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
△=(1−α)2(1−β)2(α−β)2
But given △=K(1−α)2(1−β)2(α−β)2
hence, K(1−α)2(1−β)2(α−β)2=(1−α)2(1−β)2(α−β)2
K=1