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If the line x= alpha divides the area o...

If the line `x= alpha ` divides the area of region `R={(x,y)in R^(2): x^(3) le y le x ,0 le x le 1 } ` into two equal parts, then

A

` 2 alpha^(4)-4alpha^(2)+1=0`

B

`alpha^(4)+4alpha^(2)-1=0`

C

`(1)/(2) lt alpha lt 1 `

D

`0 lt alpha le (1)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
A, C
` int_(0)^(1)(x-x^(3))dx= 2int_(0)^(alpha)(x-x^(3))dx `
`(1)/(4)=2((alpha^(2))/(2)-(alpha^(4))/(4))`
`2 alpha^(4)-4 alpha^(2)+1=0 `
` rArr alpha^(2)=(4-sqrt(16-8))/(4) " "( because alpha in (0,1))`
` alpha^(2)=1-(1)/(sqrt(2))`
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