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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

41. જો $bold C subscript bold r bold space bold equals bold space open parentheses table row bold n row bold r end table close parentheses$ તો $bold sum from bold n bold equals bold 0 to bold n of bold space open parentheses bold sum from bold r bold equals bold 0 to bold m of bold space bold Cr close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space bold.$

$bold n bold times bold 2 to the power of bold n bold minus bold 1 end exponent$
$bold left parenthesis bold n bold space bold plus bold space bold 2 bold right parenthesis bold space bold times bold space bold 2 to the power of bold n bold minus bold 1 end exponent$
$bold left parenthesis bold n bold plus bold 1 bold right parenthesis bold space bold times bold space bold 2 to the power of bold n bold minus bold 1 end exponent$
$bold left parenthesis bold n bold plus bold 2 bold right parenthesis bold space bold times bold space bold 2 to the power of bold n$

42. જો n એ 3 નો ગુણક હોય, તો $bold sum from bold r bold equals bold 0 to begin inline style bold n over bold 3 end style of bold space open parentheses table row bold n row cell bold 3 bold r end cell end table close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold space bold.$

$bold 1 over bold 3 bold thin space bold left parenthesis bold 2 to the power of bold n bold space bold plus bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold space bold 2 bold right parenthesis$
0
2ⁿ - 2 (-1)ⁿ
2ⁿ + 2 (-1)ⁿ

43. જો $bold S subscript bold k bold space bold equals bold space bold sum from bold r bold equals bold 1 to bold n of bold space bold r to the power of bold k$ તો $bold sum from bold r bold equals bold 1 to bold k bold plus bold 1 of bold space open parentheses table row cell bold k bold plus bold 1 end cell row bold r end table close parentheses bold space bold S subscript bold k bold plus bold 1 bold space bold minus bold space bold r end subscript bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space bold.$

(n + 1)^k - 1
(n + 1)^k+1 - 1
(n + 1)^k+1
n^k+1 - 1

44. $bold sum from bold r bold equals bold 0 to bold 9 of bold space open curly brackets bold left parenthesis bold x bold plus bold 4 bold right parenthesis to the power of bold 9 bold minus bold r end exponent bold space bold times bold space bold left parenthesis bold x bold space bold plus bold space bold 3 bold right parenthesis to the power of bold r close curly brackets$ માં x⁵ નો સહગુણક ......... છે.

$open parentheses table row bold 10 row bold 5 end table close parentheses$
$bold 780 bold space open parentheses table row bold 9 row bold 5 end table close parentheses$
$bold 781 bold space bold times bold space open parentheses table row bold 10 row bold 5 end table close parentheses$
781

45. $bold sum from bold r bold equals bold 0 to bold 40 of bold space bold left square bracket bold left parenthesis bold r bold space bold plus bold space bold 1 bold right parenthesis bold space bold x to the power of bold r bold space bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis to the power of bold 40 bold minus bold r end exponent bold right square bracket$ માં x⁵ નો સહગુણક ....... છે.

$open parentheses table row bold 41 row bold 5 end table close parentheses$
$open parentheses table row bold 39 row bold 5 end table close parentheses$
$open parentheses table row bold 41 row bold 5 end table close parentheses$
આપેલ પૈકી એક પણ નહી

46. $bold sum from bold m bold equals bold 1 to bold n of open parentheses bold sum from bold r bold equals bold 0 to bold m bold equals bold 1 of bold space open parentheses table row bold n row bold r end table close parentheses close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space bold.$

2^n-1 + n
$bold n bold space bold times bold space bold 2 to the power of bold n bold minus bold 1 end exponent$
$bold n bold space bold times bold space bold 2 to the power of bold n$
(n + 1) 2^n-1

$bold n bold space bold times bold space bold 2 to the power of bold n bold minus bold 1 end exponent$

Tips: -

$Error converting from MathML to accessible text.$
બંને બાજુ x ને સાપેક્ષ વિલલન કરતાં

