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Which of the following is/are correct ? ...

Which of the following is/are correct ?
(I) Density of all the nuclei is same.
(II) Heavy elements exhibit radioactivity.
(III) Amongst alpha, beta and gamma rays, `alpha`-particle has minimum penetrating power.
(IV) Isobars are the atoms of same elements having the different mass number.

A

`"^(20)C_(0)-^(20)C_(1)+^(20)C_(2)-.....-^(20)C_(15)=-^(19)C_(15)`

B

`"^(20)C_(0)-^(20)C_(1)+^(20)C_(2)-.....-^(20)C_(15)=-^(20)C_(14)`

C

`16"^(20)C_(0)-15^(20)C_(1)+14^(20)C_(2)-.....+2^(20)C_(14)-^(20)C_(15)=^(19)C_(14)`

D

`16"^(20)C_(0)-15^(20)C_(1)+14^(20)C_(2)-.....+2^(20)C_(14)-^(20)C_(15)=^(18)C_(15)`

Text Solution

Verified by Experts

The correct Answer is:
A, D
`(a,d)` `.^(20)C_(0)-^(20)C_(1)+^(20)C_(2)-......-^(20)C_(15)`
`="coefficient of" x^(15) "in the expansion of"`
`(1-x)^(20)(1+x+x^(2)+….)`
`=` coefficient of `x^(15)` in the expansion of `(1-x)^(20)(1-x)^(-1)`
`=` coefficient of `x^(15)` in the expansion of `(1-x)^(19)`
`=-^(19)C_(15)`
`16^(20)C_(0)-15^(20)C_(1)+14^(20)C_(2)-.....+2^(20)C_(14)-^(20)C_(15)`
`=` coefficient of `x^(15)` in the expansion of `(1-x)^(20)(1+2x+3x^(2)+....)`
`=` coefficient of `x^(15)` in the expansion of `(1-x)^(20)(1-x)^(-2)`
`=` coefficient of `x^(15)` in the expansion of `(1-x)^(18)`
`=^(18)C_(15)`
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