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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

31. અસમતા x + y + z ≤ 30, ના અનૃપ પૂર્ણાંક ઉકેલોની સંખ્યા ....... છે.

5646
5456
5465
5654

32. (1 + x + x² + x³ + x⁴)³ ના વિસ્તરણમાં x⁵નો સહગુણક ...... છે.

Tips: -

$bold left parenthesis bold 1 bold space bold plus bold space bold x bold space bold plus bold space bold x to the power of bold 2 bold space bold plus bold space bold x to the power of bold 3 bold space bold plus bold space bold x to the power of bold 4 bold right parenthesis to the power of bold 3 bold space bold equals bold space fraction numerator bold left parenthesis bold 1 bold minus bold x bold right parenthesis to the power of bold 3 bold space bold left parenthesis bold 1 bold space bold plus bold space bold x bold space bold plus bold space bold x to the power of bold 2 bold space bold plus bold space bold x to the power of bold 3 bold space bold plus bold space bold x to the power of bold 4 bold right parenthesis to the power of bold 3 over denominator bold left parenthesis bold 1 bold minus bold x bold right parenthesis to the power of bold 3 end fraction bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold equals bold space bold left parenthesis bold 1 bold space bold minus bold space bold x to the power of bold 5 bold right parenthesis to the power of bold 3 bold space bold left parenthesis bold 1 bold space bold minus bold space bold x bold right parenthesis to the power of bold minus bold 3 end exponent$

∴ માંગેલ x⁵ નો સહગુણક = [(1-x⁵)³ માં x⁵ નો સહગુણક] × [(1-x)^-3માં x⁰નો સહગુણક] + [(1-x⁵)³માં x⁰ નો સહગુણક] × [(1 - x)^-3 માં x⁵ નો સહગુણક ]

$bold equals bold space bold minus bold space open parentheses table row bold 3 row bold 1 end table close parentheses bold space bold cross times bold space bold left parenthesis bold 1 bold right parenthesis bold space bold plus bold space bold left parenthesis bold 1 bold right parenthesis bold space bold cross times bold space open parentheses table row cell bold 5 bold plus bold 3 bold minus bold 1 end cell row bold 5 end table close parentheses$

$bold space bold left square bracket bold because bold space bold left parenthesis bold 1 bold minus bold x bold right parenthesis to the power of bold minus bold k end exponent bold comma bold space bold k bold space bold element of bold N$ માં x^r સહગુણક $open parentheses table row cell bold k bold plus bold r bold minus bold 1 end cell row bold r end table close parentheses bold right square bracket$

$bold equals bold space bold minus bold 3 bold space bold plus bold space open parentheses table row bold 7 row bold 5 end table close parentheses bold space bold equals bold space bold minus bold 3 bold space bold plus bold space open parentheses table row bold 7 row bold 2 end table close parentheses bold space bold equals bold space bold minus bold 3 bold plus bold 21 bold space bold equals bold space bold 18$

33. જો a અને b ધન પૂર્ણાંક હોય, તો $open parentheses table row bold a row bold b end table close parentheses bold space bold plus bold space open parentheses table row bold a row cell bold r bold minus bold 1 end cell end table close parentheses bold space open parentheses table row bold b row bold 1 end table close parentheses bold space bold plus bold space open parentheses table row bold a row cell bold r bold minus bold 2 end cell end table close parentheses bold space open parentheses table row bold b row bold 2 end table close parentheses bold space bold plus bold space bold. bold. bold. bold space bold plus bold space open parentheses table row bold b row bold r end table close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold space$

$open parentheses table row cell bold a bold plus bold b end cell row bold r end table close parentheses$
$open parentheses table row cell bold a bold plus bold b end cell row cell bold r bold plus bold 1 end cell end table close parentheses$
$fraction numerator bold left parenthesis bold a bold plus bold b bold right parenthesis bold factorial over denominator bold r bold factorial end fraction$
$fraction numerator bold a bold factorial bold b bold factorial over denominator bold r bold factorial end fraction$

34.

સમીકરણ x + y + z = 20, x ≥ 1, y ≥ 2, z ≥ 3, x, y, z ∈ Wના ઉકેલોની સંખ્યા ...... છે.

124
136
210
120

35. સમીકરણ x₁ + x₂ + x₃ + x₄ = 31, x₁, x₂, x₃, x₄ ∈ W ના ઉકેલોની સંખ્યા ........ છે.

$open parentheses table row bold 34 row bold 3 end table close parentheses$
$open parentheses table row bold 35 row bold 4 end table close parentheses$
$open parentheses table row bold 33 row bold 2 end table close parentheses$
$open parentheses table row bold 34 row bold 4 end table close parentheses$

36. $bold 3 over bold 4 bold plus open parentheses bold 3 over bold 4 bold cross times bold 5 over bold 8 close parentheses bold space bold plus bold space open parentheses bold 3 over bold 4 bold cross times bold 5 over bold 8 bold cross times bold 7 over bold 12 close parentheses bold space bold plus bold space bold. bold. bold. bold space$ અનંત પદ= .......

$bold 2 bold space bold times bold space square root of bold 2 bold space bold plus bold space bold 1$
$square root of bold 2 bold space bold minus bold space bold 1$
$bold 2 bold space bold times bold space square root of bold 2 bold space bold minus bold space bold 1$
$bold 2 bold space bold times bold space square root of bold 2$

37. શબ્દ ASSASSINATION શબ્દના અક્ષરોમાંથી 4 અક્ષરો કુલ કેટલી રીતે પસંદ કરી શકાય ?

38. જો $bold c subscript bold r bold space bold equals open parentheses table row bold 30 row bold r end table close parentheses bold space bold ત ો bold space open parentheses bold c subscript bold 1 over bold c subscript bold 0 close parentheses to the power of bold 2 bold space bold plus bold space bold 8 bold space open parentheses bold c subscript bold 2 over bold c subscript bold 1 close parentheses to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold 30 to the power of bold 3 bold space open parentheses bold c subscript bold 30 over bold c subscript bold 29 close parentheses to the power of bold 2 bold space bold equals bold space bold. bold. bold. bold. bold space bold. bold space$

75880
76080
76880
76808

39. અસંખ્ય લાલ, પીળા અને વાદળી રંગના દડામાંથી 12 દડા પસંદ કરવાના પ્રકાર ....... છે.

55
91
105
અસંખ્ય

40.

15 સમાન વસ્તુઓને ત્રણ પેટીમાં કુલ કેટલી રીતે મૂકી શકાય ? (દરેક પેટીમાં વધુમાં વધુ 15 વતુ આવી શકે.)

136
3¹⁵
₁₅P₃
91

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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

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