If p,q,r are in G.P. and the equations, `px^(2) + 2qx + r = 0 and dx^2+2ex + f = 0 ` have a common root, then show that `d/p , e/q, f/r` are in A.P.

If a, b, c are in GP , then the equations ax^2 +2bx+c = 0 and dx^2 +2ex+f =0 have a common root if d/a , e/b , f/c are in

If p^(th), q^(th), r^(th) and s^(th) terms of an A.P. are in G.P, then show that (p – q), (q – r), (r – s) are also in G.P.

If the equation x^(3) +px +q =0 has three real roots then show that 4p^(3)+ 27q^(2) lt 0 .

If the equation x^(2)-px+q=0 and x^(2)-ax+b=0 have a comon root and the other root of the second equation is the reciprocal of the other root of the first, then prove that (q-b)^(2)=bq(p-a)^(2) .

If p + r = 2q and 1/q + 1/s = 2/r , then prove that p : q = r : s .

If p ,q , in {1,2,3,4}, then find the number of equations of the form p x^2+q x+1=0 having real roots.

If a and b are the roots of x^2 – 3x + p = 0 and c, d are roots of x^(2) – 12x + q = 0 , where a,b,c,d form a G.P Prove that (q+q):(q-p) = 17 : 15

The negation of p^^(q implies ~r) is …….

Let alpha and beta be the roots of equation px^2 + qx + r = 0 , p != 0 .If p,q,r are in A.P. and 1/alpha+1/beta=4 , then the value of |alpha-beta| is :

Let alpha,beta be the roots of the equation x^2-p x+r=0 and alpha/2,2beta be the roots of the equation x^2-q x+r=0 , the value of r is