quadratic equation in sin

\hspace{-16}$The Value of $\mathbf{m}$ for Which the equations\\\\ $\mathbf{(5m-m^2)^2.\sin^2 x-10.\sin x.(5m-m^2)+24=0}$\\\\ Has exactly $\mathbf{\underline{\bold{Three}}}$ Solution in $\mathbf{[0,2\pi].}$

8 Answers

262
Aditya Bhutra ·

i am getting m=6,-1

71
Vivek @ Born this Way ·

How did you do Aditya?

I'm getting m = 2,3 , infact m ε [2,3] , but I won't bid for the latter.

1
rishabh ·

im getting no value of m.
solving using quadratic formula we get sinx = 65m-m2 , 45m-m2

note that from the graph of sinx it is clear that only for sinx = 0 we have 3 solutions in [0,2pi]
but both the expressions can't be zero.

36
rahul ·

ya for m being a real no. there's no defined soln for it...
as D ≥ 0 (always) and three solns are possible only when sin x = 0 so, no soln...!!

262
Aditya Bhutra ·

we can have 3 solns. if (sinx =1 or -1) and (sinx =c where -1<c<1)

36
rahul ·

no when sinx = 1 or -1 then pi/2 and 3pi/2 are the only two solns. between 0 and 2pi.!!
and there's a difference in the two statements....
"can have three solutions" and, "exactly three solns"...!!

1
rishabh ·

@aditya we are discussing in [0,2pi]

71
Vivek @ Born this Way ·

@Rishabh Three roots do exist exactly in [0,2Î ]

See the image below for m =3, similar is for m =2.

@Aditya, for the values you gave, the 3 roots lie in [-2Î , 0] while only one in the asked range.

See here

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