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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

31.

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

124
136
210
120

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

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

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

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

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

Tips: -

$bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis to the power of bold n bold space bold minus bold space bold 1 bold space bold equals bold space bold nx bold space bold plus bold space fraction numerator bold n bold left parenthesis bold n bold minus bold 1 bold right parenthesis over denominator bold n end fraction bold space bold x to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold space$ અનંત પદ

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

$bold therefore bold space bold nx bold space bold equals bold space bold 3 over bold 4 bold space bold. bold. bold. bold. bold. bold. bold. bold. bold space bold left parenthesis bold i bold right parenthesis$ અને $fraction numerator bold n bold left parenthesis bold n bold minus bold 1 bold right parenthesis over denominator bold 2 end fraction bold space bold x to the power of bold 2 bold space bold equals bold space bold 15 over bold 32 bold space bold space bold space bold space bold space bold space bold space bold. bold. bold. bold space bold left parenthesis bold 2 bold right parenthesis$

(ii) $bold divided by$ (i) નો વર્ગ કરતાં, $fraction numerator bold n bold left parenthesis bold n bold minus bold 1 bold right parenthesis bold x to the power of bold 2 over denominator bold 2 bold n to the power of bold 2 bold x to the power of bold 2 end fraction bold space bold equals bold space bold 15 over bold 32 bold cross times bold 4 over bold 3 bold cross times bold 4 over bold 3$

$bold therefore bold space fraction numerator bold n bold minus bold 1 over denominator bold n end fraction bold space bold equals bold space bold 5 over bold 3$

$bold therefore bold space bold 3 bold n bold space bold minus bold space bold 3 bold space bold equals bold space bold 5 bold n bold.$ આથી $bold 2 bold n bold space bold equals bold space bold minus bold 3 bold space bold rightwards double arrow bold space bold n bold space bold equals bold space bold minus bold 3 over bold 2$

(i) માં $bold n bold space bold equals bold space bold minus bold 3 over bold 2$ મૂકતાં $open parentheses bold minus bold 3 over bold 2 close parentheses bold space bold italic x bold space bold equals bold space bold 3 over bold 4 bold.$ આથી $bold x bold space bold equals bold space fraction numerator bold minus bold 1 over denominator bold 2 end fraction$

$bold therefore bold space bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis to the power of bold n bold space bold minus bold space bold 1 bold space bold equals bold space open parentheses bold 1 bold minus bold 1 over bold 2 close parentheses to the power of begin inline style fraction numerator bold minus bold 3 over denominator bold 2 end fraction end style end exponent bold minus bold space bold 1 bold space bold equals bold space bold 2 to the power of begin inline style bold 3 over bold 2 end style end exponent bold space bold minus bold space bold 1 bold space bold equals bold space bold 2 bold space bold times bold space square root of bold 2 bold space bold minus bold space bold 1$

34. (1 + x + x² + x³ + x⁴)³ ના વિસ્તરણમાં x⁵નો સહગુણક ...... છે.

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

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

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

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

55
91
105
અસંખ્ય

38.

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

136
3¹⁵
₁₅P₃
91

39. 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

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

5646
5456
5465
5654

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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

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