If a,b,c,d are in A.P and ∫02fxdx=-4 where fx=x+ax+bx+a-cx+bx+cx-1x+cx+dx-b+d , then the common difference of the A.P may be

Since a,b,c,d  are in arithmatic progressionb−a=c−b=d−c=kfx=x+ax+bx+a−cx+bx+cx−1x+cx+dx−b+d          · R2→R2−R1,R3→R3−R2=x+ax+bx+a−ckk−1+2kkk1+2k           ·C2→C2−C1=x+akx+a−ck0−1+2kk01+2k=−kk+2k2+k−2k2=−2k2∫02−2k2dx=−4  4k2=4k=±1