Let `a , b , c` be nonzero real numbers such that `int_0^1(1+cos^8x)(a x^2+b x+c)dx` `=int_0^2(1+cos^8x)(a x^2+b x+c)dx=0` Then show that the equation `a x^2+b x+c=0` will have one root between 0 and 1 and other root between 1 and 2.

Let a, b, c be non zero numbers such that int_(0)^(3)sqrt(x^(2)+x+1)(ax^(2)+bx+c)dx=int_(0)^(5)sqrt(1+x^(2)+x)(ax^(2)+bx+c)dx . Then the quadratic equation ax^(2)+bx+c=0 has

Statement-1: Let a,b,c be non zero real numbers and f(x)=ax^2+bx+c satisfying int_0^1 (1+cos^8x)f(x)dx=int_0^2(1+cos^8x)f(x)dx then the equation f(x)=0 has at least one root in (0,2) .Statement-2: If int_a^b g(x)dx vanishes and g(x) is continuous then the equation g(x)=0 has at least one real root in (a,b) . (A) Both 1 and 2 are true and 2 is the correct explanation of 1 (B) Both 1 and 2 are true and 2 is not correct explanation of 1 (C) 1 is true but 2 is false (D) 1 is false but 2 is true

The roots of the equation (b-c) x^2 +(c-a)x+(a-b)=0 are

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

Let a ,b ,and c be real numbers such that 4a+2b+c=0 and a b > 0. Then the equation a x^2+b x+c = 0

If p,q be non zero real numbes and f(x)!=0, x in [0,2] also f(x)gt0 and int_0^1 f(x).(x^2+px+q)dx=int_1^2 f(x).(x^2+px+q)dx=0 then equation x^2+px+q=0 has (A) two imginary roots (B) no root in (0,2) (C) one root in (0,1) and other in (1,2) (D) one root in (-oo,0) and other in (2,oo)

Given that a x^2+b x+c=0 has no real roots and a+b+c 0 d. c=0

Given that a x^2+b x+c=0 has no real roots and a+b+c 0 d. c=0

If x^2+a x+bc=0 a n d x^2+b x+c a=0(a!=b) have a common root, then prove that their other roots satisfy the equation x^2+c x+a b=0.

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