$bold n bold space bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis to the power of bold n bold minus bold 1 end exponent bold space bold equals bold space open parentheses table row bold n row bold 1 end table close parentheses bold space bold plus bold space bold 2 bold x bold space open parentheses table row bold n row bold 2 end table close parentheses bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold nx to the power of bold n bold minus bold 1 end exponent bold space open parentheses table row bold n row bold n end table close parentheses$

x = 1 લેતાં, $bold n bold space bold times bold space bold 2 to the power of bold n bold minus bold 1 end exponent bold space bold equals bold space open parentheses table row bold n row bold 1 end table close parentheses bold space bold plus bold space bold 2 open parentheses table row bold n row bold 2 end table close parentheses bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold n bold space open parentheses table row bold n row bold n end table close parentheses$

$bold equals bold space open parentheses table row bold n row cell bold n bold minus bold 1 end cell end table close parentheses bold space bold plus bold space bold 2 open parentheses table row bold n row cell bold n bold minus bold 2 end cell end table close parentheses bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold left parenthesis bold n bold minus bold 1 bold right parenthesis bold space open parentheses table row bold n row bold 1 end table close parentheses bold space bold plus bold space bold n bold space open parentheses table row bold n row bold 0 end table close parentheses bold equals bold space bold n bold space open parentheses table row bold n row bold 0 end table close parentheses bold space bold plus bold space bold left parenthesis bold n bold space bold minus bold space bold 1 bold right parenthesis bold space open parentheses table row bold n row bold 1 end table close parentheses bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold 2 bold space open parentheses table row bold n row cell bold n bold minus bold 2 end cell end table close parentheses bold space bold plus bold space open parentheses table row bold n row cell bold n bold minus bold 1 end cell end table close parentheses bold equals bold space open parentheses table row bold n row bold 0 end table close parentheses bold space bold plus bold space open parentheses open parentheses table row bold n row bold 0 end table close parentheses bold plus open parentheses table row bold n row bold 1 end table close parentheses close parentheses bold space bold plus bold space open parentheses open parentheses table row bold n row bold 0 end table close parentheses bold plus open parentheses table row bold n row bold 1 end table close parentheses bold plus open parentheses table row bold n row bold 2 end table close parentheses close parentheses bold space bold. bold. bold. bold space bold plus bold space open parentheses open parentheses table row bold n row bold 0 end table close parentheses bold plus open parentheses table row bold n row bold 1 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 minus bold 1 end cell end table close parentheses close parentheses bold equals bold space bold sum from bold m bold equals bold 1 to bold n of bold space open parentheses open parentheses table row bold n row bold 0 end table close parentheses bold plus open parentheses table row bold n row bold 1 end table close parentheses bold plus bold space bold. bold. bold. bold space bold plus bold space open parentheses table row bold n row cell bold m bold minus bold 1 end cell end table close parentheses close parentheses bold equals bold space bold sum from bold m bold equals bold 1 to bold n of bold space open parentheses bold sum from bold r bold equals bold 0 to bold m bold minus bold 1 of bold space open parentheses table row bold n row bold r end table close parentheses close parentheses$

47. (1 + x + x² + + x³ + ...)² ના વિસ્તરણમાં xⁿ નો સહગુણક ....... છે. (જ્યાં |x| < 1)

n - 1
n+ 2
n
n + 1

48. $bold left parenthesis bold 1 bold space bold plus bold space bold 2 bold x bold space bold plus bold space bold 3 bold x to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold right parenthesis to the power of begin inline style fraction numerator bold minus bold 3 over denominator bold 2 end fraction end style end exponent$ માં x⁵ નો સહગુણક .......... છે.

26
25
21
આપેલ પૈકી એક પણ નહી

49. જો n ∈ N - {1} તો, $bold sum from bold r bold equals bold 0 to bold n of bold space open curly brackets bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold r bold space bold left parenthesis bold 100 bold space bold minus bold space bold r bold right parenthesis bold space open parentheses table row bold n row bold r end table close parentheses close curly brackets bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold space bold.$

0
2ⁿ-1
100 (2ⁿ)
100

50. $bold left parenthesis bold 1 bold space bold plus bold space bold 0 bold. bold 0001 bold right parenthesis to the power of bold 10 to the power of bold 3 end exponent$ થી મોટો પ્રથમ ધન પૂર્ણાંક ......... છે.

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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

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