Assertion (A): Equation `(x-p)(x-q)-r=0` where `p,q,r epsilon R and 0ltpltqltr` has roots in `(p,q)`, Reason(R): A polynomial equation `f(x)=0` has odd number of roots between ` a and b (altb) if f(a) and f(b)` have opposite signs

Assertion (A): Equation (x-a)(x-b)-2=0, altb has one root less than a and other root greater than b. , Reason (R): A polynomial equation f\'(x)=0 has even number of roots between a and b if f(a) and f(b) have opposite signs. .

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.

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.

Assertion (A): For 0ltaltbltc equation (x-a)(x-b)-c=0 has no roots in (a,b) Reason (R):For a continuous function f(x) equation f\'(x)=0 has at least one root between a and b if f(a) and f(b) are equal. (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.

If 3+4i is a root of equation x^(2)+px+q=0 where p, q in R then

Find p, where p,q belongs to R If p and q are the roots of equation x^2-px+q=0 then p=,

If one root of the equation x^(2)+px+q=0 is 2+sqrt(3) where p, q epsilon I and roots of the equation rx^(2)+x+q=0 are tan 268^(@) and cot 88^(@) then value of p+q+r is :

Let p, q, r in R and r gt p gt 0 . If the quadratic equation px^(2) + qx + r = 0 has two complex roots alpha and beta , then |alpha|+|beta| , is

If a polynomial function f(x) satisfies f(f(f(x))=8x+21 , where p and q are real numbers, then p+q is equal to _______

If p,q,r epsilon R and are distinct the equation (x-p)^5+(x-q)^5+(x-r)^5=0 has (A) four imaginary and one real root (B) two imaginary and three real roots (C) all the roots real (D) none of these