1.

Let rbe a real number andn in N be such that the polynomial 2x^(2)+ 2x + 1dividesthe polynomial (x + 1)^(n) - r. Then (n,r)can be-(A)(4000, 4^(1000)) (B)(4000, 1/4^(1000)) (C)(4^(1000),1/4^1000) (D)(4000,1/4000)

Answer»

`(4000, 4^(1000))`
`(4000, (1)/(4^(1000)))`
`(4^(1000),(1)/(4^(1000)))`
`(4000, (1)/(4000))`

Solution :`2X^(2) + 2x + 1 = 0`
`X = (-1+i)/(2), (-1-i)/(2)`
x satisfies `(x + 1)^(N) - r = 0`
`((-1+-i)/(2) + 1)^(n) - r = 0`
`((1+-i)/(2))^(n) -r = 0`
`((1)/(sqrt(2)))^(n) ((1+i)/(sqrt(2)))^(n) = r`
`((1)/(sqrt(2)))^(n)(e^(+-( IPI)/(4)))^(n)= r`
RHS= real
LHS = realonly when n = multiply of 4
`n = 4000`
`r = ((1)/(sqrt(2)))^(4000) = 1/(4^(1000))`


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