If p, q are positive integers, f is a function defined for positive numbers and attains only positive values such that `f(xf(y))=x^p y^q`, then prove that `p^2=q`.

If x,y,z are positive numbers in A.P., then

If (x+i y)^5=p+i q , then prove that (y+i x)^5=q+i pdot

Prove that q to p -= ~p to ~q

If (x+i y)(p+i q)=(x^2+y^2)i , prove that x=q ,y=pdot

Prove that ~(~pto ~q) -=~p ^^q

Let P be a point on the ellipse x^2/a^2+y^2/b^2=1 , 0 < b < a and let the line parallel to y-axis passing through P meet the circle x^2 +y^2=a^2 at the point Q such that P and Q are on the same side of x-axis. For two positive real numbers r and s, find the locus of the point R on PQ such that PR : RQ = r : s and P varies over the ellipse.

Let A={xinR:x" is not a positive integer "} define a function f:AtoR" such that "f(x)=(2x)/(x-1) . Then f is

P(vec p) and Q(vec q) are the position vectors of two fixed points and R(vec r) is the position vectorvariable point. If R moves such that (vec r-vec p)xx(vec r -vec q)=0 then the locus of R is

Show that any positive odd integer is of the form 4q+1 or 4q+3, where q is some integer.

If sintheta+costheta=p and sectheta+costheta=q , then prove that q(p^(2)-1)=2p