If a, b are the roots of `x^(2)+px+1=0 and c, d` are the roots of `x^(2)+qx+1=0`, show that `q^(2)-p^(2)=(a-c)(b-c)(a+d) (b+d)`.

If a and b are the roots of x^2-3x+p=0 and c,d are the 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 the ratio of the roots of ax^(2) +2bx+c=0 is same as the ratio of the px^(2)+2qx +r= 0 , then prove that (b^(2))/(ac)= (q^(2))/(pr)

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

If c, d are the roots of the equation (x-a) (x-b) - k=0 , then prove that a, b are the roots of the equation (x-c) (x-d) + k= 0

Given tan A and tan B are the roots of equation x^(2)-ax+b=0 . The value of sin^(2)(A+B) is a) (a^(2))/(a^(2)+(1-b)^(2)) b) (a^(2))/(a^(2)+b^(2)) c) (a^(2))/((a+b)^(2)) d) (b^(2))/(a^(2)+(1-b)^(2))