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If f(x)=(cosx+isinx)(cos2x+isin2x)(cos3x...

If `f(x)=(cosx+isinx)(cos2x+isin2x)(cos3x+isin3x)......(cosn x+isinn x)` and `f(1)=1`,then `f^(11)` is equal to (a) `(n(n+1))/2` (b) `{(n(n+1))/2}^2` (C) `-{(n(n+1))/2}^2` (d) none of these

Text Solution

Verified by Experts

`e^(i theta) = cos theta + i sin theta `
`f(x) = e^(ix) e^(i2x) e^(i3x) ..... e^(inx)`
`= e^(i[x+ 2x+ 3x ...... nx])`
`f(x) = e^(ixn(n+1)/2)`
`f(1) = e^((ni(n+1)/2)`
`= cos(n(n+1)/2) + isin (n(n+1)/2)= 1`
so, `(n(n+1)/2) = 2mpi`
`f'(x) = e^(nix(n+1)/2) (ni(n+1))/2`
...
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Knowledge Check

  • If f(x) is given by f(x)=(cosx+i sinx)(cos3x+isin3x) ......... ....... [cos(2n-1)x+isin(2n-1)x] , then f''(x) is equal to

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  • If x=(2+sqrt3)^n, n in N and f=x-[x], then f^2/(1-f) is

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