Show that : (a) a polynomial of an odd degreee has atleast one real root.
(b) a polynomial of an even degree has atleast two real roots if it attains atleast one value opposite in sign to the coefficient of its highest-degree term.

If f(x) is a polynomial of degree 5 with real coefficients such that f(|x|)=0 has 8 real roots, then f(x)=0 has

A polynomial of 6th degree f(x) satisfies f(x)=f(2-x), AA x in R , if f(x)=0 has 4 distinct and 2 equal roots, then sum of the roots of f(x)=0 is

Show that if p,q,r and s are real numbers and pr=2(q+s) , then atleast one of the equations x^(2)+px+q=0 and x^(2)+rx+s=0 has real roots.

If 2a+3b+6c = 0, then show that the equation a x^2 + bx + c = 0 has atleast one real root between 0 to 1.

Which one of the following functions is continuous everywhere in its domain but has atleast one point where it is not differentiable ?

Let a ,b ,c be real. If a x^2+b x+c=0 has two real roots alphaa n dbeta,w h e r ealpha >1 , then show that 1+c/a+|b/a|<0

Show that the equation x^(3)+2x^(2)+x+5=0 has only one real root, such that [alpha]=-3 , where [x] denotes the integral point of x

There is only one real value of a for which the quadratic equation ax^2+(a+3)x+a-3=0 has two positive integral solutions. The product of these two solutions is

For what values of m the equation (1+m)x^(2)-2(1+3m)x+(1+8m)=0 has (m in R) (i) both roots are imaginary? (ii) both roots are equal? (iii) both roots are real and distinct? (iv) both roots are positive? (v) both roots are negative? (vi) roots are opposite in sign? (vii)roots are equal in magnitude but opposite in sign? (viii) atleast one root is positive? (ix) atleast one root is negative? (x) roots are in the ratio 2:3 ?

A man P speaks truth with probability p and another man Q speaks truth with probability 2p. Statement-1 If P and Q contradict each other with probability (1)/(2) , then there are two values of p. Statement-2 a quadratic equation with real coefficients has two real roots.