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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

101.

bold જ ો bold space bold y bold space bold equals bold space bold sin bold space bold x bold times bold space bold sin bold space bold 2 bold x bold space bold times bold space bold sin bold space bold 3 bold x bold space bold. bold. bold. bold. bold space bold comma bold space bold sin bold space bold nx bold comma 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 space

$bold sum from bold k bold equals bold 1 to bold n of bold space bold k bold space bold cot bold space bold kx$
$bold sum from bold k bold equals bold 1 to bold n of bold space bold k bold space bold tan bold space bold kx$
$bold y bold space bold sum from bold k bold equals bold 1 to bold n of bold space bold k bold space bold cot bold space bold kx bold space bold right square bracket$
$bold y bold space bold sum from bold k bold equals bold 1 to bold n of bold space bold k bold space bold tan bold space bold kx bold space$

102.

bold જ ો bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open parentheses bold sum from bold k bold equals bold 0 to bold n of bold a to the power of bold k bold space bold vertical line bold x bold minus bold 1 bold vertical line to the power of bold k close parentheses bold space bold જ ્ ય ાં bold space bold a subscript bold k bold greater than bold 0 bold comma bold space bold ત ો bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold એ bold space

R પર વિકલનીય છે.
R પર સતત છે. પરંતુ x = 1 આગળ વિકલનીય નથી.
x = 1 આગળ સતત નથી.
R પર સતત છે, પરંતુ x = 0 આગળ વિકલનીય નથી.

R પર સતત છે. પરંતુ x = 1 આગળ વિકલનીય નથી.

Tips: -

|x - 1|, |x -1|2, ...... એ R પર સતત છે. સતત વિધેયનો સરવાળો પણ સતત હોવાથી f(x) એ R પર સતત થશે.

વળી, |x - 1|, |x -1|3, ........ એ x = 1 આગળ વિકલનીય નથી.

∴ f(x) એ R પર સતત છે પરંતુ x = 1 આગળ વિકલનીય નથી.

103.

ધારો કે, $bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell bold xnsin bold x over bold 1 end cell cell bold x bold space bold # bold space bold 0 end cell row bold 0 cell bold x bold space bold equals bold 0 end cell end table close$ જો f(x) એ x = 0 આગળ સતત હોય, પરંતુ વિકલનીય ન હોય તો

n = 0
n∈(0,1]
n∈[1, ∞)
n∈(-∞, 0)

104.

જો xy = e - e^y તો $open parentheses fraction numerator bold d to the power of bold 2 bold y over denominator bold dx to the power of bold 2 end fraction close parentheses subscript bold x bold equals bold 0 end subscript bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold.$