Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations

$x+y+z=5$    ;    $x+2 y+3 z=\mu$   ;     $x+3 y+\lambda z=1$

is constructed. If $\mathrm{p}$ is the probability that the system has a unique solution and $\mathrm{q}$ is the probability that the system has no solution, then :

Let $\alpha $ and $\beta $ be the roots of the equation $x^2 + x + 1 = 0.$ Then for $y \ne 0$ in $R,$ $\left| {\begin{array}{*{20}{c}}
{y\, + \,1}&\alpha &\beta \\
\alpha &{y\, + \,\beta }&1\\
\beta &1&{y\, + \,\alpha }
\end{array}} \right|$ is equal to

The number of values of $k $ for which the system of equations $(k + 1)x + 8y = 4k,$ $kx + (k + 3)y = 3k - 1$ has infinitely many solutions, is

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$, then det. $[( I+ B)^{50} -50B]$ is equal to

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 $........$.

Consider the system of linear equation $x+y+z=$ $4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15$, where $\lambda, \mu \in R$. Which one of the following statements is $NOT$ correct?