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

Let alpha and beta be the roots of x^2-x+p=0 and gamma and delta be the root of x^2-4x+q=0. If alpha,beta,a n dgamma,delta are in G.P., then the integral values of p and q , respectively, are

Let a, b, c, p, q be the real numbers. Suppose alpha,beta are the roots of the equation x^2+2px+ q=0 . and alpha,1/beta are the roots of the equation ax^2+2 bx+ c=0 , where beta !in {-1,0,1} . Statement I (p^2-q) (b^2-ac)>=0 Statement 2 b != pa or c != qa .

Let p,q,r be roots of cubic x^(3)+2X^(2)+3x+3=0 , then

If alpha,beta are the roots of x^2+p x+q=0a n dgamma,delta are the roots of x^2+r x+s=0, evaluate (alpha-gamma)(alpha-delta)(beta-gamma)(beta-delta) in lterms of p ,q ,r ,a n dsdot Deduce the condition that the equation has a common root.

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

If x^2+p x+q=0 is the quadratic equation whose roots are a-2 and b-2 where a and b are the roots of x^2-3x+1=0, then

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

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 3p^2 = 5p+2 and 3q^2 = 5q+2 where p!=q, pq is equal to

If the p^("th"), q^("th") and r^("th") terms of a G.P. are a, b and c, respectively. Prove that a^(q – r) b^(r-p) c^(P - q) = 1.