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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

1. $bold 1 bold space bold times bold space fraction numerator open parentheses table row bold n row bold 1 end table close parentheses over denominator open parentheses table row bold n row bold 0 end table close parentheses end fraction bold space bold plus bold space bold 2 bold space fraction numerator open parentheses table row bold n row bold 2 end table close parentheses over denominator open parentheses table row bold n row bold 1 end table close parentheses end fraction bold space bold plus bold space bold 3 bold space fraction numerator open parentheses table row bold n row bold 3 end table close parentheses over denominator open parentheses table row bold n row bold 2 end table close parentheses end fraction bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold 31 bold space fraction numerator open parentheses table row bold n row bold 31 end table close parentheses over denominator open parentheses table row bold n row bold 30 end table close parentheses end fraction bold space bold equals bold. bold. bold. bold. bold. bold. bold. bold space$

31 (n - 15)
31n - 15
15 (2n-31)
30n-465

2. જો x > 0, y > 0 તો $bold sum from bold r bold equals bold 1 to bold infinity of bold space open parentheses fraction numerator bold x over denominator bold 2 bold x bold plus bold 3 bold y end fraction close parentheses to the power of bold r bold minus bold 1 end exponent bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space$

$fraction numerator bold 2 bold x over denominator bold x bold plus bold 3 bold y end fraction$
$fraction numerator bold 2 bold x bold plus bold 3 bold y over denominator bold x bold plus bold 3 bold y end fraction$
$fraction numerator bold 3 bold y over denominator bold 2 bold x bold plus bold 3 bold y end fraction$
$fraction numerator bold 2 bold x bold plus bold 3 bold y over denominator bold 2 bold x bold plus bold y end fraction$

3. જો (1 - x + x²)^-3 = a₀ + a₁x + a₂x² + ... તો $bold sum from bold r bold equals bold 0 to bold 6 of bold space bold a subscript bold r bold space bold equals bold space bold. bold. bold. bold. bold space bold left parenthesis bold vertical line bold x bold vertical line bold space bold less than bold space bold 1 bold right parenthesis bold space$

-10
-7
10
38

-10

Tips: -

(1 - x + x²)^-3 = (1 + x)³ (1 + x)³ (1 - x + x²)^-31

= (1 + x)³ (1 + x³)^-3 1

= (1 + 3x + 3x² + x³) (1 - 3x³ + 6x⁶ - ...)

$bold therefore bold space bold sum from bold r bold space bold equals bold space bold 0 to bold 6 of bold space bold a subscript bold r bold space bold equals bold space bold a subscript bold 0 bold space bold plus bold space bold a subscript bold 1 bold space bold plus bold space bold a subscript bold 2 bold space bold plus bold space bold a subscript bold 3 bold space bold plus bold space bold a subscript bold 4 bold space bold plus bold space bold a subscript bold 5 bold space bold plus bold space bold a subscript bold 6 bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold equals bold space bold 1 bold space bold plus bold space bold 3 bold space bold plus bold space bold 3 bold space bold plus bold space bold left parenthesis bold minus bold 3 bold space bold plus bold space bold 1 bold right parenthesis bold space bold plus bold space bold 3 bold space bold left parenthesis bold minus bold 3 bold right parenthesis bold space bold plus bold space bold 3 bold left parenthesis bold minus bold 3 bold right parenthesis bold space bold plus bold space bold left parenthesis bold minus bold 3 bold space bold plus bold space bold 6 bold right parenthesis bold space bold equals bold space bold minus bold 10 bold space$

4. (1 + 2x)^-7 ના વિસ્તરણમાં x¹⁰ નો સહગુણક ....... છે.

$open parentheses table row bold 16 row bold 6 end table close parentheses$
$bold 2 to the power of bold 10 bold space open parentheses table row bold 17 row bold 10 end table close parentheses$
$bold 2 to the power of bold 10 bold space open parentheses table row bold 16 row bold 10 end table close parentheses$
$bold 2 to the power of bold 10 bold space open parentheses table row bold 16 row bold 9 end table close parentheses$

5. $bold sum from bold r bold equals bold 0 to bold 70 of bold space open parentheses table row bold 70 row bold r end table close parentheses bold space bold left parenthesis bold x bold space bold minus bold space bold 5 bold right parenthesis to the power of bold 70 bold minus bold r end exponent bold space bold 4 to the power of bold r$ વિસ્તરણમાં x⁹ નો સહગુણક ....... છે.

$open parentheses table row bold 70 row bold 9 end table close parentheses$
$bold minus open parentheses table row bold 70 row bold 61 end table close parentheses$
0
$bold minus open parentheses table row bold 70 row bold 8 end table close parentheses$

bold sum from bold r bold equals bold 0 to bold n of bold space bold r bold space open parentheses table row bold n row bold r end table close parentheses bold space open parentheses table row bold 2 row bold 7 end table close parentheses to the power of bold r bold space open parentheses bold 5 over bold 7 close parentheses to the power of bold n bold minus bold r end exponent bold space bold equals bold space bold. bold. bold. bold. bold.

0
1
$fraction numerator bold 5 bold n over denominator bold 7 end fraction$
$fraction numerator bold 2 bold n over denominator bold 7 end fraction$

7. (a + 2b)³ (2a - b)⁴ ના વિસ્તરણમાં a³b⁴ નો સહગુણક ........ છે.

593
-16
-15
15

8. (1 + x)^-n (n ∈ N) ના વિસ્તરણમાં x^r નો સહગુણક ....... છે. ( |x|<1)

$bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold r bold space open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row bold r end table close parentheses$
$open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row bold r end table close parentheses$
$open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row cell bold n bold plus bold 1 end cell end table close parentheses$
$open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row cell bold r bold minus bold 1 end cell end table close parentheses$

bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis to the power of bold 40 bold space open parentheses bold 1 bold plus bold 1 over bold x close parentheses to the power of bold 40

ના વિસ્તરણમાં x^-2 નો સહગુણક ......છે.

$open parentheses table row bold 80 row bold 37 end table close parentheses$
$fraction numerator bold 80 bold space bold factorial over denominator bold 38 bold space bold factorial end fraction$
$open parentheses table row bold 80 row bold 38 end table close parentheses$
$open parentheses table row bold 107 row bold 38 end table close parentheses$

10. (1 - x)^-n (n ∈ N)ના વિસ્તરણમાં x^r નો સહગુણક ......... છે. ( |x| < 1)

$open parentheses table row cell bold n bold minus bold r bold plus bold 1 end cell row bold r end table close parentheses$
$open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row bold r end table close parentheses$
$open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row straight n end table close parentheses$
$bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold r bold space open parentheses table row cell bold n bold minus bold r bold plus bold 1 end cell row bold r end table close parentheses$

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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

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