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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

41. 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

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

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

26
25
21
આપેલ પૈકી એક પણ નહી

45. 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

46. 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 $bold sum from bold n bold equals bold 0 to bold n of bold space open parentheses bold sum from bold r bold equals bold 0 to bold m of bold space bold Cr close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space bold.$

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

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

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

48. $bold sum from bold r bold equals bold 0 to bold 40 of bold space bold left square bracket bold left parenthesis bold r bold space bold plus bold space bold 1 bold right parenthesis bold space bold x to the power of bold r bold space bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis to the power of bold 40 bold minus bold r end exponent bold right square bracket$ 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

49. (1 + x + x² + + x³ + ...)² ના વિસ્તરણમાં xⁿ નો સહગુણક ....... છે. (જ્યાં |x| < 1)

n - 1
n+ 2
n
n + 1

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.

Tips: -

$bold m bold space bold equals bold space bold left parenthesis bold 1 bold space bold plus bold space bold 0 bold. bold 0001 bold right parenthesis to the power of bold 1000 bold space bold equals bold space bold left parenthesis bold 1 bold space bold plus bold space bold 10 to the power of bold minus bold 4 end exponent bold right parenthesis to the power of bold 1000$

$bold equals bold space bold 1 bold space bold plus bold space bold 1000 bold space bold left parenthesis bold 10 to the power of bold minus bold 4 end exponent bold right parenthesis to the power of bold 1 bold space bold plus bold space fraction numerator bold 1000 bold cross times bold 999 over denominator bold 2 end fraction bold left parenthesis bold 10 to the power of bold minus bold 4 end exponent bold right parenthesis to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold space bold 1001$ પદ સુધી

< $bold 1 bold space bold plus bold space bold 1 over bold 10 bold space bold plus bold space bold 1000 bold space bold cross times bold space bold 1000 bold space bold cross times bold space bold 10 to the power of bold minus bold 8 end exponent bold space bold plus bold space bold. bold. bold. bold space$ અનંત પદ

≤ $bold 1 bold space bold plus bold space bold 1 over bold 10 bold space bold plus bold space open parentheses bold 1 over bold 10 close parentheses to the power of bold 2 bold space bold plus bold space open parentheses bold 1 over bold 10 close parentheses to the power of bold 3 bold space bold plus bold space bold. bold. bold.$ અનંત પદ
$bold equals bold space fraction numerator bold 1 over denominator bold 1 bold minus begin display style bold 1 over bold 10 end style end fraction bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold left parenthesis bold a bold space bold equals bold space bold 1 bold comma bold space bold r bold space bold equals bold space bold 1 over bold 10 bold right parenthesis bold equals bold space bold 10 over bold 9$

∴ $bold m bold space bold less than bold space bold 10 over bold 9$ વળી m > 1 તો છે જ

∴ m થી મોટો પ્રથમ ધન પૂર્ણાંક = 2

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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

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