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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

101.

bold જ ો bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space fraction numerator bold sin to the power of bold 2 bold space bold x over denominator bold 1 bold plus bold cot bold space bold x bold space end fraction bold space bold plus bold space fraction numerator bold cos to the power of bold 2 bold space bold x over denominator bold 1 bold plus bold tan bold space bold x bold space end fraction bold space bold ત ો bold space bold f bold apostrophe bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold space

sin 2x
-sin 2x
cos 2x
-cos 2x

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.

103.

bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space fraction numerator bold vertical line bold x bold vertical line over denominator bold vertical line bold x bold vertical line bold space bold plus bold space bold 2016 end fraction

એ કેટલા બિંદુએ વિકલનીય નથી ?

0
1
2
અનંત

104.

bold જ ો bold space square root of bold r bold space bold equals bold space bold ae bold space to the power of bold theta bold space bold cot bold space bold alpha end exponent bold space bold ત ો bold space fraction numerator bold d to the power of bold 2 bold r over denominator bold dθ to the power of bold 2 end fraction bold space bold minus bold space bold 4 bold space bold r bold space bold cot to the power of bold 2 bold space bold alpha bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold space bold left parenthesis bold a bold comma bold space bold alpha bold space bold element of bold space bold R bold right parenthesis

0
1
$bold 1 over bold r$
r

105.

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

e²
$bold 1 over bold e$
$bold 1 over bold e to the power of bold 3$
$bold 1 over bold e to the power of bold 2$

bold 1 over bold e to the power of bold 2

Tips: -

xy = e = -e^y (1)

$bold x bold dy over bold dx bold space bold plus bold space bold y bold space bold equals bold space bold minus bold ey bold space bold dy over bold dx bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold left parenthesis bold 2 bold right parenthesis bold x bold dy over bold dx bold space bold plus bold space bold y bold space bold equals bold space bold minus bold e to the power of bold y bold space fraction numerator bold d to the power of bold 2 bold y over denominator bold dx to the power of bold 2 end fraction bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold left parenthesis bold 3 bold right parenthesis$

(1) માં x = 0 મૂકતાં, 0 = e -gy(0)

∴ ey(0) = e1. આથી y (0) = 1

(2)માં મૂકતાં, y(0) = $Error converting from MathML to accessible text.$

$bold 1 bold space bold equals bold space bold minus bold e open parentheses bold dy over bold dx close parentheses subscript bold x bold equals bold 0 end subscript bold space bold. bold space bold આથ ી bold space open parentheses bold dy over bold dx close parentheses subscript bold x bold equals bold 0 end subscript bold space bold equals bold space bold minus bold 1 over bold e bold left parenthesis bold 3 bold right parenthesis bold space bold મ ાં bold space bold x bold space bold equals bold space bold 0 bold space bold મ ૂ કત ાં bold comma bold space bold 2 bold space open parentheses bold dy over bold dx close parentheses subscript bold x bold equals bold 0 end subscript bold space bold equals bold space bold minus bold e to the power of bold y to the power of bold left parenthesis bold 0 bold right parenthesis end exponent end exponent open square brackets open parentheses bold dy over bold dx close parentheses subscript bold x bold equals bold 0 end subscript close square brackets to the power of bold 2 bold space bold minus bold e to the power of bold y to the power of bold left parenthesis bold 0 bold right parenthesis end exponent end exponent bold space 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 minus bold 2 over bold e bold space bold equals bold space bold minus bold e bold 1 over bold e to the power of bold 2 bold space bold minus bold e 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 therefore bold space bold 1 over bold e to the power of bold 2 bold space bold equals bold space 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 space bold space$

106.

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

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

107.

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