1)

If   α,β0   and f(n)=  αn+βn   and [31+f(1)1+f(2)1+f(1)1+f(2)1+f(3)1+f(2)1+f(3)1+f(4)]  = K(1α)2(1β)2(αβ)2  , then K is equal to

     


A) αβ

B) 1αβ

C) 1

D) -1

Answer:

Option C

Explanation:

 Use the property that, two determinants can be multipied row-to row or row -to-column to 

write the given determinant as the product of two determinants and then expand.

Given,  f(n)=  αn+βn,  f(1)=  α1+β1, f(2)=  α2+β2

 f(3)=  α3+β3, f(4)=  α4+β4

Let     =[31+f(1)1+f(2)1+f(1)1+f(2)1+f(3)1+f(2)1+f(3)1+f(4)]

       =[31+α+β1+α2+β21+α+β1+α2+β21+α3+β31+α2+β21+α3+β31+α4+β4]

=[1.1+1.1+1.11.1+1.α+1.β1.1+1.α2+1.β21.1+1.α+1.β1.1+α.α+β.β1.1+α.α2+β.β21.1+1.α2+1.β21+α2.α+β2.β1.1+α2.α2+β2.β2]

=[1111αβ1α2β2][1111αβ1α2β2]=[1111αβ1α2β2]2

On expanding , we get 

=(1α)2(1β)2(αβ)2

 But given     =K(1α)2(1β)2(αβ)2

 hence,   K(1α)2(1β)2(αβ)2=(1α)2(1β)2(αβ)2

                          K=1