The area of a triangle is $5$ and two of its vertices are $A(2, 1), B(3, -2)$. The third  vertex which lies on line $y = x + 3$ is-

If $A, B, C$ are the angles of triangle then the value of determinant $\left| {\begin{array}{*{20}{c}}
  {\sin \,2A}&{\sin \,C}&{\sin \,B} \\ 
  {\sin \,C}&{\sin \,2B}&{\sin A} \\ 
  {\sin \,B}&{\sin \,A}&{\sin \,2C} 
\end{array}} \right|$ is

Which of the following is correct?

Let $\lambda $ be a real number for which the system of linear equations $x + y + z = 6$
 ; $4x + \lambda y – \lambda z = \lambda – 2$ ; $3x + 2y -4z = -5$ Has indefinitely many solutions. Then $\lambda $ is a root of the quadratic equation

If the system of equations $x + y+  z = 5$ ; $x + 2y + 3z = 9$ ; $x + 3y + \alpha z = \beta $ has infinitely many solutions, then $\beta  – \alpha $ equals