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The figure shows a portion of the graph `y=2x-4x^(3)`. The line y=c is such that the areas of the regions marked I and II are equal. If a, b are the x-coordinates of A,B respectively, then a+b equals-

A

`(2)/sqrt(7)`

B

`(3)/sqrt(7)`

C

`(4)/sqrt(7)`

D

`(5)/sqrt(7)`

Text Solution

Verified by Experts

The correct Answer is:
A

`f_(a)^(b)(2x-4x^(3))dx=2(b-a)c`
`(x^(2)-x^(4))_(a)^(b)=2(b-a)c`
`(a+b)(1-(a^(2)+b^(2)))=2c`
`(a+b)(1-(a+b)^(2)+2ab)=ac` . . .(1)
again `2x-4x^(3)=c`

`a+b+alpha=0" ""clearly"a+b=-a` . . .(2)
`ab+(a+b)alpha=-(1)/(2)" "ab=alpha^(2)-(1)/(2)` . . .(3)
`abalpha=-(c)/(4)" "c=-4alpha(a^(2)-(1)/(2))` . . .(4)
put value of (a+b), ab, c from eq. (2), (3), (4) in equation (1) and solve it
`{1-alpha^(2)+2(alpha^(2)-(1)/(2))}=-8alpha(alpha^(2)-(1)/(2))`
`1-alpha^(2)+2alpha^(2)-1=8alpha^(2)-4`
`alpha^(2)=8alpha^(2)-4`
`7alpha^(2)=4`
`alpha=(2)/(sqrt(7))`
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