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. .

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

If bgta then show that the equation (x-a)(x-b)-1=0 has one root less than a and other root greater than b.

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): 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.

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 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 .

The values of a for which the equation 2x^(2) -2(2a+1) x+a(a+1) = 0 may have one root less them a and other root greater than a are given by

Let a and b be two distinct roots of a polynomial equation f(x) =0 Then there exist at least one root lying between a and b of the polynomial equation

If one root is greater than 2 and the other root is less than 2 for the equation x^(2)-(k+1)x+(k^(2)+k-8)=0 , then the value of k lies between

If the equation a/(x-a)+b/(x-b)=1 has two roots equal in magnitude and opposite in sign then the value of a+b is