If `a ,b ,c(a b c^2)x^2+3a^2c x+b^2c x-6a^2-a b+2b^2=0` ares rational.

If a ,b ,c are non zero rational no then prove roots of equation (a b c^2)x^2+3a^2c x+b^2c x-6a^2-a b+2b^2=0 are rational.

Suppose A, B, C are defined as A=a^2b+a b^2-a^2c-a c^2 , B=b^2c+b c^2-a^2b-a b^2,and C=a^2c +'a c^2-b^2' c-b c^2, w h e r ea > b > c >0 and the equation A x^2+B x+C=0 has equal roots, then a ,b ,c are in AdotPdot b. GdotPdot c. HdotPdot d. AdotGdotPdot

If x^2+p x+1 is a factor of the expression a x^3+b x+c , then a^2-c^2=a b b. a^2+c^2=-a b c. a^2-c^2=-a b d. none of these

If the lines (a-b-c) x + 2ay + 2a = 0 , 2bx + ( b- c - a) y + 2b = 0 and (2c+1) x + 2cy + c - a - b = 0 are concurrent , then the prove that either a+b+ c = 0 or (a+b+c)^(2) + 2a = 0

If a , b , c are real numbers, then find the intervals in which f(x)=|x+a^2a b a c a b x+b^2b c a c b c x+c^2| is increasing or decreasing.

If f(x)=|x+a^2a b a c a b x+b^2b c a c b c x+c^2|,t h e n prove that f^(prime)(x)=3x^2+2x(a^2+b^2+c^2)dot

Column I, Column II If a ,b ,c ,a n dd are four zero real numbers such that (d+a-b)^2+(d+b-c)^2=0 and he root of the equation a(b-c)x^2+b(c-a)x=c(a-b)=0 are real and equal, then, p. a+b+c=0 If the roots the equation (a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0 are real and equal, then, q. a ,b ,ca r einAdotPdot If the equation a x^2+b x+c=0a n dx^3-3x^2+3x-0 have a common real root, then, r. a ,b ,ca r einGdotPdot Let a ,b ,c be positive real number such that the expression b x^2+(sqrt((a+b)^2+b^2))x+(a+c) is non-negative AAx in R , then, s. a ,b ,ca r einHdotPdot

If a,b,c are rational numbers (a gt b gt c gt 0) and quadratic equation (a+b-2c) x ^(2) + (b+c-2a) x+ (c+a-2b)=0 has a root in the interval (-1,0) then which of the following statement (s) is/are correct ?

If the roots of the equation (c^2-a b)x^2-2(a^2-b c)x+b^2-a c=0 are equal, prove that either a=0 or a^3+b^3+c^3=3a b c dot

If a+b+c=0, a,b,c in Q then roots of the equation (b+c-a) x ^(2) + (c+a-c) =0 are: Rational Irrational Imaginary none of these .