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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

21.

bold જ ો bold space bold y bold space bold equals bold space bold tan bold minus bold 1 bold space open parentheses fraction numerator bold log bold space bold e bold space bold x to the power of bold minus bold 2 end exponent over denominator bold log bold space bold e bold space bold x to the power of bold 2 end fraction close parentheses bold space bold plus bold space bold tan to the power of bold minus bold 1 end exponent bold space open parentheses fraction numerator bold 3 bold plus bold 2 bold space bold log bold space bold x over denominator bold 1 bold space bold minus bold space bold 6 bold space bold log bold space bold x end fraction close parentheses bold space bold ત ો bold space bold dy over bold dx bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold.

22.

bold lim with bold x bold rightwards arrow bold infinity below bold space bold x bold space bold sum from bold i bold space bold equals bold space bold 1 to bold n of open parentheses fraction numerator bold 1 over denominator bold e to the power of bold x bold minus bold 1 end fraction close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold.

$fraction numerator bold n to the power of bold 2 bold left parenthesis bold n bold plus bold 1 bold right parenthesis to the power of bold 2 over denominator bold 4 end fraction$
$fraction numerator bold n bold plus bold 1 over denominator bold 2 end fraction bold space$
$fraction numerator bold n to the power of bold 2 bold plus bold 1 over denominator bold 2 end fraction$
$fraction numerator bold n bold left parenthesis bold n bold plus bold 1 bold right parenthesis over denominator bold 2 end fraction$

23.

જો $bold tan bold space bold y over bold 2 bold space bold equals bold space fraction numerator square root of bold 1 bold minus bold e end root over denominator square root of bold 1 bold plus bold e end root end fraction bold space bold tan bold space bold x over bold 2 bold space bold ત ો bold space bold dy over bold dx bold space bold equals bold space bold. bold. bold. bold. bold. bold.$

$square root of fraction numerator bold 1 bold minus bold e to the power of bold 2 over denominator bold 1 bold plus bold e bold space bold cosx end fraction end root$
$fraction numerator square root of bold 1 bold plus bold e to the power of bold 2 end root over denominator bold 1 bold space bold plus bold space bold e bold space bold cosx end fraction$
$fraction numerator square root of bold 1 bold space bold minus bold space bold e to the power of bold 2 end root over denominator bold 1 bold space bold minus bold space bold e bold space bold cosx end fraction$
$fraction numerator square root of bold 1 bold minus bold e end root to the power of bold 2 over denominator bold 1 bold plus bold e bold space bold cosx to the power of bold 2 end fraction$

24.

bold જ ો bold space bold space bold y bold space bold equals bold space fraction numerator bold 1 over denominator square root of bold cos bold space bold 2 bold x end root end fraction bold space bold ત ો bold space bold y to the power of bold 2 bold space bold plus bold space bold y bold space bold. bold. bold. bold. bold. bold. bold. bold. bold space

y⁵
2y⁵
3y⁵
4y⁵

25.

જો x = sec θ - cis θ અને y - secⁿθ - cosⁿ θ, તો (x² + 4) $open parentheses bold dy over bold dx close parentheses to the power of bold 2$ = ......... (y² + 4)

n²

Tips: -

x² = sec² θ - 2 + cos² θ અને y² = sec²n θ - 2 + cos²n θ

∴ x² + 4 = (sec θ + cos θ)², y² + 4 = (sec θ + cosn θ)²

હવે, x = sec θ - cos θ

∴ $bold dx over bold dθ$ =sec θ tan θ + sin θ = sin θ sec θ (sec θ + cos θ)

∴ $bold dx over bold dθ$ = n sec^n-1 θ sec θ tan θ + n cos^n-1θ sin θ

= n sin θ sec θ (secn θ + cosn θ)

$bold therefore bold space open parentheses bold dy over bold dx close parentheses to the power of bold 2 bold space bold equals bold space fraction numerator bold n to the power of bold 2 bold left parenthesis bold secn bold space bold theta bold space bold plus bold space bold cosn bold space bold right parenthesis to the power of bold 2 over denominator bold left parenthesis bold sec bold space bold space bold plus bold space bold cos bold space bold right parenthesis to the power of bold 2 end fraction bold space bold equals bold space fraction numerator bold n to the power of bold 2 bold space bold left parenthesis bold y to the power of bold 2 bold space bold plus bold space bold 4 bold right parenthesis over denominator bold left parenthesis bold x to the power of bold 2 bold space bold plus bold space bold 4 bold right parenthesis end fraction bold therefore bold space bold left parenthesis bold x bold 2 bold space bold plus bold space bold 4 bold right parenthesis bold space open parentheses bold dy over bold dx close parentheses to the power of bold 2 bold space bold equals bold space bold n to the power of bold 2 bold left parenthesis bold y to the power of bold 2 bold space bold plus bold space bold 4 bold right parenthesis$

26.

bold x bold space square root of bold 1 bold plus bold x end root bold space bold plus bold space bold y bold space square root of bold 1 bold plus bold y end root bold space bold equals bold space bold 0 bold space bold મ ા ટ ે bold space bold dy over bold dx bold space bold equals bold space bold. bold comma bold. bold. bold. bold. bold. bold. bold. bold space

$fraction numerator bold 1 over denominator bold left parenthesis bold 1 bold plus bold x bold right parenthesis to the power of bold 2 end fraction bold space$
$fraction numerator bold 1 over denominator bold 1 bold plus bold x to the power of bold 2 end fraction bold space$
$fraction numerator bold minus bold 1 over denominator bold 1 bold plus bold x to the power of bold 2 end fraction bold space$
$fraction numerator bold minus bold 1 over denominator bold 1 bold plus bold x bold right parenthesis to the power of bold 2 end fraction$

27.

bold lim with bold n bold rightwards arrow bold infinity below bold space bold cos bold space bold x over bold 2 bold space bold cos bold space bold x over bold 4 bold space bold cos bold space bold x over bold 8 bold space bold. bold. bold. bold. bold. bold space bold cos bold space bold x over bold 2 to the power of bold n bold space bold space bold. bold. bold. bold. bold. bold. bold. bold space

$Error converting from MathML to accessible text.$
$Error converting from MathML to accessible text.$
sin x
cos x

28.

જો f(x) = p |sin x| + qe^|x| + r|x|³ એ x = 0 આગળ વિકલનીય હોય તો.........

q = 0, r = 0, p ∈ R
r = 0, p = 0, q ∈ R
p + q = 0, p ∈ R
q + r = 0, p∈ R

29.

bold જ ો bold space bold space bold 3 bold f bold left parenthesis bold x bold right parenthesis bold space bold minus bold space bold 2 bold space bold f open parentheses bold 1 over bold x close parentheses bold space bold equals bold space bold x bold space bold હ ો ય 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

$bold 1 over bold 2$
$fraction numerator begin display style bold 2 end style over denominator begin display style bold 7 end style end fraction$
$bold 7 over bold 2$
2