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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

11. જો |x| એટલો નાનો હોય કે જેથી $open parentheses fraction numerator bold 1 bold plus bold 3 bold x over denominator bold 1 bold minus bold 2 bold x end fraction close parentheses to the power of bold 2 bold space bold equals bold space bold 1 bold space bold plus bold space bold kx bold comma$ તો k = ..... .

10
-10
-2
2

12.

(1 + x)⁵ + (1 + x)⁶ + (1 + x)⁷ + ... + (1 + x)⁵⁰ માં x²⁵ નો સહગુણક ........ છે.

$open parentheses table row bold 46 row bold 25 end table close parentheses$
$open parentheses table row bold 51 row bold 26 end table close parentheses$
$open parentheses table row bold 50 row bold 25 end table close parentheses$
$open parentheses table row bold 51 row bold 25 end table close parentheses bold minus bold 1$

13. $open parentheses table row bold n row bold 1 end table close parentheses to the power of bold 2 bold space bold plus bold space bold 2 bold space open parentheses table row bold n row bold 2 end table close parentheses to the power of bold 2 bold space bold plus bold space bold 3 bold space open parentheses table row bold n row bold 3 end table close parentheses to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold plus bold space bold n bold space open parentheses table row bold n row bold n end table close parentheses to the power of bold 2 bold space bold equals bold space bold. bold. bold. bold. bold space$

$fraction numerator bold left parenthesis bold 2 bold n bold minus bold 1 bold right parenthesis bold space bold factorial over denominator bold n bold space bold factorial end fraction$
$bold n bold space open parentheses table row cell bold 2 bold n end cell row bold n end table close parentheses$
$fraction numerator bold left parenthesis bold 2 bold n bold minus bold 1 bold right parenthesis bold factorial over denominator bold left parenthesis bold n bold minus bold 1 bold right parenthesis bold factorial end fraction$
$fraction numerator bold left parenthesis bold 2 bold n bold minus bold 1 bold right parenthesis bold space bold factorial over denominator bold left square bracket bold left parenthesis bold n bold minus bold 1 bold right parenthesis bold factorial bold right square bracket to the power of bold 2 end fraction$

14. (1 - 2x)^-2 ના વિસ્તરણમાં x⁷ નો સહગુણક ....... છે.

512
-1024
1024
128

15. જો $bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis to the power of bold n bold space bold equals bold space bold sum from bold r bold equals bold 0 to bold n of bold space bold c subscript bold r bold space bold x to the power of bold r$ તો $bold sum from bold r bold equals bold 0 to bold 200 of bold space fraction numerator bold c over denominator bold r bold plus bold 1 end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold space bold.$

$fraction numerator bold 2 to the power of bold 200 bold minus bold 1 over denominator bold 200 end fraction$
$bold 2 to the power of bold 201 over bold 201$
$fraction numerator bold 2 to the power of bold 201 bold minus bold 1 over denominator bold 201 end fraction$
$fraction numerator bold 2 to the power of bold 201 bold plus bold 1 over denominator bold 201 end fraction$

fraction numerator bold 2 to the power of bold 201 bold minus bold 1 over denominator bold 201 end fraction

Tips: -

bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis to the power of bold 200 bold space bold equals bold space bold sum from bold r bold equals bold 0 to bold 200 of bold space bold c subscript bold r bold space bold x to the power of bold r 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 equals bold space bold c subscript bold 0 bold space bold plus bold space bold c subscript bold 1 bold space bold x bold space bold plus bold space bold c subscript bold 2 bold space bold x to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold plus bold space bold c subscript bold 200 bold space bold x to the power of bold 200 bold therefore bold space bold integral from bold 0 to bold 1 of bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis to the power of bold 200 bold space bold dx bold space bold space bold space bold space bold space bold space bold equals bold space bold space bold integral from bold 0 to bold 1 of bold space bold left parenthesis bold c subscript bold 0 bold space bold plus bold space bold c subscript bold 1 bold space bold x bold space bold plus bold space bold c subscript bold 2 bold space bold x to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold c subscript bold 000 bold space bold x to the power of bold 200 bold right parenthesis bold space bold dx bold space bold therefore bold space open square brackets fraction numerator bold left parenthesis bold 1 bold plus bold x bold right parenthesis to the power of bold 201 over denominator bold 201 end fraction close square brackets subscript bold 0 superscript bold 1 bold space bold equals bold space open square brackets bold c subscript bold 0 bold x bold space bold plus bold space bold c subscript bold 1 bold space bold x to the power of bold 2 over bold 2 bold space bold plus bold space bold c subscript bold 2 bold space bold x to the power of bold 3 over bold 3 bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold plus bold space bold c subscript bold 200 bold space bold x to the power of bold 201 over bold 201 close square brackets subscript bold 0 superscript bold 1 bold therefore bold space bold 2 to the power of bold 201 over bold 201 bold space bold minus bold space bold 1 over bold 201 bold space bold equals bold space bold c subscript bold 0 bold space bold plus bold space bold c subscript bold 1 over bold 2 bold space bold plus bold space bold c subscript bold 2 over bold 3 bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold c subscript bold 200 over bold 201 bold therefore bold space bold sum from bold r bold equals bold 0 to bold 200 of bold space fraction numerator bold c subscript bold r over denominator bold r bold plus bold 1 end fraction bold space bold equals bold space fraction numerator bold 2 to the power of bold 201 bold minus bold 1 over denominator bold 201 end fraction

