Manipal Entrance Test 2020 Math Paper

Given, On equating the coefficients of $\hat{i}, \hat{j}$ and $k$ both sides, we have $x+3 y-4 z=\lambda x$ $x-3 y+5 z=\lambda y$ and $3 x+y+0=\lambda z$ Above three equations can be rewritten as $(1-\lambda) x+3 y-4 z=0$ $x-(3+\lambda) y+5 z=0$ $3 x+y-\lambda z=0$ This is homogeneous system of equations in three variables $x, y$ and $z$. It is consistent and have non-zero solution. i.e. $(x, y, z) \neq (0,0,0)$, if determinant of coefficient matrix is zero. On expanding along first row, we have $\Rightarrow -\lambda^{3}-2\lambda^{2}-\lambda=0$ $\Rightarrow \lambda(\lambda^{2}+2\lambda+1)=0$ $\Rightarrow \lambda(\lambda+1)^{2}=0$ $\Rightarrow \lambda=0,-1$