LET the equation `ax^2+bx+c=0` has no real roots Assertion (A): `c(a+b+c)gt0`, Reason (R): A quadratic expression `ax^2+bx+c ` has signs same as that of al for all real x if the roots of the corresponding equation `ax^2+bx+c=0` are imaginary. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true R is not te correct explanation of A (C) A is true but R is false. (D) A is false but R is true.

Assertion (A): For alphaltbeta equation (x-cosalpha)(x-cosbeta)-2=0 has one root less than cosbeta and other greater than cosalpha ., Reason (R): Quadratic expression ax^2+bx+c has sign opposite to that of a between the roots alpha and beta of equation ax^2+bx+c=0 if alphaltbeta (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true R is not te correct explanation of A (C) A is true but R is false. (D) A is false but R is true.

Assertion (A): Quadratic equation f(x)=0 has real and distinct roots. Reason (R): quadratic equation f(x)=0 has even number of roots between p and q(pltq) if f(p) and f(q) have same sign. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true R is not the correct explanation of A (C) A is true but R is false. (D) A is false but R is true.

Let f(x)=ax^3+bx^2+cx+d=0 have extremum of two different points of opposite signsAssertion (A): Equation ax^2+bx+c=0 has distinct real roots. , Reason (R): A differentiable function f(x) has extremum only at points where f\'(x)=0 (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true R is not the correct explanation of A (C) A is true but R is false. (D) A is false but R is true.

The quadratic equation ax^(2)+bx+c=0 has real roots if:

Let alpha be the root of the equation ax^2+bx+c=0 and beta be the root of the equation ax^2-bx-c=0 where alphaltbeta Assertion (A): Equation ax^2+2bx+2c=0 has exactly one root between alpha and beta ., Reason(R): A continuous function f(x) vanishes odd number of times between a and b if f(a) and f(b) have opposite signs. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true R is not te correct explanation of A (C) A is true but R is false. (D) A is false but R is true.

The quadratic expression ax^(2)+bx+c>0 AA x in R ,then

35.The expression ax^(2)+2bx+b has same sign as that of b for every,real x ,then the roots of equation bx^(2)+(b-c)x+b-c-a=0 are

If the equation ax^(2) + 2 bx - 3c = 0 has no real roots and (3c)/(4) lt a + b , then

If the roots of the equation ax^(2)+bx+c=0 are real and distinct,then

If the equation ax^(2)+bx+c=0 has non-real roots,prove that 1+(c)/(a)+(b)/(a)>0