Manipal Entrance Test 2020 Math Paper

Given,
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 be rewritten as
(1−λ)x+3y−4z=0
x−(3+λ)y+5z=0
3x+y−λz=0
This is homogeneous system 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.
On expanding along first row, we have
⇒‌‌−λ3−2λ2−λ=0
⇒‌‌λ(λ2+2λ+1)=0
⇒‌‌λ(λ+1)2=0
⇒‌‌λ=0,−1