Find values of $x$, if $\left|\begin{array}{ll}2 & 4 \\ 5 & 1\end{array}\right|=\left|\begin{array}{cc}2 x & 4 \\ 6 & x\end{array}\right|$

Let $A_1, A_2, A_3$ be the three A.P. with the same common difference $d$ and having their first terms as $A , A +1, A +2$, respectively. Let $a , b , c$ be the $7^{\text {th }}, 9^{\text {th }}, 17^{\text {th }}$ terms of $A_1, A_2, A_3$, respectively such that $\left|\begin{array}{lll} a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1\end{array}\right|+70=0$ If $a=29$, then the sum of first $20$ terms of an $AP$ whose first term is $c – a – b$ and common difference is $\frac{ d }{12}$, is equal to $……..$.

The number of integers $x$ satisfying $-3 x^4+\operatorname{det}\left[\begin{array}{ccc}1 & x & x^2 \\ 1 & x^2 & x^4 \\ 1 & x^3 & x^6\end{array}\right]=0$ is equal to

If $a, b, c$ are non-zero real numbers and if the system of equations $(a – 1 )x = y + z,$  $(b – 1 )y = z + x ,$ $(c – 1 )z= x + y,$ has a non-trivial solution, then $ab + bc + ca$ equals

The values of the determinant $\left| {\,\begin{array}{*{20}{c}}1&{\cos (\alpha – \beta )}&{\cos \alpha }\\{\cos (\alpha – \beta )}&1&{\cos \beta }\\{\cos \alpha }&{\cos \beta }&1\end{array}\,} \right|$ is