Question

If `A=[{:(1,-1),(2,-1):}],B=[{:(a,-1),(b,-1):}]" and "(A+B)^(2)=(A^(2)+B^(2))` then find the values of a and b.
A. `a=4,b=1`
B. `a=1b=4`
C. `a=0,b=4`
D. `a=2,b=4`

Answer 1

Correct Answer - B
We have,
`{:A+B=[(a+1,0),(b+2,-2)]:}`
`{:A^2=[(1,-1),(2,-1)][(1,-1),(2,-1)]=[(-1,0),(0,-1)]:}`
`{:B^2=[(a,1),(b,-1)][(a,1),(b,-1)]=[(a^2+b,a-1),(ab-b,b+1)]:}`
`{:(A+B)^2=[(a+1,0),(b+2,-2)][(a+1,0),(b+2,-2)]=[((a+1)^2,0),((a-1)(b+2),4)]:}`
`:. (A+B)^2=A^2+B^2`
`rArr{:[((a+1)^2,0),((a-1)(b+2),4)]=[(-1,0),(0,-1)]+[(a^2+b,a-1),(ab-b,b+1)]:}`
`rArr{:[((a+1)^2,0),((a-1)(b+2),4)]=[(a^2+b-1,a-1),(ab-b,b)]:}`
`rArr a-1=0,b=4,(a+1)^2=a^2+b-1,(a-1)(b+2)=a-b`
`rArr a=1and b=4`