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if sqrt(alpha(1)-1)+2sqrt(alpha(2)-4)+3s...

if `sqrt(alpha_(1)-1)+2sqrt(alpha_(2)-4)+3sqrt(alpha_(3)-9)+4sqrt(alpha_(4)-16)=(alpha_(1)+alpha_(2)+alpha_(3)+alpha_(4))/(2)` where `alpha_(1),alpha_(2),alpha_(3),alpha_(4)` are all real. Then

A

`alpha_(1)+alpha_(2)=10`

B

`alpha_(2)+alpha_(3)=26`

C

`alpha_(4)-alpha_(3)=12`

D

`alpha_(4)-alpha_(2)-alpha_(1)=22`

Text Solution

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The correct Answer is:
A, B, D
if `2sqrt(alpha_(1)-1)+4sqrt(alpha_(2)-4)+6sqrt(alpha_(3)-9)+8sqrt(alpha_(4)-16)`
`=(alpha_(1)-1)+1+(alpha_(2)-4)+4+(alpha_(3)-9)+9+(alpha_(4)-16)+16`
`(sqrt(alpha_(1)-1)-1)^(2)+(sqrt(alpha_(2)-4)-2)^(2)+(sqrt(alpha_(3)-9)-3)^(2)+(sqrt(alpha_(4)-16)-4)^(2)=0`
`=sqrt(alpha_(1)-1)=1,sqrt(alpha_(2)-4)=2,sqrt(alpha_(3)-9)=3`.
`sqrt(alpha_(4)-16)=4`
`impliesalpha_(1)=2,alpha_(2)=8,alpha_(3)=18,alpha_(4)=32`
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