TS EAMCET 11-Sep-2020 Shift 2 Solved Paper

Given system of equations has a non-trivial solution
Π1≡x+ay+az=0
Π2≡bx+y+bz=0
Π3≡cx+cy+z=0
It can be written as AX=0
Where, ‌‌A=[

1	a	a
b	1	b
c	c	1

]
So, the system has non-trivial solutions hence it has infinitely many solutions or unique solutions.
∴‌‌|A|=0
Now, let the system has infinitely many solutions
So, let y=t;t∈R
Then from Eqs. (i) and (ii), we get
x+az=−at⇒bx+bz=−t
Solving these, we get
z=‌

t(1−ab)

b(a−1)

,x=‌

a

b

t‌

(1−b)

(1−a)

Hence, if a≠b≠c, then x=‌

a

b

t‌

(1−b)

(1−a)

,y=t,
z=‌

t

b

‌

(1−ab)

(a−1)

,t∈R
If a=b=c
Let y=t1,z=t2,t1,t2∈R
Then by Eq. (i)
x=−a(t1+t2)
So, the system of equations has unique solution.