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

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

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 (a-x)/(p x)=(a-y)/(q y)=(a-z)/ra n dp ,q ,a n dr are in A.P., then prove that x ,y ,z are in H.P.

If sqrt(x+i y)=+-(a+i b), then find sqrt( x-i ydot)

P N is the ordinate of any point P on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 and AA ' is its transvers axis. If Q divides A P in the ratio a^2: b^2, then prove that N Q is perpendicular to A^(prime)Pdot

If lines p x+q y+r=0,q x+r y+p=0a n dr x+p y+q=0 are concurrent, then prove that p+q+r=0(w h e r ep ,q ,r are distinct )dot

If x = cos alpha + i sin alpha , y = cos beta + i sin beta , prove that x^(m)y^(n)+1/(x^m y^n) = 2 cos (m alpha + n beta) .

If x + iy = (3 + 5i)/(7-6i), they y =

Find the values of x and y so that the vectors 4 hat i + 5 hat j and x hat i + y hat j are equal.

A fair dice is thrown three times. If p, q and r are the numbers obtained on the dice, then find the probability that i^(p) + i^(q) + i^(r) = 1 , where I = sqrt(-1) .