Prove that the roots of `(a-b)^(2)x^(2)+2(a+b-2c)x+1=0, a lt b `, are real or imaginary according as c does not or does lie between a and b.

If the roots of the equation ax^2 + bx + c = 0, a != 0 (a, b, c are real numbers), are imaginary and a + c < b, then

If a lt c lt b then the roots of the equation (a−b)x^2 +2(a+b−2c)x+1=0 are

Prove that the roots of equation bx^2+(b-c)x+b-c-a=0 are real if those of equatiion ax^2+2bx+b=0 are imaginary and vice versa where a,b,c epsilon R .

Prove that the equation x^2(a^2+b^2)+2x(a c+b d)+(c^2+d^2)=0 has no real root, if a d!=bc dot

Prove that the equation x^2(a^2+b^2)+2x(a c+b d)+(c^2+d^2)=0 has no real root, if a d!=b c .

If a,b, cinR , show that roots of the equation (a-b)x^(2)+(b+c-a)x-c=0 are real and unequal,

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

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

lf 0 < a < b < c < d , then the quadratic equation ax^2 + [1-a(b+c)]x+abc-d=0 A) Real and distinct roots out of which one lies between c and d B) Real and distinct roots out of which one lies between a and b C) Real and distinct roots out of which one lies between b and c (D) non -real roots

If a,b,c,d are unequal positive numbes, then the roots of equation x/(x-a)+x/(x-b)+x/(x-c)+x+d=0 are necessarily (A) all real (B) all imaginary (C) two real and two imaginary roots (D) at least two real