If A and B are the two real values of k for which the system of equations x+2 y+z=1 , x+3 y+4 z=k x+5 y+10 z=k^2 is consistent, then A+B=

Step-by-step solution

Given system of equation is, aligned & x+2 y+z=1 & x+5 y+10 z=k^2 & D= | array lll 1 & 2 & 1 1 & 3 & 4 1 & 5 & 10 array | = & 1(30-20)-2(10-4)+1(5-3)=10-12+2=0 aligned Since, D=0 Given system of equation is consistent. Therefore, D₁=0 aligned & D₁= | array ccc 1 & 2 & 1 k & 3 & 4 k^2 & 5 & 10 array | & 1(30-20)-2 (10 k-4 k^2 )+ (5 k-3 k^2 )=0 & 10-20 k+8 k^2+5 k-3 k^2=0 & 5 k^2-15 k+10=0 & k^2-3 k+2=0 & (k-2)(k-1)=0 & k=2,1 & aligned Hence, the real values of k i.e. array rlrl A & =2 and B=1 & & A+B & =2+1=3 array