If the roots of the equation `(1-q+p^(2)/2)x^(2) + p(1+q)x+q(q-1) +p^(2)/2=0` are equal, then show that `p^(2) = 4q`

If the roots of the equation 1/ (x+p) + 1/ (x+q) = 1/r are equal in magnitude but opposite in sign, show that p+q = 2r & that the product of roots is equal to (-1/2)(p^2+q^2) .

If the roots of the equation qx^(2)+2px+2q=0 are real and unequal, prove that the roots of the equation (p+q)x^(2)+2qx+(p-q)=0 are imaginary.

If p(q-r)x^(2) + q(r-p)x+ r(p-q) = 0 has equal roots, then show that p, q, r are in H.P.

If cos^(4)theta+p, sin^(4)theta+p are the roots of the equation x^(2)+a(2x+1)=0 and cos^(2)theta+q,sin^(2)theta+q are the roots of the equation x^(2)+4x+2=0 then a is equal to

If p and q are roots of the quadratic equation x^(2) + mx + m^(2) + a = 0 , then the value of p^(2) + q^(2) + pq , is

If the roots of the equation 1/(x+p)+1/(x+q)=1/r are equal in magnitude and opposite in sign, then (A) p+q=r (B)p+q=2r (C) product of roots=- 1/2(p^2+q^2) (D) sum of roots=1

For p, q in R , the roots of (p^(2) + 2)x^(2) + 2x(p +q) - 2 = 0 are

Let alpha,beta be the roots of the equation x^(2)-px+r=0 and alpha//2,2beta be the roots of the equation x^(2)-qx+r=0 , then the value of r is (1) (2)/(9)(p-q)(2q-p) (2) (2)/(9)(q-p)(2p-q) (3) (2)/(9)(q-2p)(2q-p) (4) (2)/(9)(2p-q)(2q-p)

If alpha,beta are the roots of the equation x^2+p x+q=0,t h e n-1/alpha,-1/beta are the roots of the equation (a) x^2-p x+q=0 (b) x^2+p x+q=0 (c) q x^2+p x+1=0 (d) q x^2-p x+1=0

If p(q-r)x^2+q(r-p)x+r(p-q)=0 has equal roots, then prove that 2/q=1/p+1/rdot