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`.

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

lf r, s, t are prime numbers and p, q are the positive integers such that their LCM of p,q is r^2 t^4 s^2, then the numbers of ordered pair of (p, q) is

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

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

Show that the square of an odd positive c integer is of the form 6q + 1 or 6q + 3 for some integer q.

Let f be a function satisfying of xdot Then f(x y)=(f(x))/y for all positive real numbers xa n dydot If f(30)=20 , then find the value of f(40)dot

Prove that the function f(x)= x^(n) is continuous at x= n, where n is a positive integer

If cos theta + sin theta = p and sec theta + cosec theta = q , prove that q (p^(2) - 1) = 2p .