If a, b and c are real numbers, and `Delta=|[b+c,c+a,a+b],[c+a,a+b,b+c],[a+b,b+c,c+a]|=0`. Show that either `a+b+c=0` or `a=b=c`

If a, b, c, are real numbers such that |{:(b+c,c+a,a+b),(c+a,a+b,b+c),(a+b,b+c,c+a):}|=0, then show that either a+b+c=0 or, a=b=c.

If a , b ,c are all different and |[a,a^3,a^4-1],[b,b^3,b^4-1],[c,c^3,c^4-1]|=0, show that a b c(a b+b c+c a)=a+b+c

Solve the equation |[a-x, c, b], [ c, b-x, a], [b, a, c-x]|=0 where a+b+c!=0.

If a x^2+b x+c=0,a ,b ,c in R has no real zeros, and if c 0 (c) a >0

If a ,b ,c in R , then find the number of real roots of the equation Delta =|[x, c,-b],[-c, x, a], [b,-a,x]|=0

|{:(b+c,a,b),(c+a,c,a),(a+b,b,c):}|

The product of all values of t , for which the system of equations (a-t)x+b y+c z=0,b x+(c-t)y+a z=0,c x+a y+(b-t)z=0 has non-trivial solution, is (a) |[a, -c, -b], [-c, b, -a], [-b, -a, c]| (b) |[a, b, c], [b, c, a], [c, a, b]| (c) |[a, c, b], [b, a, c], [c, b, a]| (d) |[a, a+b, b+c], [b, b+c, c+a], [c, c+a, a+b]|

If (a+b)/(b+c) = (c+d)/(d+a) then prove that either c= a or a+b+c+d = 0.

Show that |[a+b+2c,a,b],[c,b+c+2a,b],[c,a,c+a+2b]|=2(a+b+c)^3 .

If the determinant |[b-c,c-a, a-b],[b^(prime)-c ', c^(prime)-a ', a^(prime)-b '],[ b' ' -c ' ' ,c' ' -a ' ', a' ' -b ' ']|=m|[a, b, c],[ a ', b ', c '],[ a ' ', b ' ' ,c ' ']| , then the value of m is a. 0 b. 2 c. 1 d. -1