Evaluate the following: ` |[x+lamda, x, x],[x, x+lamda, x],[x, x, x+lamda]| `

If f(x) = |(x + lamda,x,x),(x,x + lamda,x),(x,x,x + lamda)|, " then " f(3x) - f(x) =

The number of distinct real values of lamda for which the system of linear equations x + y + z = lamda x , x + y + z = lamday, x + y + z + lamda z has non - trival solution.

The vaue of lamda so that the function f defined as f = {{:(lamda x"," , x le pi),(cos x"," ,x gtpi):} is cotinuous at x = pi is :

For which value(s) of lamda , do the pair of linear equations lamda x+ y= lamda^(2) and x+ lamda y= 1 have no solution?

For which value(s) of lamda , do the pair of linear equations lamda x+ y= lamda^(2) and x+ lamda y= 1 have a unique solution?

Minimum posible number of positive root of the quadratic equation x ^(2) -(1+ lamda) x + lamda -2 =0, lamda in R:

For which value(s) of lamda , do the pair of linear equations lamda x+ y= lamda^(2) and x+ lamda y= 1 have infinitely many solutions?

The sum of all intergral values of lamda for which (lamda^(2) + lamda -2) x ^(2) + (lamda +2) x lt 1 AA x in R, is: