Which of the following systems has non-trivial solutions? If so, find these solutions.

$5x+5y+2z=0$

$2x+5y+4z=0$

$4x+5y+2z=0$.

Which of the following systems has non-trivial solutions? If so, find these solutions.

$2x-3y-z=0$

$x+3y-2z=0$

$x-3y=0$.

If the system of equations is $x-ky-z=0, kx-y-z=0,x+y-z=0$ has a non-zero solution, then find the possible values of $k$.

Find the real value(s) of a for which the system of equations

$x+ax=0,y+az=0,z+ax=0$

has infinitely many solutions.

If  $\left[x\right]$ represents the greatest integer less than or equal to the real number $x$ and $-1\le p<0, 0\le q<1, 1\le r<2,$then find the value of the determinant ;

$\left|\begin{array}{ccc}\left[p\right]+1& \left[q\right]& \left[r\right]\\ \left[p\right]& \left[q\right]+1& \left[r\right]\\ \left[p\right]& \left[q\right]& \left[r\right]+1\end{array}\right|$

If $a>0, d>0$, find the value of :

$\left|\begin{array}{ccc}\frac{1}{a}& \frac{1}{a(a+d)}& \frac{1}{(a+d)(a+2d)}\\ \frac{1}{a+d}& \frac{1}{(a+d)(a+2d)}& \frac{1}{(a+2d)(a+3d)}\\ \frac{1}{a+2d}& \frac{1}{(a+2d)(a+3d)}& \frac{1}{(a+3d)(a+4d)}\end{array}\right|$

If $a, b, c$(all positive) are the $pth, qth$ and $rth$ terms respectively of a $\mathrm{G}.\mathrm{P}.$, then prove that

$\left|\begin{array}{ccc}\log \left(a\right)& p& 1\\ \log \left(b\right)& q& 1\\ \log \left(c\right)& r& 1\end{array}\right|=0$.