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Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

11. f(x) = |3 - |3 - |x| ||એ કેટલા બિંદુ આગળ વિકલનીય નથી ?

12. જો g(x) = $open curly brackets table row cell bold k square root of bold x bold plus bold 1 end root bold comma bold space end cell cell bold 0 bold space bold less or equal than bold space bold x bold less or equal than bold space bold 3 end cell row cell bold mx bold space bold plus bold space bold 2 bold comma bold space end cell cell bold 3 bold space bold less or equal than bold space bold x bold space bold less or equal than bold space bold 5 end cell end table close$ વિકલનીય હોય, તો k + m = ........

2
4
$bold 10 over bold 3$
$bold 16 over bold 5$

13.

જો વિધેય f એ a આગળ ડાબી બાજુ તથા જમણી બાજુ વિકલનીય હોય, તો f એ ......

a આગળ સતત નથી.
a આગળ સતત હોય.
a આગળ વિકલનીય હોય.
0 આગળ વિકલનીય હોય.

14.

જો x^m yⁿ = (x + y)^m+n તો $bold dy over bold dx bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold.$

$bold x over bold y$
$bold y over bold x$
$fraction numerator bold x bold plus bold y over denominator bold xy end fraction$
xy

15.

જો f(x) = (1 + x)ⁿ, તો f(0) + f(0) + $bold 1 over bold 2$ fⁿ(0) + ..... + $fraction numerator bold 1 over denominator bold n bold factorial end fraction$ fn(0) = ......

n
1
2n
2^n-1

16.

bold lim with bold x bold rightwards arrow bold 0 below bold space fraction numerator bold left parenthesis bold 1 bold minus bold cos bold 2 bold x bold right parenthesis bold left parenthesis bold 3 bold plus bold cosx bold right parenthesis over denominator bold x bold space bold tan bold 4 bold x end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold space

4
3
2
$bold 1 over bold 2$

17.

bold lim with bold x bold rightwards arrow bold 2 below bold space fraction numerator root index bold 3 of bold 3 bold x bold plus bold space bold 2 end root begin display style bold minus end style begin display style bold 2 end style over denominator root index bold 5 of bold x bold space bold plus bold space bold 30 end root begin display style bold minus end style begin display style bold 2 end style end fraction bold equals bold space bold. bold. bold. bold. bold.

18.

bold lim with bold n bold rightwards arrow bold infinity below bold space fraction numerator bold 1 bold plus bold 2 to the power of bold 4 bold space bold plus bold space bold 3 to the power of bold 4 bold space bold plus bold space bold. bold. bold. bold. bold space bold plus bold space bold n to the power of bold 4 over denominator bold n to the power of bold 5 end fraction bold space bold minus bold space bold lim with bold n bold rightwards arrow bold infinity below bold space fraction numerator bold 1 bold plus bold 2 to the power of bold 3 bold space bold plus bold space bold 3 to the power of bold 3 bold space bold plus bold space bold. bold. bold. bold. bold. bold space bold plus bold space bold n to the power of bold 3 over denominator bold n to the power of bold 5 end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space

0
$bold 1 over bold 4$
$bold 1 over bold 30$
$fraction numerator begin display style bold 1 end style over denominator begin display style bold 5 end style end fraction$

fraction numerator begin display style bold 1 end style over denominator begin display style bold 5 end style end fraction

Tips: -

આપ ે લ space લક ્ ષ space equals space limit as straight n rightwards arrow infinity of space fraction numerator sum straight n to the power of 4 over denominator straight n to the power of 5 end fraction space minus space limit as straight n rightwards arrow infinity of fraction numerator bold sum bold n to the power of bold 4 over denominator bold n to the power of bold 5 end fraction bold space bold equals bold space bold lim with bold n bold rightwards arrow bold infinity below bold space fraction numerator bold n bold left parenthesis bold n bold plus bold 1 bold right parenthesis bold left parenthesis bold 2 bold n bold space bold plus bold space bold 1 bold right parenthesis bold left parenthesis bold 3 bold n to the power of bold 2 bold space bold plus bold space bold 3 bold n bold minus bold 1 bold right parenthesis over denominator bold 30 bold n to the power of bold 5 end fraction bold space bold minus bold space bold space bold lim with bold n bold rightwards arrow bold infinity below bold space 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 bold times bold n to the power of bold 5 end fraction bold equals bold space bold lim with bold n bold rightwards arrow bold infinity below bold space fraction numerator bold n to the power of bold 5 open parentheses bold 1 bold plus begin display style bold 1 over bold n end style close parentheses open parentheses bold 2 bold plus begin display style bold 1 over bold n end style close parentheses open parentheses bold 3 bold plus begin display style bold 3 over bold n end style bold plus begin display style bold 1 over bold n to the power of bold 2 end style close parentheses over denominator bold 30 bold n to the power of bold 5 end fraction bold space bold minus bold space bold lim with bold n bold rightwards arrow bold infinity below bold space fraction numerator open parentheses bold 1 bold plus begin display style bold 1 over bold n end style close parentheses to the power of bold 2 over denominator bold 4 bold n end fraction bold space bold equals bold space bold 1 over bold 5 bold space bold minus bold space bold 0 bold space bold equals bold space bold 1 over bold 5

19.

જો α, β એ દ્વિઘાત સમીકરણ ax² + bx + c = 0 નાં ભિન્ન વાસ્તવિક બીજ હોય, તો $bold lim with bold x bold rightwards arrow bold alpha below bold space fraction numerator bold 1 bold minus bold cos bold left parenthesis bold ax bold 2 bold space bold plus bold space bold bx bold space bold plus bold space bold c bold right parenthesis bold space over denominator bold left parenthesis bold x bold space bold minus bold space bold alpha bold right parenthesis to the power of bold 2 end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold.$

$fraction numerator begin display style bold b to the power of bold 2 bold space bold minus bold space bold 4 bold ac end style over denominator begin display style bold 2 end style end fraction$
$fraction numerator bold b to the power of bold 2 bold space bold plus bold space bold 4 bold ac over denominator bold 2 end fraction bold space$
1
0

20. નીચેનામાંથી કયા બિંદુગણમાં f(x) = $fraction numerator bold x over denominator bold 1 bold plus bold 1 bold plus bold 1 end fraction$ વિકલનીય થશે ?

(-∞, ∞)
(0, ∞)
(-∞, 0) ∪ (0, ∞)
(-∞, -1) ∪ (-1, ∞)

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Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

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