space bold plus bold space bold 1 end root$ ) છે, જ્યાં n ∈ N
$bold dn bold space bold equals bold space open vertical bar table row cell fraction numerator square root of bold n to the power of bold 2 bold space bold plus bold space bold 1 end root bold minus bold n over denominator square root of bold 2 end fraction end cell end table close vertical bar bold space bold equals bold space fraction numerator square root of bold n to the power of bold 2 bold space bold plus bold space bold 1 end root bold minus bold n over denominator square root of bold 2 end fraction bold lim with bold n bold rightwards arrow bold infinity below bold space bold n bold times bold d subscript bold n bold space bold equals bold space fraction numerator bold 1 over denominator square root of bold 2 end fraction bold space bold lim with bold n bold rightwards arrow bold infinity below bold n open parentheses square root of bold n to the power of bold 2 bold space bold plus bold space bold 1 end root bold minus bold n close parentheses bold equals bold space fraction numerator bold 1 over denominator square root of bold 2 end fraction bold space bold lim with bold n bold rightwards arrow bold infinity below bold space fraction numerator bold n open parentheses square root of bold n to the power of bold 2 bold space bold plus bold space bold 1 end root bold minus bold n to the power of bold 2 close parentheses over denominator open parentheses square root of bold n to the power of bold 2 bold space bold plus bold space bold 1 end root bold plus bold n close parentheses end fraction bold equals bold space fraction numerator bold 1 over denominator square root of bold 2 end fraction bold space bold lim with bold n bold rightwards arrow bold infinity below bold space fraction numerator bold 1 over denominator square root of bold 1 bold plus begin display style bold 1 over bold n to the power of bold 2 end style bold plus bold 1 end root end fraction bold equals bold space fraction numerator bold 1 over denominator bold 2 square root of bold 2 end fraction$

132.

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

133.

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

134.

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

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

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

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.

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

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

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.

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