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If f(x) satisfies the relation f((5x-3y)...

If `f(x)` satisfies the relation `f((5x-3y)/2)=(5f(x)-3f(y))/2AAx ,y in R ,` and `f(0)=3a n df^(prime)(0)=2,` then the period of `"sin"(f(x))` is `2pi` (b) `pi` (c) `3pi` (d) `4pi`

A

`2pi`

B

`pi`

C

`3pi`

D

`4pi`

Text Solution

Verified by Experts

`"Given "f((5x-3y)/(2))=(5f(x)-3f(y))/(2)`
`"or "f((5x-3y)/(5-3))=(5f(x)-3f(y))/(5-3)`
which satisfies section formula for abscissa on L.H.S. and for ordinate on R.H.S. Hence, f(x) must be the linear function (as only straight line satisfies such section formula).
`"Hence, "f(x)=ax+b`
But f(0)=3 or b=3, f'(0)=2 or a=2
Thus, f(x)=2x+3 or Period of sin (f(x))= sin (2x+3) is `pi`.
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