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+p):(q-p)=17:15

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 GP. Prove that (q + p) : (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,and gamma,delta are in G.P., then the integral values of p and q , respectively, are

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 a ,b , c ,p ,q be real numbers. Suppose alpha,beta are the roots of the equation x^2+2p x+q=0,alphaa n d1//beta are the roots of the equation a x^2+2b x+c=0,w h e r ebeta^2 !in {-1,0,1}dot Statement 1: (p^2-q)(b^2-a c)geq0 Statement 2: b!=p aorc!=q a

If p and q are the roots of the equation x^2-p x+q=0 , then (a) p=1,\ q=-2 (b) p=1,\ q=0 (c) p=-2,\ q=0 (d) p=-2,\ q=1

Sum of the first p, q and r terms of an A.P. are a, b and c, respectively. Prove that a/p(q-r)+b/q(r-p)+c/r(p-q)=0

Sum of first p,q and r terms of an A.P. are a,b,c respectively. Prove that a/p(q-r) +b/q(r-p) + c/r(p-q)=0 .

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 .

If p, q, r are in A.P. while x, y, z are in G.P., prove that x^(q-r).y^(r-p).z^(p-q) =1 .

Let S_p and S_q be the coefficients of x^p and x^q respectively in (1 + x)^(p+q) , then :