If in the expansion of (1+x)^ p (1-x)^ q , the coefficients of x and x^2 are 1 and -2 , respectively, then p ^2+ q ^2 is equal to :

Step-by-step solution

aligned & (1+ x )^ p (1- x )^ q = ( ^ p C ₀+ ^ p C ₁ x + ^ p C ₂ x ^2+ ) ( ^ q C ₀- ^ q C ₁ x + ^ q C ₂ x ^2+ ) & coff of x ^ p C ₀ ^ q C ₁+ ^ p C ₁ ^ q C ₀=1 & p - q =1 & coff of x ^2 ^ p C ₀ ^ q C ₂- ^ p C ₁ ^ q C ₁+ ^ p C ₂ ^ q C ₀=-2 & q ( q -1) 2 - pq + p ( p -1) 2 =-2 & q ^2- q -2 pq + p ^2- p =-4 & 1-( p + q )=-4 & p + q =5 & p =3 & q =2 & so p ^2+ q ^2=13 aligned