16. $fraction numerator bold 1 over denominator open parentheses table row bold 101 row bold 0 end table close parentheses end fraction bold space bold minus bold space fraction numerator bold 1 over denominator open parentheses table row bold 101 row bold 1 end table close parentheses end fraction bold space bold plus bold space fraction numerator bold 1 over denominator open parentheses table row bold 101 row bold 3 end table close parentheses end fraction bold space bold plus bold space bold. bold. bold. bold space bold plus bold space fraction numerator bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold 101 over denominator open parentheses table row bold 101 row bold 101 end table close parentheses end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold space$

1
0
-1
2

17. $bold 2 bold space open parentheses table row bold n row bold 2 end table close parentheses bold space bold plus bold space bold 4 bold space open parentheses table row bold n row bold 4 end table close parentheses bold space bold plus bold 6 bold space open parentheses table row bold n row bold 6 end table close parentheses bold space bold plus bold space bold. bold. bold. bold space bold equals bold space bold. bold. bold. bold. bold. bold space$

n × 2^n-2
n × 2ⁿ⁺²
n 2^n-1
n × 2ⁿ⁺¹

18. જો ....... તો $fraction numerator bold 1 over denominator scriptbase square root of bold 7 bold minus bold 3 bold x end root end scriptbase presuperscript bold 5 end fraction$ નું x ની ઘાતના સ્વરૂપમાં વિસ્તરણ થઈ શકે.

$bold vertical line bold x bold vertical line bold space bold less than bold space bold 7 over bold 3$
$bold vertical line bold x bold vertical line bold space bold less than bold space 11 over bold 3$
$bold vertical line bold x bold vertical line bold space bold greater than bold space bold 7 over bold 3$
$bold vertical line bold x bold vertical line bold space bold greater than bold space fraction numerator 3 space factorial over denominator 7 end fraction$

19. $fraction numerator open parentheses table row cell bold 2 bold n end cell row bold 1 end table close parentheses over denominator bold n end fraction bold plus fraction numerator open parentheses table row cell bold 2 bold n end cell row bold 2 end table close parentheses over denominator bold n to the power of bold 2 end fraction bold plus fraction numerator open parentheses table row cell bold 2 bold n end cell row bold 3 end table close parentheses over denominator bold n to the power of bold 3 end fraction bold plus bold space bold. bold. bold. bold space bold plus bold space bold 2 bold n$ પદ = ..........

$open parentheses fraction numerator bold n bold plus bold 1 over denominator bold n end fraction close parentheses to the power of bold 2 bold n bold plus bold 1 end exponent bold space bold minus bold space bold 1$
$open parentheses fraction numerator bold n over denominator bold n bold plus bold 1 end fraction close parentheses to the power of bold 2 bold n end exponent bold space bold minus bold space bold 1$
$open parentheses fraction numerator bold n bold plus bold 1 over denominator bold n end fraction close parentheses to the power of bold 2 bold n end exponent bold space bold minus bold space bold 1$
આપેલ પૈકી એક પણ નહી

20. (1 + x)ⁿ ના વિસ્તરણમાં $open parentheses table row bold n row bold 0 end table close parentheses bold space open parentheses table row bold n row bold r end table close parentheses bold space bold plus bold space open parentheses table row bold n row bold 1 end table close parentheses bold space open parentheses table row bold n row cell bold r bold plus bold 1 end cell end table close parentheses bold space bold plus bold space bold. bold. bold. bold space bold plus bold space open parentheses table row bold n row cell bold n bold space bold minus bold space bold r end cell end table close parentheses bold space open parentheses table row bold n row bold n end table close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold space bold. bold space$

$open parentheses table row cell bold 2 bold n end cell row cell bold n bold minus bold r bold plus bold 1 end cell end table close parentheses$
$open parentheses table row cell bold 2 bold n end cell row cell bold n bold minus bold r end cell end table close parentheses$
$open parentheses table row cell bold 2 bold n end cell row bold r end table close parentheses$
$open parentheses table row cell bold 2 bold n end cell row cell bold r bold minus bold 1 end cell end table close parentheses$

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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

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