Let $\mathrm{A}(-1,1)$ and $\mathrm{B}(2,3)$ be two points and $\mathrm{P}$ be a variable point above the line $A B$ such that the area of $\triangle \mathrm{PAB}$ is $10$ . If the locus of $\mathrm{P}$ is $\mathrm{ax}+\mathrm{by}=15$, then $5 a+2 b$ is :

The number of distinct real roots of $\left| {\,\begin{array}{*{20}{c}}{\sin x}&{\cos x}&{\cos x}\\{\cos x}&{\sin x}&{\cos x}\\{\cos x}&{\cos x}&{\sin x}\end{array}\,} \right| = 0$ in the interval $ - \frac{\pi }{4} \le x \le \frac{\pi }{4}$ is

The number of solutions of equations $x + y - z = 0$, $3x - y - z = 0, \,x - 3y + z = 0$ is

Number of values of $m$ for which the lines $x + y - 1 = 0$, $(m - 1) x + (m^2 - 7) y - 5 = 0 \,\,\&\,\, (m - 2) x + (2m - 5) y = 0$ are concurrent, are

The system of linear equation $x + y + z = 2, 2x + 3y + 2z = 5$, $2x + 3y + (a^2 -1)\,z = a + 1$ then

If $\omega $ is cube root of unity, then root of the equation $\left| {\begin{array}{*{20}{c}}
  {x + 2}&\omega &{{\omega ^2}} \\ 
  \omega &{x + 1 + {\omega ^2}}&1 \\ 
  {{\omega ^2}}&1&{x + 1 + \omega } 
\end{array}} \right| = 0$ is