Doubt in integration

bro by applying by parts ,
we get a=0
then option 1 satisfies

I think C also should be correct

Not getting, can you share your working?

Please share your approach

To be atleast one root f(x)dx=0 in 0 to 1 but since it is given 1.How can you conclude it

Apply by part for 2 info(xf(x)dx=a). You will get a=0

U have taken limits wrong @Venkat_2020 @Ayush_2020_3

You are right @akash_2020. Pls ignore post 7.
But shouldn't D be right too

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Not necessary that f(x) has no roots

As \int\limits_{0}^{1}{x^2}f(x)dx \ = \ 0
we can conclude that f has atleast one real root in (0,1)

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a is not equal to 0


Sorry I missed -a but then also

Sorry for wrong thought f \ is \ not \ continous
So we cant say anything about real roots