If p,q,r are not in geometric progression and `|[1,q/p,alpha+q/p],[1,r/q,alpha+r/q],[palpha+q,qalpha+r,0]|=0` then prove that `palpha^2+2q alpha^2+r=0`

If sec alpha and cosec alpha are the roots of x^2-px+q=0 then prove that p^2=q(q+2)

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

If a/(q -r) = b/(r-p) = c/(p-q) , then prove that a + b + c = 0 =pa+ qb + rc .

If p sin alpha = q sin (120^(@) + alpha) = r sin (240^(@) + alpha),"prove that", pq + qr +rp = 0

If sec alpha and "cosec" alpha are the roots of x^(2)- px+q=0 then show that p^(2)=q(q+2)

Let p and q be real numbers such that p!=0,p^3!=q ,and p^3!=-qdot If alpha and beta are nonzero complex numbers satisfying alpha+beta=-p and alpha^3+beta^3=q , then a quadratic equation having alpha//beta and beta//alpha as its roots is A. (p^3+q)x^2-(p^3+2q)x+(p^3+q)=0 B. (p^3+q)x^2-(p^3-2q)x+(p^3+q)=0 C. (p^3+q)x^2-(5p^3-2q)x+(p^3-q)=0 D. (p^3+q)x^2-(5p^3+2q)x+(p^3+q)=0

If the equation x^2+px+qr=0 and x^2+qx+pr=0(p!=q,r!=0) has same root then prove that p+q+r=0

Let A=[[0,2q,r] , [p,q,-r] , [p,-q,r]] If A A^T=I_3 then |p|=

Ifa,b,c,p,q,r be six complex numbers such that a/p+b/q+c/r=0 and p/a+q/b+r/c=1-i then prove that (p^2/a^2)+(q^2/b^2)+(r^2/c^2)=-2i

If p,q and r are simple propositions such that (p^^q)^^(q^^r) is true, then