Inequalities

Here
A=
We get A² = A and similarly, for
B=A−I=
We get,
B² = – B
⇒B³ = B
∴ Aⁿ + (ωB)ⁿ = A + (ωB)ⁿ for n∈ N
For ωⁿ to be unity n shall be multiple of 3 and for Bⁿ to be B.n shell be 3, 5, 7, … 99
∴ n = {3, 9, 15,….. 99}
Number of elements = 17.

General Term
=¹⁵Cr( )^15−r( )^r
for term independent on t
2(15 – r) – r = 0
⇒ r = 10
∴ T₁₁ = ¹⁵C₁₀x(1 – x)
Maximum value of x (1 – x) occur at
x=
i.e.,(15(1−x))_max=
⇒K=¹⁵C₁₀×
⇒8K=2(¹⁵C₁₀)=6006