If `(x+iy) (p+iq)= (x^(2) + y^(2))i`, prove that x= q, y= p

If (a+i)^2/(2a-i) = p+iq , prove that p^2+q^2 = (a^2+1)^2/(4a^2+1)

In the polynomial p(x)= a x^ 3 + b x^ 2 + c x + d if p(2)=p(-2), prove that 4a+c=0.

If x y=a x^2+b/x , prove that x^2 y_2+2(x (dy/dx)-y)=0

If x-iy=sqrt((a-ib)/(c-id) , prove that (x^2+y^2)^2 = (a^2+b^2)/(c^2+d^2)

If y=a cos(log x)+b sin(logx) , Prove that x^2y_2+xy_1+y=0 .

If y=5 cos x-3 sin x , prove that (d^2y)/(dx^2)+y=0

If AM and GM of x and y are in the ratio p: q, then x: y is : a) p - sqrt(p^(2)+ q^(2)) : p +sqrt (p^(2)+ q^(2)) b) p +sqrt(p^(2) - q^ (2) ) : p- sqrt(p^(2)-q^(2)) c) p : q d) p + sqrt(p^(2) + q^(2)) : p-sqrt(p^(2) + q^(2))

p(x)=x^2-4 x+4 a) Prove that (x-2) is a factor of p(x) . b) Prove that for any number x , p(x) is always non negative.