CGPET 2015 Solved Paper

Given,
(

∧

i

+

∧

j

+3

∧

k

)‌x+(3

∧

i

−3

∧

j

+

∧

k

)‌y+(−4

∧

i

+5‌

∧

j

)‌z =λ(

∧

i

x+

∧

j

y+

∧

k

z)
On equating the coefficients of

∧

i

,

∧

j

‌and‌

∧

k

both sides, we have
x+3y−4z=λx,
x−3y+5z=λy
and 3x+y+0=λz
Above three equations can b e rewritten as
(1−λ)x+3y−4z=0
x−(3+λ)y+5z=0
3x+y−λz=0
This is homogeneous systemn of equations in three variables x, y and z.
It is consistent and have non - zero solution .
i.e (x, y, z) ≠ (0, 0, 0), if determinant of coefficient matrix is zero.
⇒[

1−λ	3	−4
1	−(3+λ)	5
3	1	−λ

]
On expanding along first row, we have
(1−λ)[λ(3+λ)−5]−3(−λ−15)−4(1+9+3λ)=0
⇒(1−λ)(λ2+3λ−5)+3λ+45−40−12λ=0
⇒λ2+3λ−5−λ3−3λ2+5λ−9λ+5=0
⇒−λ3−2λ2−λ=0
⇒λ(λ2+2λ+1)=0
⇒λ(λ+1)2=0
⇒λ=0,−1