Q.

Let limn1+Cn2(2n+3+2sinn)2=12. If Cαβ where α and β are the roots of the quadratic equation x22px+p21=0, then the minimum integral value of P is___

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answer is 3.

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Detailed Solution

limn1+cn2(2n+3+2sinn)2=12limn1n2+c(2+3n+2sinnn)2=12

c4=12c=2cα <βΔ0,b2a>2 and f(2)0

Where f(x)=x22px+p21

Δ=4p24p21>0b2a=p21f(2)=44p+p210p24p+30

(p1)(p3)02

From 1 and 2 , p3

Min p = 3

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Let limn→∞1+Cn2(2n+3+2sin⁡n)2=12. If C≤α≤β where α and β are the roots of the quadratic equation x2−2px+p2−1=0, then the minimum integral value of P is___